On Further Refinements of Numerical Radius Inequalities

Abstract

This paper introduces several generalized extensions of some recent numerical radius inequalities of Hilbert space operators. More preciously, these inequalities refine the recent inequalities that were proved in literature. It has already been demonstrated that some inequalities can be improved or restored by concatenating some into one inequality. The main idea of this paper is to extend the existing numerical radius inequalities by providing a unified framework. We also present a numerical example to demonstrate the effectiveness of the proposed approach. Roughly, our approach combines the existing inequalities, proved in literature, into a single inequality that can be used to obtain improved or restored results. This unified approach allows us to extend the existing numerical radius inequalities and show their effectiveness through numerical experiments.

1. Introduction

The concepts of numerical range and numerical radius play a crucial role in linear algebra. The numerical range of a matrix is the set of all its eigenvalues, while the numerical radius is defined as the largest absolute value of the eigenvalues. These concepts are used to measure the stability and accuracy of a given matrix. They have applications in solving problems such as the numerical solution of differential equations, the numerical solution of integral equations and the approximation of functions. They also play a role in the study of linear operators, such as the spectral theory of linear operators. Numerical range and numerical radius have also been used to study the stability of solutions to linear and nonlinear differential equations. The numerical radius of a matrix is the largest magnitude of its eigenvalues, while the numerical range is the set of all values obtained by taking the inner product of the matrix with a unit vector. Both of these concepts are used to measure the stability of a matrix. Numerical range and numerical radius are used to analyze matrix behavior and can be utilized to determine the convergence or divergence of an iterative process. For example, the numerical radius of a matrix can be used to determine the convergence rate of the power method, which is a popular iterative algorithm used to calculate eigenvalues. They also provide insight into the structure of a matrix, its eigenvectors, and the nature of its eigenvalues. For example, if the numerical radius of a matrix is zero, then all eigenvalues of the matrix are real and the matrix is stable [1,2,3,4,5].

We let $\mathcal{A}\left(\mathcal{J}\right)$ be the Banach algebra of all bounded linear operators defined on a complex Hilbert space $\left(\mathcal{J};〈·,·〉\right)$ with the identity operator $1_{\mathcal{J}}$ in $\mathcal{A}\left(\mathcal{J}\right)$. Then, for a bounded linear operator $\mathfrak{F}$ on a Hilbert space $\mathcal{J}$, the numerical range $W\left(\mathfrak{F}\right)$ of a bounded operator $\mathfrak{F}\in \mathcal{A}\left(\mathcal{J}\right)$ is defined by $W\left(\mathfrak{F}\right)=\left\{〈\mathfrak{F}\mu ,\mu 〉:\mu \in \mathcal{J},∥\mu ∥=1\right\}.$ Additionally, the numerical radius is defined to be

$$\begin{array}{c}\hfill \omega \left(\mathfrak{F}\right)=\underset{\beta \in W\left(\mathfrak{F}\right)}{sup}\left|\beta \right|=\underset{∥\mu ∥=1}{sup}\left|〈\mathfrak{F}\mu ,\mu 〉\right|.\end{array}$$

We recall that the usual operator norm of an operator $\mathfrak{F}$ is defined to be

$$\begin{array}{c}\hfill ∥\mathfrak{F}∥=sup\left\{∥\mathfrak{F}\mu ∥:\mu \in \mathcal{J},∥\mu ∥=1\right\}.\end{array}$$

It is well known that the numerical radius $\omega \left(·\right)$ defines an operator norm on $\mathcal{A}\left(\mathcal{J}\right)$, which is equivalent to the operator norm $\parallel ·\parallel$. Moreover, we have

$$\begin{array}{c}\hfill \frac{1}{2}\parallel \mathfrak{F}\parallel \le \omega \left(\mathfrak{F}\right)\le \parallel \mathfrak{F}\parallel \end{array}$$

(1)

for any $\mathfrak{F}\in \mathcal{A}\left(\mathcal{J}\right)$.

In 2003, Kittaneh [6] refined the right-hand side of (1) by obtaining that

$$\begin{array}{c}\hfill \omega \left(\mathfrak{F}\right)\le \frac{1}{2}∥\left|\mathfrak{F}\right|+|{\mathfrak{F}}^{*}|∥\le \frac{1}{2}\left(\parallel \mathfrak{F}\parallel +\parallel {\mathfrak{F}}^2{\parallel }^{1/2}\right)\end{array}$$

(2)

for any $\mathfrak{F}\in \mathcal{A}\left(\mathcal{J}\right)$.

Two years later, Kittaneh [7] proved his celebrated two-sided inequality

$$\begin{array}{c}\hfill \frac{1}{4}\parallel {\mathfrak{F}}^{*}\mathfrak{F}+\mathfrak{F}{\mathfrak{F}}^{*}\parallel \le {\omega }^2\left(\mathfrak{F}\right)\le \frac{1}{2}\parallel {\mathfrak{F}}^{*}\mathfrak{F}+\mathfrak{F}{\mathfrak{F}}^{*}\parallel \end{array}$$

(3)

for any $\mathfrak{F}\in \mathcal{A}\left(\mathcal{J}\right)$. These inequalities are sharp.

In [8], Dragomir established an upper bound for the numerical radius of the product of two Hilbert space operators, as follows:

$$\begin{array}{c}\hfill {\omega }^r\left({\mathfrak{G}}^{*}\mathfrak{H}\right)\le \frac{1}{2}∥{\left|\mathfrak{H}\right|}^{2r}+{\left|\mathfrak{G}\right|}^{2r}∥(r\ge 1).\end{array}$$

(4)

In his recent work [1], Alomari refined the right-hand side of (3) and the recent results of Kittaneh and Moradi [2], as follows:

$$\begin{array}{cc}\hfill {\omega }^{2p}\left(\mathfrak{F}\right)& \le \frac{1}{4}\gamma {∥{\left|\mathfrak{F}\right|}^{2p\eta }+{\left|{\mathfrak{F}}^{*}\right|}^{2p\left(1-\eta \right)}∥}^2+\frac{1}{2}\left(1-\gamma \right){\omega }^p\left(\mathfrak{F}\right)∥{\left|\mathfrak{F}\right|}^{2p\eta }+{\left|{\mathfrak{F}}^{*}\right|}^{2p\left(1-\eta \right)}∥\hfill \\ & \le \frac{1}{2}\gamma ∥{\left|\mathfrak{F}\right|}^{4p\eta }+{\left|{\mathfrak{F}}^{*}\right|}^{4p\left(1-\eta \right)}∥+\frac{1}{2}\left(1-\gamma \right){\omega }^p\left(\mathfrak{F}\right)∥{\left|\mathfrak{F}\right|}^{2p\eta }+{\left|{\mathfrak{F}}^{*}\right|}^{2p\left(1-\eta \right)}∥\hfill \\ & \le \frac{1}{2}∥{\left|\mathfrak{F}\right|}^{4p\eta }+{\left|{\mathfrak{F}}^{*}\right|}^{4p\left(1-\eta \right)}∥\hfill \end{array}$$

(5)

for any operator $\mathfrak{F}\in \mathcal{A}\left(\mathcal{J}\right)$, $p\ge 1$, and $\eta ,\gamma \in \left[0,1\right]$. In particular, it was shown that

$$\begin{array}{cc}\hfill {\omega }^2\left(\mathfrak{F}\right)& \le \frac{1}{12}{∥\left|\mathfrak{F}\right|+\left|{\mathfrak{F}}^{*}\right|∥}^2+\frac{1}{3}\omega \left(\mathfrak{F}\right)∥\left|\mathfrak{F}\right|+\left|{\mathfrak{F}}^{*}\right|∥\hfill \\ \hfill & \le \frac{1}{6}∥{\left|\mathfrak{F}\right|}^2+{\left|{\mathfrak{F}}^{*}\right|}^2∥+\frac{1}{3}\omega \left(\mathfrak{F}\right)∥\left|\mathfrak{F}\right|+\left|{\mathfrak{F}}^{*}\right|∥\hfill \\ \hfill & \le \frac{1}{4}{∥\left|\mathfrak{F}\right|+\left|{\mathfrak{F}}^{*}\right|∥}^2.\hfill \end{array}$$

(6)

The first inequality in (6) was proven by Alomari in [1] and the second inequality by Kittaneh and Moradi in [2].

In the same work [1], a refinement of (4) was proven as follows:

$$\begin{array}{cc}\hfill {\omega }^{2r}\left({\mathfrak{G}}^{*}\mathfrak{H}\right)& \le \frac{1}{4}\gamma {∥{\left|\mathfrak{H}\right|}^{2p}+{\left|\mathfrak{G}\right|}^{2p}∥}^2+\frac{1}{2}\left(1-\gamma \right){\omega }^p\left(\mathfrak{H}\right)∥{\left|\mathfrak{H}\right|}^{2p}+{\left|\mathfrak{G}\right|}^{2p}∥\hfill \\ \hfill & \le \frac{1}{2}\gamma ∥{\left|\mathfrak{F}\right|}^{4p}+{\left|\mathfrak{G}\right|}^{4p}∥+\frac{1}{2}\left(1-\gamma \right){\omega }^p\left(\mathfrak{H}\right)∥{\left|\mathfrak{H}\right|}^{2p}+{\left|\mathfrak{G}\right|}^{2p}∥\hfill \\ \hfill & \le \frac{1}{2}∥{\left|\mathfrak{H}\right|}^{4p}+{\left|\mathfrak{G}\right|}^{4p}∥.\hfill \end{array}$$

(7)

In particular, it was shown that

$$\begin{array}{cc}\hfill {\omega }^2\left({\mathfrak{G}}^{*}\mathfrak{H}\right)& \le \frac{1}{12}{∥{\left|\mathfrak{H}\right|}^2+{\left|\mathfrak{G}\right|}^2∥}^2+\frac{1}{3}\omega \left({\mathfrak{G}}^{*}\mathfrak{H}\right)∥{\left|\mathfrak{H}\right|}^2+{\left|\mathfrak{G}\right|}^2∥\hfill \\ \hfill & \le \frac{1}{6}∥{\left|\mathfrak{H}\right|}^4+{\left|\mathfrak{G}\right|}^4∥+\frac{1}{3}\omega \left({\mathfrak{G}}^{*}\mathfrak{H}\right)∥{\left|\mathfrak{H}\right|}^2+{\left|\mathfrak{G}\right|}^2∥\hfill \\ \hfill & \le \frac{1}{2}∥{\left|\mathfrak{H}\right|}^4+{\left|\mathfrak{G}\right|}^4∥.\hfill \end{array}$$

(8)

In [3], Sababheh and Moradi presented some new numerical radius inequalities. Among others, the well-known Hermite–Hadamard inequality was used to perform the following result:

$$\begin{array}{c}\hfill \phi \left(\omega \left(\mathfrak{F}\right)\right)\le ∥{\int }_0^1\phi \left(\left(1-s\right)\left|\mathfrak{F}\right|+s\left|{\mathfrak{F}}^{*}\right|\right)ds∥\le \frac{1}{2}∥\phi \left(\left|\mathfrak{F}\right|\right)+\phi \left(\left|{\mathfrak{F}}^{*}\right|\right)∥\end{array}$$

(9)

for every $\mathfrak{F}\in \mathcal{A}\left(\mathcal{J}\right)$ and increasing the operator convex function $\phi :\left[0,\infty \right)\to \left[0,\infty \right)$.

On the other hand, Moradi and Sababheh, in [4], proved the following refinement of (9):

$$\begin{array}{c}\hfill \phi \left(\omega \left(\mathfrak{F}\right)\right)\le \frac{1}{2}∥\phi \left(\frac{3\left|\mathfrak{F}\right|+\left|{\mathfrak{F}}^{*}\right|}{4}\right)+\phi \left(\frac{\left|\mathfrak{F}\right|+3\left|{\mathfrak{F}}^{*}\right|}{4}\right)∥\end{array}$$

(10)

for all increasing convex functions $\phi :\left[0,\infty \right)\to \left[0,\infty \right)$. In particular, they proved

$$\begin{array}{c}\hfill {\omega }^2\left(\mathfrak{F}\right)\le \frac{1}{32}∥{\left(3\left|\mathfrak{F}\right|+\left|{\mathfrak{F}}^{*}\right|\right)}^2+{\left(\left|\mathfrak{F}\right|+3\left|{\mathfrak{F}}^{*}\right|\right)}^2∥.\end{array}$$

(11)

The constant $\frac{1}{32}$ is the best possible.

For more generalizations and recent related results concerning numerical radius, the reader may refer to [9,10,11,12,13,14,15,16,17,18,19] and the references therein. In addition, a survey of the numerical radius can be found in [20]. Other topics related to the numerical radius are discussed in [20,21,22]. Finally, readers should consult [23] for an overview of the most recent results and applications.

In this paper, we present several generalized extensions of numerical radius inequalities for Hilbert space operators that have been recently established. These extensions refine inequalities that were previously proven in [1,2]. It has been demonstrated that combining certain inequalities can lead to improvements or restorations of other inequalities. The main objective of this study is to provide a unified framework for certain numerical radius inequalities by extending the existing ones. Additionally, a numerical example is provided to showcase the effectiveness of the proposed approach.

Our approach involves combining the existing inequalities proven in [1,2] into a single inequality that can yield improved or restored results. This unified approach enables us to extend the existing numerical radius inequalities and validate their effectiveness through numerical examples. Specifically, this work explores several new extensions and generalizations of the inequalities stated in Equation (9), building upon the Sababheh–Moradi inequality (9). Notably, we prove that the middle term in Equation (9) is bounded by a generalized extension of the first inequality in Equation (5), thereby improving upon the second bound of the same inequality. Furthermore, Equation (9) itself is extended and refined. Similarly, we provide generalized extensions for Equations (10) and (11), as well as establish extensions for Equations (7) and (8).

2. Refinement of the Cauchy–Schwarz Inequality

To prove our results, we need a sequence of lemmas.

Lemma 1

([24]). (Theorem 1.4) Let $\mathfrak{P}\in \mathcal{J}\left(\mathcal{G}\right)$ be a positive operator. Then,

$$\begin{array}{c}\hfill {〈\mathfrak{P}u,u〉}^r\le 〈{\mathfrak{P}}^{r}u,u〉,r\ge 1\end{array}$$

(12)

for any vector $u\in \mathcal{G}$. Inequality (12) is reversed if $0\le r\le 1$.

Lemma 2

([25]). (Theorem 2.3) Let g be a non-negative convex function on $\left[0,\infty \right)$, and let $\mathfrak{P},\mathfrak{Q}\in \mathcal{J}\left(\mathcal{G}\right)$ be two positive operators. Then,

$$\begin{array}{c}\hfill ∥g\left(\frac{\mathfrak{P}+\mathfrak{Q}}{2}\right)∥\le ∥\frac{g\left(\mathfrak{P}\right)+g\left(\mathfrak{Q}\right)}{2}∥.\end{array}$$

(13)

Lemma 3

([24]). (Theorem 1.4) Let g be a convex function on a real interval J, then

$$\begin{array}{c}\hfill g\left(〈\mathfrak{P}u,u〉\right)\le 〈g\left(\mathfrak{P}\right)u,u〉\end{array}$$

(14)

for every self-adjoint operator $\mathfrak{P}\in \mathcal{J}\left(\mathcal{G}\right)$ whose spectrum contained in J.

The following two lemmas are key lemmas that are used as primary results in the whole of the presented results of this work. The author of [1] proved the following refinement of the mixed Cauchy–Schwarz inequality:

Lemma 4

([1]). Let $\mathfrak{P}\in \mathcal{J}\left(\mathcal{G}\right)$. Then,

$$\begin{array}{cc}\hfill & {\left|〈\mathfrak{P}v,v〉\right|}^2\hfill \\ \hfill & \le \gamma 〈{\left|\mathfrak{P}\right|}^{2\eta }u,u〉〈{\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}v,v〉+\left(1-\gamma \right)\left|〈\mathfrak{P}u,v〉\right|\sqrt{〈{\left|\mathfrak{P}\right|}^{2\eta }u,u〉〈{\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}v,v〉}\hfill \\ \hfill & \le 〈{\left|\mathfrak{P}\right|}^{2\eta }u,u〉〈{\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}v,v〉\hfill \end{array}$$

(15)

for all $0\le \eta ,\gamma \le 1$.

Lemma 5

([1]). Let $\mathfrak{P},\mathfrak{Q}\in \mathcal{J}\left(\mathcal{G}\right)$, then

$$\begin{array}{cc}\hfill {\left|〈\mathfrak{P}u,\mathfrak{Q}v〉\right|}^2& \le \eta 〈{\left|\mathfrak{P}\right|}^{2}u,u〉〈{\left|\mathfrak{Q}\right|}^{2}v,v〉+\left(1-\eta \right)\left|〈\mathfrak{P}u,\mathfrak{Q}v〉\right|\sqrt{〈{\left|\mathfrak{P}\right|}^{2}u,u〉〈{\left|\mathfrak{Q}\right|}^{2}v,v〉}\hfill \\ \hfill & \le {∥\mathfrak{P}u∥}^2{∥\mathfrak{Q}v∥}^2\hfill \end{array}$$

(16)

for any vectors $u,v\in \mathcal{G}$ and all $\eta \in \left[0,1\right]$.

The following two results generalize the main results in [26].

Proposition 1.

Let $\mathfrak{P}\in \mathcal{J}\left(\mathcal{G}\right)$. If $g:\left[0,\infty \right)\to \left[0,\infty \right)$ is an increasing and operator convex function, then

$$\begin{array}{c}\hfill g\left(\omega \left(K\right)\right)\le \frac{1}{2}∥g\left({\left|\mathfrak{P}\right|}^{2\eta }\right)+g\left({\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}\right)∥\end{array}$$

(17)

for all $0\le \eta \le 1$.

Proof. 

By utilizing the classical mixed Schwarz inequality (15) (with $\gamma =1$), we obtain

$$\begin{array}{c}\hfill \left|〈\mathfrak{P}u,u〉\right|\le \sqrt{〈{\left|\mathfrak{P}\right|}^{2\eta }u,u〉〈{\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}u,u〉},u\in \mathcal{G}.\end{array}$$

(18)

Since g is increasing,

$$\begin{array}{cc}\hfill g\left(\left|〈\mathfrak{P}u,u〉\right|\right)& \le g\left(\sqrt{〈{\left|\mathfrak{P}\right|}^{2\eta }u,u〉〈{\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}u,u〉}\right)\hfill \\ \hfill & \le g\left(〈\frac{{\left|\mathfrak{P}\right|}^{2\eta }+{\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}}{2}u,u〉\right)(\mathrm{by}\mathrm{AM}–\mathrm{GM}\mathrm{inequality})\hfill \\ \hfill & \le 〈g\left(\frac{{\left|\mathfrak{P}\right|}^{2\eta }+{\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}}{2}\right)u,u〉\hfill \\ \hfill & \le 〈\frac{g\left({\left|\mathfrak{P}\right|}^{2\eta }\right)+g\left({\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}\right)}{2}u,u〉(g\mathrm{is}\mathrm{operator}\mathrm{convex}),\hfill \end{array}$$

taking the supremum over all unit vector $u\in \mathcal{G}$ in all previous inequalities, we obtain the required result. □

Proposition 2.

Let $\mathfrak{P}\in \mathcal{J}\left(\mathcal{G}\right)$. If $g:\left[0,\infty \right)\to \left[0,\infty \right)$ is an increasing and convex function, then

$$\begin{array}{c}\hfill g\left({\omega }^2\left(\mathfrak{P}\right)\right)\le ∥\eta g\left({\left|\mathfrak{P}\right|}^2\right)+\left(1-\eta \right)g\left({\left|{\mathfrak{P}}^{*}\right|}^2\right)∥\end{array}$$

(19)

for all $0\le \eta \le 1$.

Proof. 

Since g is increasing, by (18), we have

$$\begin{array}{cc}\hfill g\left({\left|〈\mathfrak{P}u,u〉\right|}^2\right)& \le g\left(〈{\left|\mathfrak{P}\right|}^{2\eta }u,u〉〈{\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}u,u〉\right)\hfill \\ \hfill & \le g\left({〈{\left|\mathfrak{P}\right|}^{2}u,u〉}^{\eta }{〈{\left|{\mathfrak{P}}^{*}\right|}^{2}u,u〉}^{\left(1-\eta \right)}\right)(\mathrm{by}\mathrm{the}\mathrm{reverse}\mathrm{of}(12\left)\right)\hfill \\ \hfill & \le g\left(\eta 〈{\left|\mathfrak{P}\right|}^{2}u,u〉+\left(1-\eta \right)〈{\left|{\mathfrak{P}}^{*}\right|}^{2}u,u〉\right)(\mathrm{by}\mathrm{AM}–\mathrm{GM}\mathrm{inequality})\hfill \\ \hfill & \le \eta g\left(〈{\left|\mathfrak{P}\right|}^{2}u,u〉\right)+\left(1-\eta \right)g\left(〈{\left|{\mathfrak{P}}^{*}\right|}^{2}u,u〉\right)(g\mathrm{is}\mathrm{convex})\hfill \\ \hfill & \le \eta 〈g\left({\left|\mathfrak{P}\right|}^2\right)u,u〉+\left(1-\eta \right)〈g\left({\left|{\mathfrak{P}}^{*}\right|}^2\right)u,u〉(\mathrm{by}\mathrm{Lemma}3)\hfill \\ \hfill & =〈\left(\eta g\left({\left|\mathfrak{P}\right|}^2\right)+\left(1-\eta \right)g\left({\left|{\mathfrak{P}}^{*}\right|}^2\right)\right)u,u〉.\hfill \end{array}$$

Taking the supremum over all unit vector $u\in \mathcal{G}$ in all previous inequalities, we obtain the required result. □

The following result establishes a generalized extension of [1] (Theorem 6) that refines both the second inequality in (5) and (9) as well as merges the second inequality in (5) with (9).

Theorem 1.

Let $\mathfrak{P}\in \mathcal{J}\left(\mathcal{G}\right)$. If $g:\left[0,\infty \right)\to \left[0,\infty \right)$ is an increasing, submultiplicative, and operator convex function, then

$$\begin{array}{cc}\hfill g\left({\omega }^2\left(\mathfrak{P}\right)\right)& \le \gamma ∥{\int }_0^{1}g\left(s{\left|\mathfrak{P}\right|}^{4\eta }+\left(1-s\right){\left|{\mathfrak{P}}^{*}\right|}^{4\left(1-\eta \right)}\right)ds∥\hfill \\ \hfill & +\left(1-\gamma \right)g\left(\omega \left(\mathfrak{P}\right)\right)∥{\int }_0^{1}g\left(s{\left|\mathfrak{P}\right|}^{2\eta }+\left(1-s\right){\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}\right)ds∥\hfill \\ \hfill & \le \frac{1}{2}\gamma ∥g\left({\left|\mathfrak{P}\right|}^{4\eta }\right)+g\left({\left|{\mathfrak{P}}^{*}\right|}^{4\left(1-\eta \right)}\right)∥\hfill \\ \hfill & +\frac{1}{2}\left(1-\gamma \right)g\left(\omega \left(\mathfrak{P}\right)\right)∥g\left({\left|\mathfrak{P}\right|}^{2\eta }\right)+g\left({\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}\right)∥\hfill \\ \hfill & \le \frac{1}{2}∥g^2\left({\left|\mathfrak{P}\right|}^{2\eta }\right)+g^2\left({\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}\right)∥\hfill \end{array}$$

(20)

for all $\eta ,\gamma \in \left[0,1\right]$.

Proof. 

Since g is an increasing and operator convex, then, by Jensen’s inequality, we have

$$\begin{array}{cc}\hfill & g\left({\left|〈\mathfrak{P}u,u〉\right|}^2\right)\hfill \\ \hfill & \le g\left(\gamma 〈{\left|\mathfrak{P}\right|}^{2\eta }u,u〉〈{\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}u,u〉\right)(\mathrm{by}\mathrm{Lemma}4)\hfill \\ \hfill & +\left(1-\gamma \right)\left|〈\mathfrak{P}u,u〉\right|\sqrt{〈{\left|\mathfrak{P}\right|}^{2\eta }u,u〉〈{\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}u,u〉}\hfill \\ \hfill & \le \gamma g\left(〈{\left|\mathfrak{P}\right|}^{2\eta }u,u〉〈{\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}u,u〉\right)(g\mathrm{is}\mathrm{convex})\hfill \\ \hfill & +\left(1-\gamma \right)g\left(\left|〈\mathfrak{P}u,u〉\right|\sqrt{〈{\left|\mathfrak{P}\right|}^{2\eta }u,u〉〈{\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}u,u〉}\right)\hfill \\ \hfill & \le \gamma g\left({\left[\frac{〈{\left|\mathfrak{P}\right|}^{2\eta }u,u〉+〈{\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}u,u〉}{2}\right]}^2\right)(\mathrm{by}\mathrm{AM}–\mathrm{GM}\mathrm{inequality})\hfill \\ \hfill & +\left(1-\gamma \right)g\left(\left|〈\mathfrak{P}u,u〉\right|\right)g\left(\sqrt{〈{\left|\mathfrak{P}\right|}^{2\eta }u,u〉〈{\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}u,u〉}\right)\left(g\mathrm{is}\mathrm{submultiplicative}\right)\hfill \\ \hfill & \le \gamma g\left(\frac{{〈{\left|\mathfrak{P}\right|}^{2\eta }u,u〉}^2+{〈{\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}u,u〉}^2}{2}\right)(t^2\mathrm{is}\mathrm{convex})\hfill \\ \hfill & +\left(1-\gamma \right)g\left(\left|〈\mathfrak{P}u,u〉\right|\right)g\left(〈\frac{{\left|\mathfrak{P}\right|}^{2\eta }+{\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}}{2}u,u〉\right)(\mathrm{by}\mathrm{AM}–\mathrm{GM}\mathrm{inequality})\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le \gamma g\left(\frac{〈{\left|\mathfrak{P}\right|}^{4\eta }u,u〉+〈{\left|{\mathfrak{P}}^{*}\right|}^{4\left(1-\eta \right)}u,u〉}{2}\right)(\mathrm{by}\mathrm{Lemma}1)\hfill \\ \hfill & +\left(1-\gamma \right)g\left(\left|〈\mathfrak{P}u,u〉\right|\right)g\left(〈\frac{{\left|\mathfrak{P}\right|}^{2\eta }+{\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}}{2}u,u〉\right)\hfill \\ \hfill & (\mathrm{by}\mathrm{the}\mathrm{Hermite}–\mathrm{Hadamard}\mathrm{inequality})\hfill \\ \hfill & \le \gamma {\int }_0^{1}g\left(〈\left(s{\left|\mathfrak{P}\right|}^{4\eta }+\left(1-s\right){\left|{\mathfrak{P}}^{*}\right|}^{4\left(1-\eta \right)}\right)u,u〉\right)ds\hfill \\ \hfill & +\left(1-\gamma \right)g\left(\left|〈\mathfrak{P}u,u〉\right|\right){\int }_0^{1}g\left(〈\left(s{\left|\mathfrak{P}\right|}^{2\eta }+\left(1-s\right){\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}\right)u,u〉\right)ds.\hfill \end{array}$$

On the other hand, we have

$$\begin{array}{cc}\hfill & {\int }_0^{1}g\left(〈\left(s{\left|\mathfrak{P}\right|}^{2\eta }+\left(1-s\right){\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}\right)u,u〉\right)ds\hfill \\ \hfill & \le {\int }_0^1〈g\left(s{\left|\mathfrak{P}\right|}^{2\eta }+\left(1-s\right){\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}\right)u,u〉ds(\mathrm{by}\mathrm{Lemma}3)\hfill \\ \hfill & \le {\int }_0^1〈\left(\eta g\left({\left|\mathfrak{P}\right|}^{2\eta }\right)+\left(1-s\right)g\left({\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}\right)\right)u,u〉ds(g\mathrm{is}\mathrm{operator}\mathrm{convex})\hfill \\ \hfill & =〈\frac{g\left({\left|\mathfrak{P}\right|}^{2\eta }\right)+g\left({\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}\right)}{2}u,u〉.\hfill \end{array}$$

Similarly, we have

$$\begin{array}{c}\hfill {\int }_0^{1}g\left(〈\left(s{\left|\mathfrak{P}\right|}^{4\eta }+\left(1-s\right){\left|{\mathfrak{P}}^{*}\right|}^{4\left(1-\eta \right)}\right)u,u〉\right)ds\le 〈\frac{g\left({\left|\mathfrak{P}\right|}^{4\eta }\right)+g\left({\left|{\mathfrak{P}}^{*}\right|}^{4\left(1-\eta \right)}\right)}{2}u,u〉.\end{array}$$

Taking the supremum over all unit vector $u\in \mathcal{G}$ in all previous inequalities, we obtain the required result. To obtain the third inequality, from (20), we have

$$\begin{array}{cc}\hfill & g\left({\omega }^2\left(\mathfrak{P}\right)\right)\hfill \\ \hfill & \le \frac{1}{2}\gamma ∥g\left({\left|\mathfrak{P}\right|}^{4\eta }\right)+g\left({\left|{\mathfrak{P}}^{*}\right|}^{4\left(1-\eta \right)}\right)∥\hfill \\ \hfill & +\frac{1}{2}\left(1-\gamma \right)g\left(\omega \left(\mathfrak{P}\right)\right)∥g\left({\left|\mathfrak{P}\right|}^{2\eta }\right)+g\left({\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}\right)∥\hfill \\ \hfill & \le \frac{1}{2}\gamma ∥g\left({\left({\left|\mathfrak{P}\right|}^{2\eta }\right)}^2\right)+g\left({\left({\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}\right)}^2\right)∥(\mathrm{by}(17\left)\right)\hfill \\ \hfill & +\frac{1}{4}\left(1-\gamma \right){∥g\left({\left|\mathfrak{P}\right|}^{2\eta }\right)+g\left({\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}\right)∥}^2\hfill \\ \hfill & \le \frac{1}{2}\gamma ∥g^2\left({\left|\mathfrak{P}\right|}^{2\eta }\right)+g\left({\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}\right)∥(g\mathrm{is}\mathrm{submultiplicative})\hfill \\ \hfill & +\left(1-\gamma \right)∥{\left(\frac{g\left({\left|\mathfrak{P}\right|}^{2\eta }\right)+g\left({\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}\right)}{2}\right)}^2∥\hfill \\ \hfill & \le \frac{1}{2}\gamma ∥g^2\left({\left|\mathfrak{P}\right|}^{2\eta }\right)+g\left({\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}\right)∥(t^2\mathrm{is}\mathrm{operator}\mathrm{convex})\hfill \\ \hfill & +\left(1-\gamma \right)∥\frac{g^2\left({\left|\mathfrak{P}\right|}^{2\eta }\right)+g^2\left({\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}\right)}{2}∥\hfill \\ \hfill & =\frac{1}{2}\gamma ∥g^2\left({\left|\mathfrak{P}\right|}^{2\eta }\right)+g\left({\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}\right)∥+\left(1-\gamma \right)∥\frac{g^2\left({\left|\mathfrak{P}\right|}^{2\eta }\right)+g^2\left({\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}\right)}{2}∥\hfill \\ \hfill & =\frac{1}{2}\gamma ∥g^2\left({\left|\mathfrak{P}\right|}^{2\eta }\right)+g\left({\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}\right)∥+\frac{1}{2}\left(1-\gamma \right)∥g^2\left({\left|\mathfrak{P}\right|}^{2\eta }\right)+g\left({\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}\right)∥\hfill \\ \hfill & =\frac{1}{2}∥g^2\left({\left|\mathfrak{P}\right|}^{2\eta }\right)+g^2\left({\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}\right)∥,\hfill \end{array}$$

and this completes the proof of the Theorem. □

Corollary 1.

Let $\mathfrak{P}\in \mathcal{J}\left(\mathcal{G}\right)$. Then, for any $1\le r\le 2$,

$$\begin{array}{cc}\hfill {\omega }^{2r}\left(\mathfrak{P}\right)& \le \gamma ∥{\int }_0^1{\left(s{\left|\mathfrak{P}\right|}^{4\eta }+\left(1-s\right){\left|{\mathfrak{P}}^{*}\right|}^{4\left(1-\eta \right)}\right)}^{r}ds∥\hfill \\ \hfill & +\left(1-\gamma \right){\omega }^r\left(\mathfrak{P}\right)∥{\int }_0^1{\left(s{\left|\mathfrak{P}\right|}^{2\eta }+\left(1-s\right){\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}\right)}^{r}ds∥\hfill \\ \hfill & \le \frac{1}{2}\gamma ∥{\left|\mathfrak{P}\right|}^{4r\eta }+{\left|{\mathfrak{P}}^{*}\right|}^{4r\left(1-\eta \right)}∥+\frac{1}{2}\left(1-\gamma \right){\omega }^r\left(\mathfrak{P}\right)∥{\left|\mathfrak{P}\right|}^{2r\eta }+{\left|{\mathfrak{P}}^{*}\right|}^{2r\left(1-\eta \right)}∥\hfill \\ \hfill & \le \frac{1}{2}∥{\left|\mathfrak{P}\right|}^{4r\eta }+{\left|{\mathfrak{P}}^{*}\right|}^{4r\left(1-\eta \right)}∥\hfill \end{array}$$

(21)

for all $\eta ,\gamma \in \left[0,1\right]$.

Proof. 

Applying Theorem 1 for the increasing, submultiplicative, and operator convex function, $g\left(t\right)=t^r$, $r\in \left[1,2\right]$. □

Remark 1.

Setting $\eta =\frac{1}{2}$ and $r=2$ in Corollary 1, we obtain

$$\begin{array}{cc}\hfill {\omega }^4\left(\mathfrak{P}\right)& \le \gamma ∥{\int }_0^1{\left(s{\left|\mathfrak{P}\right|}^2+\left(1-s\right){\left|{\mathfrak{P}}^{*}\right|}^2\right)}^{2}ds∥\hfill \\ \hfill & +\left(1-\gamma \right){\omega }^2\left(\mathfrak{P}\right)∥{\int }_0^1{\left(s\left|\mathfrak{P}\right|+\left(1-s\right)\left|{\mathfrak{P}}^{*}\right|\right)}^{2}ds∥\hfill \\ \hfill & \le \frac{1}{2}\gamma ∥{\left|\mathfrak{P}\right|}^4+{\left|{\mathfrak{P}}^{*}\right|}^4∥+\frac{1}{2}\left(1-\gamma \right){\omega }^2\left(\mathfrak{P}\right)∥{\left|\mathfrak{P}\right|}^2+{\left|{\mathfrak{P}}^{*}\right|}^2∥\hfill \\ \hfill & \le \frac{1}{2}∥{\left|\mathfrak{P}\right|}^4+{\left|{\mathfrak{P}}^{*}\right|}^4∥,\hfill \end{array}$$

while when $r=1$, inequality (21) reduces to

$$\begin{array}{cc}\hfill {\omega }^2\left(\mathfrak{P}\right)& \le \frac{1}{2}\gamma ∥{\left|\mathfrak{P}\right|}^2+{\left|{\mathfrak{P}}^{*}\right|}^2∥+\frac{1}{2}\left(1-\gamma \right)\omega \left(\mathfrak{P}\right)∥\left|\mathfrak{P}\right|+\left|{\mathfrak{P}}^{*}\right|∥\hfill \\ \hfill & \le \frac{1}{2}∥{\left|\mathfrak{P}\right|}^2+{\left|{\mathfrak{P}}^{*}\right|}^2∥.\hfill \end{array}$$

A generalized extension that refines [1] (Theorem 5) and refines the middle term in (9) is incorporated in the following result.

Theorem 2.

Let $\mathfrak{P}\in \mathcal{J}\left(\mathcal{G}\right)$. If $g:\left[0,\infty \right)\to \left[0,\infty \right)$ is an increasing, submultiplicative and operator convex function, then

$$\begin{array}{cc}\hfill g\left({\omega }^2\left(\mathfrak{P}\right)\right)& \le \gamma ∥{\int }_0^{1}g\left(\eta {\left|\mathfrak{P}\right|}^2+\left(1-\eta \right){\left|{\mathfrak{P}}^{*}\right|}^2\right)d\eta ∥\hfill \\ \hfill & +\left(1-\gamma \right)g\left(\omega \left(\mathfrak{P}\right)\right){∥{\int }_0^{1}g\left(\eta {\left|\mathfrak{P}\right|}^2+\left(1-\eta \right){\left|{\mathfrak{P}}^{*}\right|}^2\right)d\eta ∥}^{\frac{1}{2}}\hfill \\ \hfill & \le \frac{1}{2}\gamma ∥g\left({\left|\mathfrak{P}\right|}^2\right)+g\left({\left|{\mathfrak{P}}^{*}\right|}^2\right)∥+\frac{1}{\sqrt{2}}\left(1-\gamma \right)g\left(\omega \left(\mathfrak{P}\right)\right){∥g\left({\left|\mathfrak{P}\right|}^2\right)+g\left({\left|{\mathfrak{P}}^{*}\right|}^2\right)∥}^{\frac{1}{2}}\hfill \end{array}$$

(22)

for all $\gamma \in \left[0,1\right]$.

Proof. 

Since g is increasing, (15) implies that

$$\begin{array}{cc}\hfill & g\left({\left|〈\mathfrak{P}u,u〉\right|}^2\right)\hfill \\ \hfill & \le g\left(\gamma 〈{\left|\mathfrak{P}\right|}^{2\eta }u,u〉〈{\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}u,u〉+\left(1-\gamma \right)\left|〈\mathfrak{P}u,u〉\right|\sqrt{〈{\left|\mathfrak{P}\right|}^{2\eta }u,u〉〈{\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}u,u〉}\right)\hfill \\ \hfill & \le \gamma g\left(〈{\left|\mathfrak{P}\right|}^{2\eta }u,u〉〈{\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}u,u〉\right)\hfill \\ \hfill & +\left(1-\gamma \right)g\left(\left|〈\mathfrak{P}u,u〉\right|\sqrt{〈{\left|\mathfrak{P}\right|}^{2\eta }u,u〉〈{\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}u,u〉}\right)(g\mathrm{is}\mathrm{convex})\hfill \\ \hfill & \le \gamma g\left({〈{\left|\mathfrak{P}\right|}^{2}u,u〉}^{\eta }{〈{\left|{\mathfrak{P}}^{*}\right|}^{2}u,u〉}^{\left(1-\eta \right)}\right)(\mathrm{by}(12\left)\right)\hfill \\ \hfill & +\left(1-\gamma \right)g\left(\left|〈\mathfrak{P}u,u〉\right|\right)g\left(\sqrt{{〈{\left|\mathfrak{P}\right|}^{2}u,u〉}^{\eta }{〈{\left|{\mathfrak{P}}^{*}\right|}^{2}u,u〉}^{\left(1-\eta \right)}}\right)(g\mathrm{is}\mathrm{submultiplicative})\hfill \\ \hfill & \le \gamma g\left(\eta 〈{\left|\mathfrak{P}\right|}^{2}u,u〉+\left(1-\eta \right)〈{\left|{\mathfrak{P}}^{*}\right|}^{2}u,u〉\right)(\mathrm{by}\mathrm{AM}–\mathrm{GM}\mathrm{inequality})\hfill \\ \hfill & +\left(1-\gamma \right)g\left(\left|〈\mathfrak{P}u,u〉\right|\right)g\left(\sqrt{\eta 〈{\left|\mathfrak{P}\right|}^{2}u,u〉+\left(1-\eta \right)〈{\left|{\mathfrak{P}}^{*}\right|}^{2}u,u〉}\right)(\mathrm{by}(12\left)\right)\hfill \\ \hfill & =\gamma g\left(〈\left(\eta {\left|\mathfrak{P}\right|}^2+\left(1-\eta \right){\left|{\mathfrak{P}}^{*}\right|}^2\right)u,u〉\right)\hfill \\ \hfill & +\left(1-\gamma \right)g\left(\left|〈\mathfrak{P}u,u〉\right|\right)g\left(\sqrt{〈\left(\eta {\left|\mathfrak{P}\right|}^2+\left(1-\eta \right){\left|{\mathfrak{P}}^{*}\right|}^2\right)u,u〉}\right)\hfill \\ \hfill & \le \gamma g\left(〈\left(\eta {\left|\mathfrak{P}\right|}^2+\left(1-\eta \right){\left|{\mathfrak{P}}^{*}\right|}^2\right)u,u〉\right)\hfill \\ \hfill & +\left(1-\gamma \right)g\left(\left|〈\mathfrak{P}u,u〉\right|\right)\sqrt{g\left(〈\left(\eta {\left|\mathfrak{P}\right|}^2+\left(1-\eta \right){\left|{\mathfrak{P}}^{*}\right|}^2\right)u,u〉\right)}(g\mathrm{is}\mathrm{submultiplicative})\hfill \\ \hfill & \le \gamma 〈g\left(\eta {\left|\mathfrak{P}\right|}^2+\left(1-\eta \right){\left|{\mathfrak{P}}^{*}\right|}^2\right)u,u〉(\mathrm{by}\mathrm{Lemma}3)\hfill \\ \hfill & +\left(1-\gamma \right)g\left(\left|〈\mathfrak{P}u,u〉\right|\right)\sqrt{〈g\left(\eta {\left|\mathfrak{P}\right|}^2+\left(1-\eta \right){\left|{\mathfrak{P}}^{*}\right|}^2\right)u,u〉}.\hfill \end{array}$$

Integrating with respect to $\eta$ over $\left[0,1\right]$, we obtain

$$\begin{array}{cc}\hfill & g\left({\left|〈\mathfrak{P}u,u〉\right|}^2\right)\hfill \\ \hfill & \le \gamma {\int }_0^1\left(〈g\left(\eta {\left|\mathfrak{P}\right|}^2+\left(1-\eta \right){\left|{\mathfrak{P}}^{*}\right|}^2\right)u,u〉\right)d\eta \hfill \\ \hfill & +\left(1-\gamma \right)g\left(\left|〈\mathfrak{P}u,u〉\right|\right){\int }_0^1\left(\sqrt{〈g\left(\eta {\left|\mathfrak{P}\right|}^2+\left(1-\eta \right){\left|{\mathfrak{P}}^{*}\right|}^2\right)u,u〉}\right)d\eta \hfill \\ \hfill & \le \gamma 〈{\int }_0^1\left(g\left(\eta {\left|\mathfrak{P}\right|}^2+\left(1-\eta \right){\left|{\mathfrak{P}}^{*}\right|}^2\right)d\eta \right)u,u〉(\mathrm{by}\mathrm{H}ö\mathrm{lder}\mathrm{inequality})\hfill \\ \hfill & +\left(1-\gamma \right)g\left(\left|〈\mathfrak{P}u,u〉\right|\right)\sqrt{{\int }_0^1\left(〈g\left(\eta {\left|\mathfrak{P}\right|}^2+\left(1-\eta \right){\left|{\mathfrak{P}}^{*}\right|}^2\right)u,u〉\right)d\eta }\hfill \\ \hfill & =\gamma 〈{\int }_0^1\left(g\left(\eta {\left|\mathfrak{P}\right|}^2+\left(1-\eta \right){\left|{\mathfrak{P}}^{*}\right|}^2\right)d\eta \right)u,u〉\hfill \\ \hfill & +\left(1-\gamma \right)g\left(\left|〈\mathfrak{P}u,u〉\right|\right)\sqrt{〈{\int }_0^1\left(g\left(\eta {\left|\mathfrak{P}\right|}^2+\left(1-\eta \right){\left|{\mathfrak{P}}^{*}\right|}^2\right)d\eta \right)u,u〉}.\hfill \end{array}$$

Taking the supremum over all unit vector $u\in \mathcal{G}$ in all previous inequalities, we obtain the first two inequalities, i.e.,

$$\begin{array}{cc}\hfill g\left({\omega }^2\left(\mathfrak{P}\right)\right)& \le \gamma ∥{\int }_0^{1}g\left(\eta {\left|\mathfrak{P}\right|}^2+\left(1-\eta \right){\left|{\mathfrak{P}}^{*}\right|}^2\right)d\eta ∥\hfill \\ \hfill & +\left(1-\gamma \right)g\left(\omega \left(\mathfrak{P}\right)\right){∥{\int }_0^{1}g\left(\eta {\left|\mathfrak{P}\right|}^2+\left(1-\eta \right){\left|{\mathfrak{P}}^{*}\right|}^2\right)d\eta ∥}^{\frac{1}{2}}\hfill \\ \hfill & \le \frac{1}{2}\gamma ∥g\left({\left|\mathfrak{P}\right|}^2\right)+g\left({\left|{\mathfrak{P}}^{*}\right|}^2\right)∥+\frac{1}{\sqrt{2}}\left(1-\gamma \right)g\left(\omega \left(\mathfrak{P}\right)\right){∥g\left({\left|\mathfrak{P}\right|}^2\right)+g\left({\left|{\mathfrak{P}}^{*}\right|}^2\right)∥}^{\frac{1}{2}},\hfill \end{array}$$

as desired, and this completes the proof of the Theorem. □

In the following result, we establish a generalized extension of [1] (Theorem 6) for the first inequality in (5) and the second inequality in (9), as well as merging the second inequality of (5) and (9). The following result simply refines Theorem 1.

Corollary 2.

Let $\mathfrak{P}\in \mathcal{J}\left(\mathcal{G}\right)$. If $g:\left[0,\infty \right)\to \left[0,\infty \right)$ is an increasing, submultiplicative, and operator convex function, then

$$\begin{array}{cc}\hfill g^2\left({\omega }^2\left(\mathfrak{P}\right)\right)& \le \gamma {∥{\int }_0^{1}g\left(\eta {\left|\mathfrak{P}\right|}^2+\left(1-\eta \right){\left|{\mathfrak{P}}^{*}\right|}^2\right)d\eta ∥}^2\hfill \\ \hfill & +\left(1-\gamma \right)g^2\left(\omega \left(\mathfrak{P}\right)\right)∥{\int }_0^{1}g\left(\eta {\left|\mathfrak{P}\right|}^2+\left(1-\eta \right){\left|{\mathfrak{P}}^{*}\right|}^2\right)d\eta ∥\hfill \\ \hfill & \le \frac{1}{4}\gamma {∥g\left({\left|\mathfrak{P}\right|}^2\right)+g\left({\left|{\mathfrak{P}}^{*}\right|}^2\right)∥}^2+\frac{1}{2}\left(1-\gamma \right)g^2\left(\omega \left(\mathfrak{P}\right)\right)∥g\left({\left|\mathfrak{P}\right|}^2\right)+g\left({\left|{\mathfrak{P}}^{*}\right|}^2\right)∥\hfill \\ \hfill & \le \frac{1}{4}{∥g\left({\left|\mathfrak{P}\right|}^2\right)+g\left({\left|{\mathfrak{P}}^{*}\right|}^2\right)∥}^2\hfill \end{array}$$

(23)

for all $\eta \in \left[0,1\right]$.

Proof. 

From the proof of the Theorem 2, we have

$$\begin{array}{cc}\hfill g\left({\left|〈\mathfrak{P}u,u〉\right|}^2\right)& \le \gamma 〈{\int }_0^1\left(g\left(\eta {\left|\mathfrak{P}\right|}^2+\left(1-\eta \right){\left|{\mathfrak{P}}^{*}\right|}^2\right)d\eta \right)u,u〉\hfill \\ \hfill & +\left(1-\gamma \right)g\left(\left|〈\mathfrak{P}u,u〉\right|\right)\sqrt{〈{\int }_0^1\left(g\left(\eta {\left|\mathfrak{P}\right|}^2+\left(1-\eta \right){\left|{\mathfrak{P}}^{*}\right|}^2\right)d\eta \right)u,u〉}.\hfill \end{array}$$

Applying the increasing operator convex function $t\mapsto t^2$, we obtain

$$\begin{array}{cc}\hfill & g^2\left({\left|〈\mathfrak{P}u,u〉\right|}^2\right)\hfill \\ \hfill & \le \left(\gamma 〈\left({\int }_0^{1}g\left(\eta {\left|\mathfrak{P}\right|}^2+\left(1-\eta \right){\left|{\mathfrak{P}}^{*}\right|}^2\right)d\eta \right)u,u〉\right.\hfill \\ \hfill & {\left.+\left(1-\gamma \right)g\left(\left|〈\mathfrak{P}u,u〉\right|\right)\sqrt{〈\left({\int }_0^{1}g\left(\eta {\left|\mathfrak{P}\right|}^2+\left(1-\eta \right){\left|{\mathfrak{P}}^{*}\right|}^2\right)d\eta \right)u,u〉}\right)}^2\hfill \\ \hfill & \le \gamma {〈\left({\int }_0^{1}g\left(\eta {\left|\mathfrak{P}\right|}^2+\left(1-\eta \right){\left|{\mathfrak{P}}^{*}\right|}^2\right)d\eta \right)u,u〉}^2\hfill \\ \hfill & +\left(1-\gamma \right)g^2\left(\left|〈\mathfrak{P}u,u〉\right|\right)〈\left({\int }_0^{1}g\left(\eta {\left|\mathfrak{P}\right|}^2+\left(1-\eta \right){\left|{\mathfrak{P}}^{*}\right|}^2\right)d\eta \right)u,u〉\hfill \\ \hfill & \le \gamma 〈{\left({\int }_0^{1}g\left(\eta {\left|\mathfrak{P}\right|}^2+\left(1-\eta \right){\left|{\mathfrak{P}}^{*}\right|}^2\right)d\eta \right)}^{2}u,u〉(\mathrm{by}(12)\mathrm{and}g\mathrm{is}\mathrm{operator}\mathrm{convex})\hfill \\ \hfill & +\left(1-\gamma \right)g^2\left(\left|〈\mathfrak{P}u,u〉\right|\right)〈\left({\int }_0^1\left(\eta g\left({\left|\mathfrak{P}\right|}^2\right)+\left(1-\eta \right)g\left({\left|{\mathfrak{P}}^{*}\right|}^2\right)\right)d\eta \right)u,u〉\hfill \\ \hfill & =\gamma 〈{\left(\frac{g\left({\left|\mathfrak{P}\right|}^2\right)+g\left({\left|{\mathfrak{P}}^{*}\right|}^2\right)}{2}\right)}^{2}u,u〉\hfill \\ \hfill & +\left(1-\gamma \right)g^2\left(\left|〈\mathfrak{P}u,u〉\right|\right)〈\left(\frac{g\left({\left|\mathfrak{P}\right|}^2\right)+g\left({\left|{\mathfrak{P}}^{*}\right|}^2\right)}{2}\right)u,u〉.\hfill \end{array}$$

Taking the supremum over all unit vector $u\in \mathcal{G}$, we obtain the desired result. The third inequality follows by applying (17) to the second term in the second inequality. □

Example 1.

Consider $\mathfrak{P}=\left[\begin{array}{cc}0& 2\\ 1& 0\end{array}\right]$. It is easy to observe that $\omega \left(\mathfrak{P}\right)=1.5$. Applying the inequalities in (28) with $\gamma =\frac{1}{3}$ for the increasing, submultiplicative, and operator convex function $g\left(t\right)=t^2$$(t\ge 0)$, we obtain

$$\begin{array}{cc}\hfill 11.390625={\omega }^6\left(\mathfrak{P}\right)& \le \gamma {∥{\int }_0^1{\left(\eta {\left|\mathfrak{P}\right|}^2+\left(1-\eta \right){\left|{\mathfrak{P}}^{*}\right|}^2\right)}^{2}d\eta ∥}^2\hfill \\ \hfill & +\left(1-\gamma \right){\omega }^4\left(\mathfrak{P}\right)∥{\int }_0^1{\left(\eta {\left|\mathfrak{P}\right|}^2+\left(1-\eta \right){\left|{\mathfrak{P}}^{*}\right|}^2\right)}^{2}d\eta ∥\hfill \\ \hfill & =39.9583\hfill \\ \hfill & \le \frac{1}{4}\gamma {∥{\left|\mathfrak{P}\right|}^4+{\left|{\mathfrak{P}}^{*}\right|}^4∥}^2+\frac{1}{2}\left(1-\gamma \right){\omega }^4\left(\mathfrak{P}\right)∥{\left|\mathfrak{P}\right|}^4+{\left|{\mathfrak{P}}^{*}\right|}^4∥\hfill \\ \hfill & =52.7708\hfill \\ \hfill & \le \frac{1}{4}{∥{\left|\mathfrak{P}\right|}^4+{\left|{\mathfrak{P}}^{*}\right|}^4∥}^2\hfill \\ \hfill & =72.25.\hfill \end{array}$$

We may rewrite this in a more appropriate manner as follows:

$$\begin{array}{cc}\hfill 1.5=\omega \left(\mathfrak{P}\right)& \le \left(\gamma {∥{\int }_0^1{\left(\eta {\left|\mathfrak{P}\right|}^2+\left(1-\eta \right){\left|{\mathfrak{P}}^{*}\right|}^2\right)}^{2}d\eta ∥}^2\right.\hfill \\ \hfill & {\left.+\left(1-\gamma \right){\omega }^4\left(\mathfrak{P}\right)∥{\int }_0^1{\left(\eta {\left|\mathfrak{P}\right|}^2+\left(1-\eta \right){\left|{\mathfrak{P}}^{*}\right|}^2\right)}^{2}d\eta ∥\right)}^{\frac{1}{6}}\hfill \\ \hfill & =1.8489\hfill \\ \hfill & \le {\left(\frac{1}{4}\gamma {∥{\left|\mathfrak{P}\right|}^4+{\left|{\mathfrak{P}}^{*}\right|}^4∥}^2+\frac{1}{2}\left(1-\gamma \right){\omega }^4\left(\mathfrak{P}\right)∥{\left|\mathfrak{P}\right|}^4+{\left|{\mathfrak{P}}^{*}\right|}^4∥\right)}^{\frac{1}{6}}\hfill \\ \hfill & =1.9367\hfill \\ \hfill & \le {\left(\frac{1}{4}{∥{\left|\mathfrak{P}\right|}^4+{\left|{\mathfrak{P}}^{*}\right|}^4∥}^2\right)}^{1/6}\hfill \\ \hfill & =2.04082.\hfill \end{array}$$

As a result, the first inequality in (23) is a nontrivial refinement of the first inequality in (5). This implies that (23) can be used to obtain a tighter upper bound than the one obtained with (5).

A generalization of (23) could be extended in the following result.

Corollary 3.

Let $\mathfrak{P}\in \mathcal{J}\left(\mathcal{G}\right)$. If $g:\left[0,\infty \right)\to \left[0,\infty \right)$ is an increasing, submultiplicative and operator convex function, then

$$\begin{array}{cc}\hfill & g^2\left({\omega }^2\left(\mathfrak{P}\right)\right)\hfill \\ \hfill & \le \gamma {∥\eta g\left({\left|\mathfrak{P}\right|}^2\right)+\left(1-\eta \right)g\left({\left|{\mathfrak{P}}^{*}\right|}^2\right)∥}^2\hfill \\ \hfill & +\left(1-\gamma \right)g^2\left(\omega \left(\mathfrak{P}\right)\right)∥\eta g\left({\left|\mathfrak{P}\right|}^2\right)+\left(1-\eta \right)g\left({\left|{\mathfrak{P}}^{*}\right|}^2\right)∥\hfill \\ \hfill & \le \gamma ∥\eta g^2\left({\left|\mathfrak{P}\right|}^2\right)+\left(1-\eta \right)g^2\left({\left|{\mathfrak{P}}^{*}\right|}^2\right)∥\hfill \\ \hfill & +\left(1-\gamma \right)g^2\left(\omega \left(\mathfrak{P}\right)\right)∥\eta g\left({\left|\mathfrak{P}\right|}^2\right)+\left(1-\eta \right)g\left({\left|{\mathfrak{P}}^{*}\right|}^2\right)∥\hfill \end{array}$$

(24)

for all $\eta \in \left[0,1\right]$.

Proof. 

From the proof of the Theorem 2, we have

$$\begin{array}{cc}\hfill & g\left({\left|〈\mathfrak{P}u,u〉\right|}^2\right)\hfill \\ \hfill & \le \gamma 〈g\left(\eta {\left|\mathfrak{P}\right|}^2+\left(1-\eta \right){\left|{\mathfrak{P}}^{*}\right|}^2\right)u,u〉\hfill \\ \hfill & +\left(1-\gamma \right)g\left(\left|〈\mathfrak{P}u,u〉\right|\right)\sqrt{〈g\left(\eta {\left|\mathfrak{P}\right|}^2+\left(1-\eta \right){\left|{\mathfrak{P}}^{*}\right|}^2\right)u,u〉}.\hfill \end{array}$$

Applying the operator convex function $t\mapsto t^2$, we obtain

$$\begin{array}{cc}\hfill & g^2\left({\left|〈\mathfrak{P}u,u〉\right|}^2\right)\hfill \\ \hfill & \le \left(\gamma 〈g\left(\eta {\left|\mathfrak{P}\right|}^2+\left(1-\eta \right){\left|{\mathfrak{P}}^{*}\right|}^2\right)u,u〉\right.\hfill \\ \hfill & {\left.+\left(1-\gamma \right)g\left(\left|〈\mathfrak{P}u,u〉\right|\right)\sqrt{〈g\left(\eta {\left|\mathfrak{P}\right|}^2+\left(1-\eta \right){\left|{\mathfrak{P}}^{*}\right|}^2\right)u,u〉}\right)}^2\hfill \\ \hfill & \le \gamma {〈g\left(\eta {\left|\mathfrak{P}\right|}^2+\left(1-\eta \right){\left|{\mathfrak{P}}^{*}\right|}^2\right)u,u〉}^2\hfill \\ \hfill & +\left(1-\gamma \right)g^2\left(\left|〈\mathfrak{P}u,u〉\right|\right)〈g\left(\eta {\left|\mathfrak{P}\right|}^2+\left(1-\eta \right){\left|{\mathfrak{P}}^{*}\right|}^2\right)u,u〉\hfill \\ \hfill & \le \gamma 〈{\left(\eta g\left({\left|\mathfrak{P}\right|}^2\right)+\left(1-\eta \right)g\left({\left|{\mathfrak{P}}^{*}\right|}^2\right)\right)}^{2}u,u〉\hfill \\ \hfill & +\left(1-\gamma \right)g^2\left(\left|〈\mathfrak{P}u,u〉\right|\right)〈\left(\eta g\left({\left|\mathfrak{P}\right|}^2\right)+\left(1-\eta \right)g\left({\left|{\mathfrak{P}}^{*}\right|}^2\right)\right)u,u〉\hfill \\ \hfill & \le \gamma 〈\left(\eta g^2\left({\left|\mathfrak{P}\right|}^2\right)+\left(1-\eta \right)g^2\left({\left|{\mathfrak{P}}^{*}\right|}^2\right)\right)u,u〉\hfill \\ \hfill & +\left(1-\gamma \right)g^2\left(\left|〈\mathfrak{P}u,u〉\right|\right)〈\left(\eta g\left({\left|\mathfrak{P}\right|}^2\right)+\left(1-\eta \right)g\left({\left|{\mathfrak{P}}^{*}\right|}^2\right)\right)u,u〉.\hfill \end{array}$$

Taking the supremum over all unit vector $u\in \mathcal{G}$, we obtain the desired result. □

Another approach leads us to the following generalization of (23) and (24).

Corollary 4.

Let $\mathfrak{P}\in \mathcal{J}\left(\mathcal{G}\right)$. If $g:\left[0,\infty \right)\to \left[0,\infty \right)$ is an increasing, submultiplicative and convex, then

$$\begin{array}{cc}\hfill g\left({\omega }^2\left(\mathfrak{P}\right)\right)& \le \frac{1}{4}\gamma {∥g\left({\left|\mathfrak{P}\right|}^{2\eta }\right)+g\left({\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}\right)∥}^2\hfill \\ \hfill & +\frac{1}{2}\left(1-\gamma \right)g\left(\omega \left(\mathfrak{P}\right)\right)∥g\left({\left|\mathfrak{P}\right|}^{2\eta }\right)+g\left({\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}\right)∥\hfill \\ \hfill & \le \frac{1}{2}\gamma ∥g^2\left({\left|\mathfrak{P}\right|}^{2\eta }\right)+g^2\left({\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}\right)∥\hfill \\ \hfill & +\frac{1}{2}\left(1-\gamma \right)g\left(\omega \left(\mathfrak{P}\right)\right)∥g\left({\left|\mathfrak{P}\right|}^{2\eta }\right)+g\left({\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}\right)∥\hfill \\ \hfill & \le \frac{1}{2}∥g^2\left({\left|\mathfrak{P}\right|}^{2\eta }\right)+g^2\left({\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}\right)∥\hfill \end{array}$$

(25)

for all $\eta ,\gamma \in \left[0,1\right]$.

Proof. 

Since g is increasing, from (5) with $p=1$, we have

$$\begin{array}{cc}\hfill & g\left({\omega }^2\left(\mathfrak{P}\right)\right)\hfill \\ \hfill & \le g\left(\frac{1}{4}\gamma {∥{\left|\mathfrak{P}\right|}^{2\eta }+{\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}∥}^2+\frac{1}{2}\left(1-\gamma \right)\omega \left(\mathfrak{P}\right)∥{\left|\mathfrak{P}\right|}^{2\eta }+{\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}∥\right)\hfill \\ \hfill & =g\left(\gamma {∥\frac{{\left|\mathfrak{P}\right|}^{2\eta }+{\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}}{2}∥}^2+\left(1-\gamma \right)\omega \left(\mathfrak{P}\right)∥\frac{{\left|\mathfrak{P}\right|}^{2\eta }+{\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}}{2}∥\right)\hfill \\ \hfill & \le \gamma g\left({∥\frac{{\left|\mathfrak{P}\right|}^{2\eta }+{\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}}{2}∥}^2\right)(g\mathrm{is}\mathrm{convex})\hfill \\ \hfill & +\left(1-\gamma \right)g\left(\omega \left(\mathfrak{P}\right)∥\frac{{\left|\mathfrak{P}\right|}^{2\eta }+{\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}}{2}∥\right)\hfill \\ \hfill & \le \gamma \left(g{\left(∥\frac{{\left|\mathfrak{P}\right|}^{2\eta }+{\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}}{2}∥\right)}^2\right)\hfill \\ \hfill & +\left(1-\gamma \right)g\left(\omega \left(\mathfrak{P}\right)\right)g\left(∥\frac{{\left|\mathfrak{P}\right|}^{2\eta }+{\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}}{2}∥\right)(g\mathrm{is}\mathrm{submultiplicative})\hfill \\ \hfill & \le \gamma {∥\frac{g\left({\left|\mathfrak{P}\right|}^{2\eta }\right)+g\left({\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}\right)}{2}∥}^2(\mathrm{by}\mathrm{Lemma}(13\left)\right)\hfill \\ \hfill & +\left(1-\gamma \right)g\left(\omega \left(\mathfrak{P}\right)\right)∥\frac{g\left({\left|\mathfrak{P}\right|}^{2\eta }\right)+g\left({\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}\right)}{2}∥\hfill \\ \hfill & \le \frac{1}{4}\gamma {∥g\left({\left|\mathfrak{P}\right|}^{2\eta }\right)+g\left({\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}\right)∥}^2\hfill \\ \hfill & +\frac{1}{2}\left(1-\gamma \right)g\left(\omega \left(\mathfrak{P}\right)\right)∥g\left({\left|\mathfrak{P}\right|}^{2\eta }\right)+g\left({\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}\right)∥,\hfill \end{array}$$

and this completes the proof of the result. □

An improved extension of (10) and thus (11) and all previous inequalities could be stated as follows:

Theorem 3.

Let $\mathfrak{P}\in \mathcal{J}\left(\mathcal{G}\right)$. If $g:\left[0,\infty \right)\to \left[0,\infty \right)$ is an increasing and convex, then

$$\begin{array}{cc}\hfill g\left({\omega }^2\left(\mathfrak{P}\right)\right)& \le \frac{1}{2}\gamma ∥g\left(\frac{3{\left|\mathfrak{P}\right|}^{4\eta }+{\left|{\mathfrak{P}}^{*}\right|}^{4\left(1-\eta \right)}}{4}\right)+g\left(\frac{{\left|\mathfrak{P}\right|}^{4\eta }+3{\left|{\mathfrak{P}}^{*}\right|}^{4\left(1-\eta \right)}}{4}\right)∥\hfill \\ \hfill & +\frac{1}{2}\left(1-\gamma \right)g\left(\omega \left(\mathfrak{P}\right)\right)∥g\left(\frac{3{\left|\mathfrak{P}\right|}^{2\eta }+{\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}}{4}\right)+g\left(\frac{{\left|\mathfrak{P}\right|}^{2\eta }+3{\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}}{4}\right)∥\hfill \\ \hfill & \le \frac{1}{2}\gamma ∥g\left({\left|\mathfrak{P}\right|}^{4\eta }\right)+g\left({\left|{\mathfrak{P}}^{*}\right|}^{4\left(1-\eta \right)}\right)∥+\frac{1}{4}\left(1-\gamma \right){∥g\left({\left|\mathfrak{P}\right|}^{2\eta }\right)+g\left({\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}\right)∥}^2\hfill \\ \hfill & \le \frac{1}{2}∥g^2\left({\left|\mathfrak{P}\right|}^{2\eta }\right)+g^2\left({\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}\right)∥\hfill \end{array}$$

(26)

for all $\eta ,\gamma \in \left[0,1\right]$. In a particular case, we have

$$\begin{array}{cc}\hfill g\left({\omega }^2\left(\mathfrak{P}\right)\right)& \le \frac{1}{3}∥g\left(\frac{3{\left|\mathfrak{P}\right|}^{4\eta }+{\left|{\mathfrak{P}}^{*}\right|}^{4\left(1-\eta \right)}}{4}\right)+g\left(\frac{{\left|\mathfrak{P}\right|}^{4\eta }+3{\left|{\mathfrak{P}}^{*}\right|}^{4\left(1-\eta \right)}}{4}\right)∥\hfill \\ \hfill & +\frac{1}{6}g\left(\omega \left(\mathfrak{P}\right)\right)∥g\left(\frac{3{\left|\mathfrak{P}\right|}^{2\eta }+{\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}}{4}\right)+g\left(\frac{{\left|\mathfrak{P}\right|}^{2\eta }+3{\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}}{4}\right)∥\hfill \\ \hfill & \le \frac{1}{3}∥g\left({\left|\mathfrak{P}\right|}^{4\eta }\right)+g\left({\left|{\mathfrak{P}}^{*}\right|}^{4\left(1-\eta \right)}\right)∥+\frac{1}{12}{∥g\left({\left|\mathfrak{P}\right|}^{2\eta }\right)+g\left({\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}\right)∥}^2\hfill \\ \hfill & \le \frac{1}{2}∥g^2\left({\left|\mathfrak{P}\right|}^{2\eta }\right)+g^2\left({\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}\right)∥.\hfill \end{array}$$

(27)

Proof. 

Since g is increasing and convex, then, by Jensen’s inequality, we have

$$\begin{array}{cc}\hfill & g\left({\left|〈\mathfrak{P}u,u〉\right|}^2\right)\hfill \\ \hfill & \le g\left(\gamma 〈{\left|\mathfrak{P}\right|}^{2\eta }u,u〉〈{\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}u,u〉+\left(1-\gamma \right)\left|〈\mathfrak{P}u,u〉\right|\sqrt{〈{\left|\mathfrak{P}\right|}^{2\eta }u,u〉〈{\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}u,u〉}\right)\hfill \\ \hfill & \le \gamma g\left(〈{\left|\mathfrak{P}\right|}^{2\eta }u,u〉〈{\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}u,u〉\right)\hfill \\ \hfill & +\left(1-\gamma \right)g\left(\left|〈\mathfrak{P}u,u〉\right|\sqrt{〈{\left|\mathfrak{P}\right|}^{2\eta }u,u〉〈{\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}u,u〉}\right)\hfill \\ \hfill & \le \gamma g\left({\left[\frac{〈{\left|\mathfrak{P}\right|}^{2\eta }u,u〉+〈{\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}u,u〉}{2}\right]}^2\right)\hfill \\ \hfill & +\left(1-\gamma \right)g\left(\left|〈\mathfrak{P}u,u〉\right|\right)g\left(\sqrt{〈{\left|\mathfrak{P}\right|}^{2\eta }u,u〉〈{\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}u,u〉}\right)\hfill \\ \hfill & \le \gamma g\left(\frac{{〈{\left|\mathfrak{P}\right|}^{2\eta }u,u〉}^2+{〈{\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}u,u〉}^2}{2}\right)\hfill \\ \hfill & +\left(1-\gamma \right)g\left(\left|〈\mathfrak{P}u,u〉\right|\right)g\left(〈\frac{{\left|\mathfrak{P}\right|}^{2\eta }+{\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}}{2}u,u〉\right)\hfill \\ \hfill & =\gamma g\left(\frac{〈{\left|\mathfrak{P}\right|}^{4\eta }u,u〉+〈{\left|{\mathfrak{P}}^{*}\right|}^{4\left(1-\eta \right)}u,u〉}{2}\right)\hfill \\ \hfill & +\left(1-\gamma \right)g\left(\left|〈\mathfrak{P}u,u〉\right|\right)g\left(\frac{〈{\left|\mathfrak{P}\right|}^{2\eta }u,u〉+〈{\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}u,u〉}{2}\right)\hfill \\ \hfill & \le \gamma g\left(\frac{1}{2}\left[\frac{3〈{\left|\mathfrak{P}\right|}^{4\eta }u,u〉+〈{\left|{\mathfrak{P}}^{*}\right|}^{4\left(1-\eta \right)}u,u〉}{4}+\frac{〈{\left|\mathfrak{P}\right|}^{4\eta }u,u〉+3〈{\left|{\mathfrak{P}}^{*}\right|}^{4\left(1-\eta \right)}u,u〉}{4}\right]\right)\hfill \\ \hfill & +\left(1-\gamma \right)g\left(\left|〈\mathfrak{P}u,u〉\right|\right)\hfill \\ \hfill & \times g\left(\frac{1}{2}\left[\frac{3〈{\left|\mathfrak{P}\right|}^{2\eta }u,u〉+〈{\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}u,u〉}{4}+\frac{〈{\left|\mathfrak{P}\right|}^{2\eta }u,u〉+3〈{\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}u,u〉}{4}\right]\right)\hfill \\ \hfill & \le \frac{1}{2}\left[g\left(\frac{3〈{\left|\mathfrak{P}\right|}^{4\eta }u,u〉+〈{\left|{\mathfrak{P}}^{*}\right|}^{4\left(1-\eta \right)}u,u〉}{4}\right)+g\left(\frac{〈{\left|\mathfrak{P}\right|}^{4\eta }u,u〉+3〈{\left|{\mathfrak{P}}^{*}\right|}^{4\left(1-\eta \right)}u,u〉}{4}\right)\right]\hfill \\ \hfill & +\frac{1}{2}\left(1-\gamma \right)g\left(\left|〈\mathfrak{P}u,u〉\right|\right)\left[g\left(\frac{3〈{\left|\mathfrak{P}\right|}^{2\eta }u,u〉+〈{\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}u,u〉}{4}\right)\right.\hfill \\ \hfill & \left.+g\left(\frac{〈{\left|\mathfrak{P}\right|}^{2\eta }u,u〉+3〈{\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}u,u〉}{4}\right)\right].\hfill \end{array}$$

Taking the supremum over all unit vector $u\in \mathcal{G}$ in all previous inequalities, we obtain the first and second inequalities (16).

To obtain the second, third, and fourth inequalities, since g is increasing and operator convex, the inequalities in (16) imply that

$$\begin{array}{cc}\hfill & g\left({\omega }^2\left(\mathfrak{P}\right)\right)\hfill \\ \hfill & \le \frac{1}{2}\gamma ∥g\left(\frac{3{\left|\mathfrak{P}\right|}^{4\eta }+{\left|{\mathfrak{P}}^{*}\right|}^{4\left(1-\eta \right)}}{4}\right)+g\left(\frac{{\left|\mathfrak{P}\right|}^{4\eta }+3{\left|{\mathfrak{P}}^{*}\right|}^{4\left(1-\eta \right)}}{4}\right)∥\hfill \\ \hfill & +\frac{1}{2}\left(1-\gamma \right)g\left(\omega \left(\mathfrak{P}\right)\right)∥g\left(\frac{3{\left|\mathfrak{P}\right|}^{2\eta }+{\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}}{4}\right)+g\left(\frac{{\left|\mathfrak{P}\right|}^{2\eta }+3{\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}}{4}\right)∥\hfill \\ \hfill & \le \frac{1}{2}\gamma ∥g\left(\frac{3{\left|\mathfrak{P}\right|}^{4\eta }+{\left|{\mathfrak{P}}^{*}\right|}^{4\left(1-\eta \right)}}{4}\right)+g\left(\frac{{\left|\mathfrak{P}\right|}^{4\eta }+3{\left|{\mathfrak{P}}^{*}\right|}^{4\left(1-\eta \right)}}{4}\right)∥\hfill \\ \hfill & +\frac{1}{4}\left(1-\gamma \right){∥g\left(\frac{3{\left|\mathfrak{P}\right|}^{2\eta }+{\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}}{4}\right)+g\left(\frac{{\left|\mathfrak{P}\right|}^{2\eta }+3{\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}}{4}\right)∥}^2\hfill \\ \hfill & \le \frac{1}{2}\gamma ∥\frac{3g\left({\left|\mathfrak{P}\right|}^{4\eta }\right)+g\left({\left|{\mathfrak{P}}^{*}\right|}^{4\left(1-\eta \right)}\right)}{4}+\frac{g\left({\left|\mathfrak{P}\right|}^{4\eta }\right)+3g\left({\left|{\mathfrak{P}}^{*}\right|}^{4\left(1-\eta \right)}\right)}{4}∥(g\mathrm{is}\mathrm{convex})\hfill \\ \hfill & +\frac{1}{4}\left(1-\gamma \right){∥\frac{3g\left({\left|\mathfrak{P}\right|}^{2\eta }\right)+g\left({\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}\right)}{4}+\frac{g\left({\left|\mathfrak{P}\right|}^{2\eta }\right)+3g\left({\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}\right)}{4}∥}^2\hfill \\ \hfill & =\frac{1}{2}\gamma ∥g\left({\left|\mathfrak{P}\right|}^{4\eta }\right)+g\left({\left|{\mathfrak{P}}^{*}\right|}^{4\left(1-\eta \right)}\right)∥+\frac{1}{4}\left(1-\gamma \right){∥g\left({\left|\mathfrak{P}\right|}^{2\eta }\right)+g\left({\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}\right)∥}^2\hfill \\ \hfill & \le \frac{1}{2}\gamma ∥g^2\left({\left|\mathfrak{P}\right|}^{2\eta }\right)+g^2\left({\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}\right)∥(g\mathrm{is}\mathrm{submultiplicative})\hfill \\ \hfill & +\left(1-\gamma \right){∥\frac{g\left({\left|\mathfrak{P}\right|}^{2\eta }\right)+g\left({\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}\right)}{2}∥}^2\hfill \\ \hfill & =\frac{1}{2}\gamma ∥g^2\left({\left|\mathfrak{P}\right|}^{2\eta }\right)+g^2\left({\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}\right)∥+\left(1-\gamma \right)∥{\left(\frac{g\left({\left|\mathfrak{P}\right|}^{2\eta }\right)+g\left({\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}\right)}{2}\right)}^2∥\hfill \\ \hfill & \le \frac{1}{2}\gamma ∥g^2\left({\left|\mathfrak{P}\right|}^{2\eta }\right)+g^2\left({\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}\right)∥\hfill \\ \hfill & +\frac{1}{2}\left(1-\gamma \right)∥g^2\left({\left|\mathfrak{P}\right|}^{2\eta }\right)+g^2\left({\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}\right)∥(t^2\mathrm{is}\mathrm{convex})\hfill \\ \hfill & \le \frac{1}{2}∥g^2\left({\left|\mathfrak{P}\right|}^{2\eta }\right)+g^2\left({\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}\right)∥,\hfill \end{array}$$

which proves the second inequality. The inequalities in (27) follows from (26) by setting $\gamma =\frac{1}{3}$. □

The following result refines the chain of inequalities stated in Theorem 3. The new chain of inequalities is tighter and yields a better bound. It improves the accuracy of the numerical radius and yields more accurate predictions.

Corollary 5.

Let $\mathfrak{P}\in \mathcal{J}\left(\mathcal{G}\right)$. If $g:\left[0,\infty \right)\to \left[0,\infty \right)$ is an increasing, submultiplicative and operator convex, then

$$\begin{array}{cc}\hfill g\left({\omega }^2\left(\mathfrak{P}\right)\right)& \le \frac{1}{2}\gamma ∥g\left(\frac{3{\left|\mathfrak{P}\right|}^{4\eta }+{\left|{\mathfrak{P}}^{*}\right|}^{4\left(1-\eta \right)}}{4}\right)+g\left(\frac{{\left|\mathfrak{P}\right|}^{4\eta }+3{\left|{\mathfrak{P}}^{*}\right|}^{4\left(1-\eta \right)}}{4}\right)∥\hfill \\ \hfill & +\frac{1}{2}\left(1-\gamma \right)g\left(\omega \left(\mathfrak{P}\right)\right)∥g\left(\frac{3{\left|\mathfrak{P}\right|}^{2\eta }+{\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}}{4}\right)+g\left(\frac{{\left|\mathfrak{P}\right|}^{2\eta }+3{\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}}{4}\right)∥\hfill \\ \hfill & \le \gamma ∥{\int }_0^{1}g\left(s{\left|\mathfrak{P}\right|}^{4\eta }+\left(1-s\right){\left|{\mathfrak{P}}^{*}\right|}^{4\left(1-\eta \right)}\right)ds∥\hfill \\ \hfill & +\left(1-\gamma \right)g\left(\omega \left(\mathfrak{P}\right)\right)∥{\int }_0^{1}g\left(s{\left|\mathfrak{P}\right|}^{2\eta }+\left(1-s\right){\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}\right)ds∥\hfill \\ \hfill & \le \frac{1}{2}\gamma ∥g\left({\left|\mathfrak{P}\right|}^{4\eta }\right)+g\left({\left|{\mathfrak{P}}^{*}\right|}^{4\left(1-\eta \right)}\right)∥\hfill \\ \hfill & +\frac{1}{2}\left(1-\gamma \right)g\left(\omega \left(\mathfrak{P}\right)\right)∥g\left({\left|\mathfrak{P}\right|}^{2\eta }\right)+g\left({\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}\right)∥\hfill \\ \hfill & \le \frac{1}{2}∥g^2\left({\left|\mathfrak{P}\right|}^{2\eta }\right)+g^2\left({\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}\right)∥\hfill \end{array}$$

(28)

for all $\eta ,\gamma \in \left[0,1\right]$.

Proof. 

The first two inequalities follow from the proof of Theorem 3. The last two inequalities follow from Theorem 1. □

Example 2.

Consider $\mathfrak{P}=\left[\begin{array}{cc}0& 2\\ 1& 0\end{array}\right]$. It is easy to observe that $\omega \left(\mathfrak{P}\right)=1.5$. Applying the inequalities in (28) with $\eta =\frac{1}{2}$ and $\gamma =\frac{1}{3}$ for the increasing, submultiplicative, and operator convex function $g\left(t\right)=t^2$$(t\ge 0)$, we obtain

$$\begin{array}{cc}\hfill 5.0625={\omega }^4\left(\mathfrak{P}\right)& \lneqq \frac{1}{32}\gamma ∥{\left(3{\left|\mathfrak{P}\right|}^2+{\left|{\mathfrak{P}}^{*}\right|}^2\right)}^2+{\left({\left|\mathfrak{P}\right|}^2+3{\left|{\mathfrak{P}}^{*}\right|}^2\right)}^2∥\hfill \\ \hfill & +\frac{1}{32}\left(1-\gamma \right){\omega }^2\left(\mathfrak{P}\right)∥{\left(3\left|\mathfrak{P}\right|+\left|{\mathfrak{P}}^{*}\right|\right)}^2+{\left(\left|\mathfrak{P}\right|+3\left|{\mathfrak{P}}^{*}\right|\right)}^2∥\hfill \\ \hfill & =5.7395\hfill \\ \hfill & \lneqq \gamma ∥{\int }_0^1{\left(s{\left|\mathfrak{P}\right|}^2+\left(1-s\right){\left|{\mathfrak{P}}^{*}\right|}^2\right)}^{2}ds∥\hfill \\ \hfill & +\left(1-\gamma \right){\omega }^2\left(\mathfrak{P}\right)∥{\int }_0^1{\left(s\left|\mathfrak{P}\right|+\left(1-s\right)\left|{\mathfrak{P}}^{*}\right|\right)}^{2}ds∥\hfill \\ \hfill & =5.8333\hfill \\ \hfill & \lneqq \frac{1}{2}\gamma ∥{\left|\mathfrak{P}\right|}^4+{\left|{\mathfrak{P}}^{*}\right|}^4∥+\frac{1}{2}\left(1-\gamma \right){\omega }^2\left(\mathfrak{P}\right)∥{\left|\mathfrak{P}\right|}^2+{\left|{\mathfrak{P}}^{*}\right|}^2∥\hfill \\ \hfill & =6.5833\hfill \\ \hfill & \lneqq \frac{1}{2}∥{\left|\mathfrak{P}\right|}^4+{\left|{\mathfrak{P}}^{*}\right|}^4∥\hfill \\ \hfill & =8.5,\hfill \end{array}$$

which shows that (28) is a non-trivial refinement of (5) and thus (6). Indeed, this example shows that the main results in [1,2] are refined and improved.

In the following two results, we prove two nontrivial generalized extensions of the numerical radius inequalities for Hilbert space operators. In the next result, a refined generalization of (4) and (7) could be extended as in the following result.

Theorem 4.

Let $\mathfrak{P},\mathfrak{Q}\in \mathcal{J}\left(\mathcal{G}\right)$. If g is an increasing, submultiplicative and operator convex function, then

$$\begin{array}{cc}\hfill g\left({\omega }^2\left({\mathfrak{Q}}^{*}\mathfrak{P}\right)\right)& \le \gamma ∥{\int }_0^{1}g\left(\left(1-s\right){\left|\mathfrak{P}\right|}^2+s{\left|\mathfrak{Q}\right|}^2\right)ds∥\hfill \\ \hfill & +\left(1-\gamma \right)g\left(\omega \left(S^{*}\mathfrak{P}\right)\right)∥{\int }_0^{1}g\left(\left(1-s\right){\left|\mathfrak{P}\right|}^2+s{\left|\mathfrak{Q}\right|}^2\right)ds∥\hfill \\ \hfill & \le \frac{1}{2}\gamma ∥g\left({\left|\mathfrak{P}\right|}^4\right)+g\left({\left|\mathfrak{Q}\right|}^4\right)∥+\frac{1}{2}\left(1-\gamma \right)g\left(\omega \left({\mathfrak{Q}}^{*}\mathfrak{P}\right)\right)∥g\left({\left|\mathfrak{P}\right|}^2\right)+g\left({\left|\mathfrak{Q}\right|}^2\right)∥\hfill \end{array}$$

(29)

for all $\gamma \in \left[0,1\right]$.

Proof. 

Let $u\in \mathcal{G}$ be a unit vector since g is increasing; then, by the refined Cauchy–Schwarz inequality (16), we have

$$\begin{array}{cc}\hfill & g\left({\left|〈\mathfrak{P}u,\mathfrak{Q}u〉\right|}^2\right)\hfill \\ \hfill & \le g\left(\gamma 〈{\left|\mathfrak{P}\right|}^{2}u,u〉〈{\left|\mathfrak{Q}\right|}^{2}u,u〉+\left(1-\gamma \right)\left|〈\mathfrak{P}u,\mathfrak{Q}u〉\right|\sqrt{〈{\left|\mathfrak{P}\right|}^{2}u,u〉〈{\left|\mathfrak{Q}\right|}^{2}u,u〉}\right)\hfill \\ \hfill & \le \gamma g\left(〈{\left|\mathfrak{P}\right|}^{2}u,u〉〈{\left|\mathfrak{Q}\right|}^{2}u,u〉\right)\hfill \\ \hfill & +\left(1-\gamma \right)g\left(\left|〈\mathfrak{P}u,\mathfrak{Q}u〉\right|\sqrt{〈{\left|\mathfrak{P}\right|}^{2}u,u〉〈{\left|\mathfrak{Q}\right|}^{2}u,u〉}\right)\left(g\mathrm{is}\mathrm{convex}\right)\hfill \\ \hfill & \le \gamma g\left({\left[\frac{〈{\left|\mathfrak{P}\right|}^{2}u,u〉+〈{\left|\mathfrak{Q}\right|}^{2}u,u〉}{2}\right]}^2\right)\left(\mathrm{By}\mathrm{AM}–\mathrm{GM}\mathrm{inequality}\right)\hfill \\ \hfill & +\left(1-\gamma \right)g\left(\left|〈\mathfrak{P}u,\mathfrak{Q}u〉\right|\right)g\left(\sqrt{〈{\left|\mathfrak{P}\right|}^{2}u,u〉〈{\left|\mathfrak{Q}\right|}^{2}u,u〉}\right)\left(g\mathrm{is}\mathrm{submultiplicative}\right)\hfill \\ \hfill & \le \gamma g\left(\frac{{〈{\left|\mathfrak{P}\right|}^{2}u,u〉}^2+{〈{\left|\mathfrak{Q}\right|}^{2}u,u〉}^2}{2}\right)\hfill \\ \hfill & +\left(1-\gamma \right)g\left(\left|〈\mathfrak{P}u,\mathfrak{Q}u〉\right|\right)g\left(〈\frac{{\left|\mathfrak{P}\right|}^2+{\left|\mathfrak{Q}\right|}^2}{2}u,u〉\right)\left(\mathrm{By}\mathrm{AM}–\mathrm{GM}\mathrm{inequality}\right)\hfill \\ \hfill & \le \gamma g\left(\frac{〈{\left|\mathfrak{P}\right|}^{4}u,u〉+〈{\left|\mathfrak{Q}\right|}^{4}u,u〉}{2}\right)\left(\mathrm{By}\right(12\left)\right)\hfill \\ \hfill & +\left(1-\gamma \right)g\left(\left|〈\mathfrak{P}u,\mathfrak{Q}u〉\right|\right)g\left(〈\frac{{\left|\mathfrak{P}\right|}^2+{\left|\mathfrak{Q}\right|}^2}{2}u,u〉\right)\hfill \\ \hfill & \le \gamma {\int }_0^{1}g\left(〈\left(\left(1-s\right){\left|\mathfrak{P}\right|}^4+s{\left|\mathfrak{Q}\right|}^4\right)u,u〉\right)ds\left(\mathrm{By}\mathrm{the}\mathrm{Hermite}–\mathrm{Hadamard}\mathrm{inequality}\right)\hfill \\ \hfill & +\left(1-\gamma \right)g\left(\left|〈\mathfrak{P}u,\mathfrak{Q}u〉\right|\right){\int }_0^{1}g\left(〈\left(\left(1-s\right){\left|\mathfrak{P}\right|}^2+s{\left|\mathfrak{Q}\right|}^2\right)u,u〉\right)ds.\hfill \end{array}$$

(30)

On the other hand,

$$\begin{array}{cc}\hfill & {\int }_0^{1}g\left(〈\left(\left(1-s\right){\left|\mathfrak{P}\right|}^2+s{\left|\mathfrak{Q}\right|}^2\right)u,u〉\right)ds\hfill \\ \hfill & \le {\int }_0^1〈g\left(\left(\left(1-s\right){\left|\mathfrak{P}\right|}^2+s{\left|\mathfrak{Q}\right|}^2\right)\right)u,u〉ds\left(\mathrm{By}\right(14\left)\right)\hfill \\ \hfill & \le {\int }_0^1〈\left(\left(1-s\right)g\left({\left|\mathfrak{P}\right|}^2\right)+sg\left({\left|\mathfrak{Q}\right|}^2\right)\right)u,u〉ds\left(g\mathrm{is}\mathrm{operator}\mathrm{convex}\right)\hfill \\ \hfill & =〈\left({\int }_0^1\left(\left(1-s\right)g\left({\left|\mathfrak{P}\right|}^2\right)+sg\left({\left|\mathfrak{Q}\right|}^2\right)\right)ds\right)u,u〉\hfill \\ \hfill & =〈\frac{g\left({\left|\mathfrak{P}\right|}^2\right)+g\left({\left|\mathfrak{Q}\right|}^2\right)}{2}u,u〉.\hfill \end{array}$$

(31)

Also, and similarly, we have

$$\begin{array}{c}\hfill {\int }_0^{1}g\left(〈\left(\left(1-s\right){\left|\mathfrak{P}\right|}^4+s{\left|\mathfrak{Q}\right|}^4\right)u,u〉\right)ds\le 〈\frac{g\left({\left|\mathfrak{P}\right|}^4\right)+g\left({\left|\mathfrak{Q}\right|}^4\right)}{2}u,u〉.\end{array}$$

(32)

Combining inequalities (31) and (32) with the last inequality in (30) and taking the supremum over all unit vector $u\in \mathcal{G}$ in all previous inequalities, we obtain the required result. □

The following result generalizes and extends the second inequality in (29).

Theorem 5.

Let $\mathfrak{P},\mathfrak{Q}\in \mathcal{J}\left(\mathcal{G}\right)$. If $g:\left[0,\infty \right)\to \left[0,\infty \right)$ is an increasing and operator convex, then

$$\begin{array}{cc}\hfill & g\left({\omega }^2\left({\mathfrak{Q}}^{*}\mathfrak{P}\right)\right)\hfill \\ \hfill & \le \frac{1}{4}\gamma {∥g\left({\left|\mathfrak{P}\right|}^2\right)+g\left({\left|\mathfrak{Q}\right|}^2\right)∥}^2+\frac{1}{2}\left(1-\gamma \right)g\left(\omega \left({\mathfrak{Q}}^{*}\mathfrak{P}\right)\right)∥g\left({\left|\mathfrak{P}\right|}^2\right)+g\left({\left|\mathfrak{Q}\right|}^2\right)∥\hfill \\ \hfill & \le \frac{1}{2}\gamma ∥g^2\left({\left|\mathfrak{P}\right|}^2\right)+g^2\left({\left|\mathfrak{Q}\right|}^2\right)∥+\frac{1}{2}\left(1-\gamma \right)g\left(\omega \left({\mathfrak{Q}}^{*}\mathfrak{P}\right)\right)∥g\left({\left|\mathfrak{P}\right|}^2\right)+g\left({\left|\mathfrak{Q}\right|}^2\right)∥\hfill \\ \hfill & \le \frac{1}{2}∥g^2\left({\left|\mathfrak{P}\right|}^2\right)+g^2\left({\left|\mathfrak{Q}\right|}^2\right)∥\hfill \end{array}$$

(33)

for all $\eta ,\gamma \in \left[0,1\right]$.

Proof. 

Employing (7) with $r=1$, and since g is increasing, we have

$$\begin{array}{cc}\hfill g\left({\omega }^2\left({\mathfrak{Q}}^{*}\mathfrak{P}\right)\right)& \le g\left(\frac{1}{4}\gamma {∥{\left|\mathfrak{P}\right|}^2+{\left|\mathfrak{Q}\right|}^2∥}^2+\frac{1}{2}\left(1-\gamma \right)\omega \left({\mathfrak{Q}}^{*}\mathfrak{P}\right)∥{\left|\mathfrak{P}\right|}^2+{\left|\mathfrak{Q}\right|}^2∥\right)\hfill \\ \hfill & =g\left(\gamma {∥\frac{{\left|\mathfrak{P}\right|}^2+{\left|\mathfrak{Q}\right|}^2}{2}∥}^2+\left(1-\gamma \right)\omega \left({\mathfrak{Q}}^{*}\mathfrak{P}\right)∥\frac{{\left|\mathfrak{P}\right|}^2+{\left|\mathfrak{Q}\right|}^2}{2}∥\right)\hfill \\ \hfill & \le \gamma g\left({∥\frac{{\left|\mathfrak{P}\right|}^2+{\left|\mathfrak{Q}\right|}^2}{2}∥}^2\right)+\left(1-\gamma \right)g\left(\omega \left({\mathfrak{Q}}^{*}\mathfrak{P}\right)\right)g\left(∥\frac{{\left|\mathfrak{P}\right|}^2+{\left|\mathfrak{Q}\right|}^2}{2}∥\right)\hfill \\ \hfill & \le \gamma {∥g\left(\frac{{\left|\mathfrak{P}\right|}^2+{\left|\mathfrak{Q}\right|}^2}{2}\right)∥}^2+\left(1-\gamma \right)g\left(\omega \left({\mathfrak{Q}}^{*}\mathfrak{P}\right)\right)∥g\left(\frac{{\left|\mathfrak{P}\right|}^2+{\left|\mathfrak{Q}\right|}^2}{2}\right)∥\hfill \\ \hfill & \le \gamma {∥\frac{g\left({\left|\mathfrak{P}\right|}^2\right)+g\left({\left|\mathfrak{Q}\right|}^2\right)}{2}∥}^2+\left(1-\gamma \right)g\left(\omega \left({\mathfrak{Q}}^{*}\mathfrak{P}\right)\right)∥\frac{g\left({\left|\mathfrak{P}\right|}^2\right)+g\left({\left|\mathfrak{Q}\right|}^2\right)}{2}∥\hfill \\ \hfill & \le \frac{1}{4}\gamma {∥g\left({\left|\mathfrak{P}\right|}^2\right)+g\left({\left|\mathfrak{Q}\right|}^2\right)∥}^2+\frac{1}{2}\left(1-\gamma \right)g\left(\omega \left({\mathfrak{Q}}^{*}\mathfrak{P}\right)\right)∥g\left({\left|\mathfrak{P}\right|}^2\right)+g\left({\left|\mathfrak{Q}\right|}^2\right)∥\hfill \\ \hfill & \le \frac{1}{4}\gamma {∥g\left({\left|\mathfrak{P}\right|}^2\right)+g\left({\left|\mathfrak{Q}\right|}^2\right)∥}^2+\frac{1}{4}\left(1-\gamma \right){∥g\left({\left|\mathfrak{P}\right|}^2\right)+g\left({\left|\mathfrak{Q}\right|}^2\right)∥}^2\hfill \\ \hfill & ={∥\frac{g\left({\left|\mathfrak{P}\right|}^2\right)+g\left({\left|\mathfrak{Q}\right|}^2\right)}{2}∥}^2\hfill \\ \hfill & =∥{\left(\frac{g\left({\left|\mathfrak{P}\right|}^2\right)+g\left({\left|\mathfrak{Q}\right|}^2\right)}{2}\right)}^2∥\hfill \\ \hfill & \le \frac{1}{2}∥g^2\left({\left|\mathfrak{P}\right|}^2\right)+g^2\left({\left|\mathfrak{Q}\right|}^2\right)∥,\hfill \end{array}$$

and this proves the desired result. □

Remark 2.

Further results concerning the numerical radius inequalities for the product of two Hilbert space operators could be deduced from Theorems 1–3 and Corollaries 1–5. For instance, setting $\eta =\frac{1}{2}$ and replacing ${\mathfrak{P}}^{*}$ by $\mathfrak{Q}$ in Theorem 1, we obtain

$$\begin{array}{cc}\hfill g\left({\omega }^2\left({\mathfrak{Q}}^{*}\mathfrak{P}\right)\right)& \le \gamma ∥{\int }_0^{1}g\left(s{\left|\mathfrak{P}\right|}^2+\left(1-s\right){\left|\mathfrak{Q}\right|}^2\right)ds∥\hfill \\ \hfill & +\left(1-\gamma \right)g\left(\omega \left(\mathfrak{P}\right)\right)∥{\int }_0^{1}g\left(s\left|\mathfrak{P}\right|+\left(1-s\right)\left|\mathfrak{Q}\right|\right)ds∥\hfill \\ \hfill & \le \frac{1}{2}\gamma ∥g\left({\left|\mathfrak{P}\right|}^2\right)+g\left({\left|\mathfrak{Q}\right|}^2\right)∥\hfill \\ \hfill & +\frac{1}{2}\left(1-\gamma \right)g\left(\omega \left(\mathfrak{P}\right)\right)∥g\left(\left|\mathfrak{P}\right|\right)+g\left(\left|\mathfrak{Q}\right|\right)∥\hfill \\ \hfill & \le \frac{1}{2}∥g^2\left(\left|\mathfrak{P}\right|\right)+g^2\left(\left|\mathfrak{Q}\right|\right)∥\hfill \end{array}$$

for all $\gamma \in \left[0,1\right]$. We leave further construction and the details to the interested reader.

3. Conclusions

This work provides several generalized extensions of some recent numerical radius inequalities of Hilbert space operators. The presented inequalities refine some recently proven inequalities in [1,2]. It has already been shown that concatenating certain inequalities can improve or restore some. By extending the existing numerical radius inequalities, the paper aims to provide a unified framework for numerical radius inequalities. We also present a numerical example to demonstrate the effectiveness of the proposed method. Our method combines the existing inequalities proven in [1,2] into a single inequality that can be used to obtain improved or restored results. This unified approach allows us extending the existing numerical radius inequalities and showing their effectiveness through numerical examples. Namely, in light of Sababheh–Moradi inequality (9), this work discusses several new extensions and generalizations of inequalities (5). Namely, it is shown that

$$\begin{array}{cc}\hfill g\left({\omega }^2\left(\mathfrak{P}\right)\right)& \le \frac{1}{2}\gamma ∥g\left(\frac{3{\left|\mathfrak{P}\right|}^{4\eta }+{\left|{\mathfrak{P}}^{*}\right|}^{4\left(1-\eta \right)}}{4}\right)+g\left(\frac{{\left|\mathfrak{P}\right|}^{4\eta }+3{\left|{\mathfrak{P}}^{*}\right|}^{4\left(1-\eta \right)}}{4}\right)∥\hfill \\ \hfill & +\frac{1}{2}\left(1-\gamma \right)g\left(\omega \left(\mathfrak{P}\right)\right)∥g\left(\frac{3{\left|\mathfrak{P}\right|}^{2\eta }+{\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}}{4}\right)+g\left(\frac{{\left|\mathfrak{P}\right|}^{2\eta }+3{\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}}{4}\right)∥\hfill \\ \hfill & \le \gamma ∥{\int }_0^{1}g\left(s{\left|\mathfrak{P}\right|}^{4\eta }+\left(1-s\right){\left|{\mathfrak{P}}^{*}\right|}^{4\left(1-\eta \right)}\right)ds∥\hfill \\ \hfill & +\left(1-\gamma \right)g\left(\omega \left(\mathfrak{P}\right)\right)∥{\int }_0^{1}g\left(s{\left|\mathfrak{P}\right|}^{2\eta }+\left(1-s\right){\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}\right)ds∥\hfill \\ \hfill & \le \frac{1}{2}\gamma ∥g\left({\left|\mathfrak{P}\right|}^{4\eta }\right)+g\left({\left|{\mathfrak{P}}^{*}\right|}^{4\left(1-\eta \right)}\right)∥\hfill \\ \hfill & +\frac{1}{2}\left(1-\gamma \right)g\left(\omega \left(\mathfrak{P}\right)\right)∥g\left({\left|\mathfrak{P}\right|}^{2\eta }\right)+g\left({\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}\right)∥\hfill \\ \hfill & \le \frac{1}{2}∥g^2\left({\left|\mathfrak{P}\right|}^{2\eta }\right)+g^2\left({\left|{\mathfrak{P}}^{*}\right|}^{2\left(1-\eta \right)}\right)∥\hfill \end{array}$$

for all $\eta ,\gamma \in \left[0,1\right]$ and $\mathfrak{P}\in \mathcal{J}\left(\mathcal{G}\right)$, such that $g:\left[0,\infty \right)\to \left[0,\infty \right)$ is an increasing, submultiplicative and operator convex. Similar inequalities for the numerical radius of a product of two Hilbert space operators are established.