Fractional Multiplicative Bullen-Type Inequalities for Multiplicative Differentiable Functions

Abstract

Various scholars have lately employed a wide range of strategies to resolve specific types of symmetrical fractional differential equations. In this paper, we propose a new fractional identity for multiplicatively differentiable functions; based on this identity, we establish some new fractional multiplicative Bullen-type inequalities for multiplicative differentiable convex functions. Some applications of the obtained results are given.

1. Introduction

Fractional calculus has been one of the main axes of mathematical analysis over the last three decades, both theoretically and in terms of practical applications. Basically, this theory, together with the qualitative theory of fractional differential and fractional integro-differential equations, their numerical simulations, and their symmetry represent a tool of mathematical analysis used to study integrals and derivatives of an arbitrary order, which unifies and generalizes the notions traditional ways of differentiation and integration. Fractional-order nonlinear operators are more convenient than classical formulations. Many scientific disciplines, including fluid mechanics, signal processing, and entropy theory, may involve the qualitative theory of fractional differential equations, fractional integro-differential equations, and fractional order operators. For this reason, the applications of the aforementioned fractional calculus theory and qualitative theory of equations have attracted the attention of scholars around the world.

Integrals play an essential and vital role in certain pure and applied fields combined with convexity, and their generalizations have been the subject of permanent research. As consequence, several papers dealing with convex inequalities in different types of computation have been published (in classical computation, see [1,2,3,4,5,6], for fractional computation [7,8,9,10] and for non-Newtonian calculus [11,12,13,14,15,16]).

Recently, Abdeljawad and Grossman [17] introduced the multiplicative Riemann–Liouville fractional integrals as follows:

Definition 1.

The multiplicative Riemann–Liouville fractional integrals of order $\alpha \in \mathbb{C}$, $\mathrm{Re}\left(\alpha \right)>0$, respectively, are defined by

$$\left({}_f{I}_{*}^{\alpha }\rho \right)\left(u\right)=e^{\left(J_{f^{+}}^{\alpha }\left(ln \circ \rho \right)\right)\left(u\right)}$$

and

$$\left({}_{*}{I}_k^{\alpha }\rho \right)\left(u\right)=e^{\left(J_{k^{-}}^{\alpha }\left(ln \circ \rho \right)\right)\left(u\right)},$$

where $J_{l^{+}}^{\alpha }$ and $J_{k^{-}}^{\alpha }$ represents the left and the right Riemann–Liouville fractional integral:

$$\left(J_{f^{+}}^{\alpha }\rho \right)\left(u\right)=\frac{1}{\Gamma \left(\alpha \right)}\stackrel{u}{\underset{f}{\int }}{\left(u-\varsigma \right)}^{\alpha -1}\rho \left(\varsigma \right)d\varsigma ,f<u$$

and

$$\left(J_{k^{-}}^{\alpha }\rho \right)\left(u\right)=\frac{1}{\Gamma \left(\alpha \right)}\stackrel{k}{\underset{u}{\int }}{\left(\varsigma -u\right)}^{\alpha -1}\rho \left(\varsigma \right)d\varsigma ,u<k.$$

Budak and Özçelik [18] used the above operator and established some Hermite–Hadamard-type inequalities for multiplicatively fractional integrals. Regarding some papers related to the applications concerning the non-Newtonian calculus, one can see [19,20,21,22,23,24,25,26,27,28].

The following inequality is known as the Bullen inequality:

$$\stackrel{k}{\underset{l}{\int }}\rho \left(x\right)dx\le \frac{1}{2}\left[f\left(\frac{l+k}{2}\right)+\frac{f\left(l\right)+f\left(k\right)}{2}\right].$$

(1)

In this paper, we propose the fractional multiplicative analogue of inequality (1). For this, we first prove a new fractional identity for multiplicative differentiable functions. Based on this equality, we provide some fractional Bullen-type inequalities for multiplicatively convex functions. We also give some examples of applications of the obtained results to analytical inequalities.

The non-Newtonian calculus was first presented by Grossman and Katz, where they created and studied the first non-Newtonian calculation system, called geometric calculation. Next, they created an infinite family of non-Newtonian calculi, thus modifying the classical calculus introduced by Newton and Leibniz in the 17th century, each of which differed markedly from the classical calculus of Newton and Leibniz known today as the non-Newtonian calculus or the multiplicative calculus, where the ordinary product and ratio are used, respectively, as the sum and exponential difference over the domain of positive real numbers see [29]. This calculation is useful for dealing with exponentially varying functions.

It is worth noting that the complete mathematical description of multiplicative calculus was given by Bashirov et al. [30]. The reader can also refer to the recent monographs [31,32].

Definition 2.

([30]). Let $\rho :\mathbb{R}\to {\mathbb{R}}^{+}$ be a positive function. The multiplicative derivative ${\rho }^{*}$ of the function ρ is defined as follows:

$$\frac{d^{*}\rho }{dt}={\rho }^{*}\left(\varsigma \right)=\underset{h\to 0}{lim}{\left(\frac{\rho \left(\varsigma +h\right)}{\rho \left(\varsigma \right)}\right)}^{\frac{1}{h}}.$$

Remark 1.

The relation between the multiplicative derivative ${\rho }^{*}$ and the ordinary derivative ${\rho }^{\prime }$ is as follows:

$${\rho }^{*}\left(\varsigma \right)=e^{{\left(ln\rho \left(\varsigma \right)\right)}^{\prime }}=e^{\frac{{\rho }^{\prime }\left(\varsigma \right)}{\rho \left(\varsigma \right)}}.$$

The multiplicative derivative admits the following properties:

Theorem 1

([30]). Let c be an arbitrary constant, and let ρ and ϑ be two multiplicatively differentiable functions. Then, functions $c\rho ,\rho \vartheta ,\rho +\vartheta ,\rho /\vartheta$ and ${\rho }^{\vartheta }$ are ${}^{*}$ differentiable, and they satisfy

• ${\left(c\rho \right)}^{*}\left(\varsigma \right)={\rho }^{*}\left(\varsigma \right),$
• ${\left(\rho \vartheta \right)}^{*}\left(\varsigma \right)={\rho }^{*}\left(\varsigma \right){\vartheta }^{*}\left(\varsigma \right),$
• ${\left(\rho +\vartheta \right)}^{*}\left(\varsigma \right)={\rho }^{*}{\left(\varsigma \right)}^{\frac{\rho \left(\varsigma \right)}{\rho \left(\varsigma \right)+\vartheta \left(\varsigma \right)}}{\vartheta }^{*}{\left(\varsigma \right)}^{\frac{\vartheta \left(\varsigma \right)}{\rho \left(\varsigma \right)+\vartheta \left(\varsigma \right)}},$
• ${\left(\frac{\rho }{\vartheta }\right)}^{*}\left(\varsigma \right)=\frac{{\rho }^{*}\left(\varsigma \right)}{{\vartheta }^{*}\left(\varsigma \right)},$
• ${\left({\rho }^{\vartheta }\right)}^{*}\left(\varsigma \right)={\rho }^{*}{\left(\varsigma \right)}^{\vartheta \left(\varsigma \right)}\rho {\left(\varsigma \right)}^{{\vartheta }^{\prime }\left(\varsigma \right)}.$

In [30], Bashirov et al. introduced the concept of the ${}^{*}$ integral called the multiplicative integral, which is written as $\underset{l}{\stackrel{k}{\int }}{\left(\rho \left(\varsigma \right)\right)}^{d\varsigma }$. It is clear that the sum in the classical Riemann integral of $\rho$ over $[l,k]$, is replaced in the multiplicative integral of $\rho$ over $[l,k]$ by the product. However, the product is represented by the raising to power.

The relationship between the Riemann integral and the multiplicative integral is as follows:

Proposition 1

([30]). If ρ is a Riemann integrable on $[l,k]$, then ρ is a multiplicative integrable on $[l,k]$ and

$$\stackrel{k}{\underset{l}{\int }}{\left(\rho \left(\varsigma \right)\right)}^{d\varsigma }=exp\left\{\stackrel{k}{\underset{l}{\int }}ln\left(\rho \left(\varsigma \right)\right)d\varsigma \right\}.$$

Some properties of the multiplicative integral are given by the following theorem.

Theorem 2

([30]). Let ρ be a positive and a Riemann integrable on $[l,k]$; then, ρ is a multiplicative integrable on $[l,k]$ and

• $\underset{l}{\stackrel{k}{\int }}{\left({\left(\rho \left(\varsigma \right)\right)}^p\right)}^{d\varsigma }={\left(\underset{l}{\stackrel{k}{\int }}{\left(\rho \left(\varsigma \right)\right)}^{d\varsigma }\right)}^p,$
• $\underset{l}{\stackrel{k}{\int }}{\left(\rho \left(\varsigma \right)\vartheta \left(\varsigma \right)\right)}^{d\varsigma }=\underset{l}{\stackrel{k}{\int }}{\left(\rho \left(\varsigma \right)\right)}^{d\varsigma }\underset{l}{\stackrel{k}{\int }}{\left(\vartheta \left(\varsigma \right)\right)}^{d\varsigma },$
• $\underset{l}{\stackrel{k}{\int }}{\left(\frac{\rho \left(\varsigma \right)}{\vartheta \left(\varsigma \right)}\right)}^{d\varsigma }=\frac{\underset{l}{\stackrel{k}{\int }}{\left(\rho \left(\varsigma \right)\right)}^{d\varsigma }}{\underset{l}{\stackrel{k}{\int }}{\left(\vartheta \left(\varsigma \right)\right)}^{d\varsigma }},$
• $\underset{l}{\stackrel{k}{\int }}{\left(\rho \left(\varsigma \right)\right)}^{d\varsigma }=\underset{l}{\stackrel{c}{\int }}{\left(\rho \left(\varsigma \right)\right)}^{d\varsigma }\underset{c}{\stackrel{k}{\int }}{\left(\rho \left(\varsigma \right)\right)}^{d\varsigma },l<c<k,$
• $\underset{l}{\stackrel{l}{\int }}{\left(\rho \left(\varsigma \right)\right)}^{d\varsigma }=1$ and $\underset{l}{\stackrel{k}{\int }}{\left(\rho \left(\varsigma \right)\right)}^{d\varsigma }={\left(\underset{k}{\stackrel{l}{\int }}{\left(\rho \left(\varsigma \right)\right)}^{d\varsigma }\right)}^{-1}.$

Theorem 3

([30], multiplicative integration by parts). Let $\rho :[l,k]\to \mathbb{R}$ be multiplicative differentiable, let $\vartheta :[l,k]\to \mathbb{R}$ be a differentiable so the function ${\rho }^{\vartheta }$ is a multiplicative integrable, and

$$\stackrel{k}{\underset{l}{\int }}{\left({\rho }^{*}{\left(\varsigma \right)}^{\vartheta \left(\varsigma \right)}\right)}^{d\varsigma }=\frac{\rho {\left(k\right)}^{\vartheta \left(k\right)}}{\rho {\left(l\right)}^{\vartheta \left(l\right)}}\times \frac{1}{\underset{l}{\stackrel{k}{\int }}{\left(\rho {\left(\varsigma \right)}^{{\vartheta }^{\prime }\left(\varsigma \right)}\right)}^{d\varsigma }}.$$

Lemma 1.

([33]). Let $\rho :[l,k]\to \mathbb{R}$ be multiplicative differentiable, let $h:[l,k]\to \mathbb{R}$, and let $\vartheta :J\subset \mathbb{R}\to \mathbb{R}$ be two differentiable functions. Then, we have

$$\stackrel{k}{\underset{l}{\int }}{\left({\rho }^{*}{\left(h\left(\varsigma \right)\right)}^{h^{\prime }\left(\varsigma \right)\vartheta \left(\varsigma \right)}\right)}^{d\varsigma }=\frac{\rho {\left(h\left(k\right)\right)}^{\vartheta \left(k\right)}}{\rho {\left(h\left(l\right)\right)}^{\vartheta \left(l\right)}}\times \frac{1}{\underset{l}{\stackrel{k}{\int }}{\left(\rho {\left(h\left(\varsigma \right)\right)}^{{\vartheta }^{\prime }\left(\varsigma \right)}\right)}^{d\varsigma }}.$$

Definition 3.

([33]). A function $\rho :I\to \left[0,+\infty \right)$ is said to be multiplicatively convex or log-convex, if

$$\rho \left(\varsigma x+\left(1-\varsigma \right)y\right)\le {\left[\rho \left(x\right)\right]}^{\varsigma }{\left[\rho \left(y\right)\right]}^{1-\varsigma }$$

holds for all $x,y\in I$ and all $\varsigma \in [0,1]$.

2. Main Results

In order to prove our results, we need the following lemma.

Lemma 2.

Let $\rho :$$\left[l,k\right]$ $\to {\mathbb{R}}^{+}$ be a multiplicative differentiable mapping on $\left[l,k\right]$ with $l<k$. If ${\rho }^{*}$ is a multiplicative integrable on $[l,k]$, then we have the following identity for multiplicative integrals:

$$\begin{array}{cc}\hfill & {\left(\left(\rho \left(l\right)\right){\left(\rho \left(\frac{l+k}{2}\right)\right)}^2\left(\rho \left(k\right)\right)\right)}^{\frac{1}{4}}{\left(\left({}_l{I}_{*}^{\alpha }\rho \right)\left(\frac{l+k}{2}\right)\left({}_{*}{I}_{\frac{l+k}{2}}^{\alpha }\rho \right)\left(k\right)\right)}^{-\frac{2^{\alpha -1}\Gamma \left(\alpha +1\right)}{{\left(k-l\right)}^{\alpha }}}\hfill \\ \hfill =& {\left(\underset{0}{\stackrel{1}{\int }}{\left({\rho }^{*}{\left(\left(1-\varsigma \right)l+\varsigma \frac{l+k}{2}\right)}^{{\varsigma }^{\alpha }-\frac{1}{2}}\right)}^{d\varsigma }\right)}^{\frac{k-l}{4}}\hfill \\ \hfill & \times {\left(\underset{0}{\stackrel{1}{\int }}{\left({\rho }^{*}{\left(\left(1-\varsigma \right)\frac{l+k}{2}+\varsigma k\right)}^{\frac{1}{2}-{\left(1-\varsigma \right)}^{\alpha }}\right)}^{d\varsigma }\right)}^{\frac{k-l}{4}}.\hfill \end{array}$$

Proof. 

Let

$$I_1={\left(\underset{0}{\stackrel{1}{\int }}{\left({\rho }^{*}{\left(\left(1-\varsigma \right)l+\varsigma \frac{l+k}{2}\right)}^{{\varsigma }^{\alpha }-\frac{1}{2}}\right)}^{d\varsigma }\right)}^{\frac{k-l}{4}}$$

and

$$I_2={\left(\underset{0}{\stackrel{1}{\int }}{\left({\rho }^{*}{\left(\left(1-\varsigma \right)\frac{l+k}{2}+\varsigma k\right)}^{\frac{1}{2}-{\left(1-\varsigma \right)}^{\alpha }}\right)}^{d\varsigma }\right)}^{\frac{k-l}{4}}.$$

Using the integration by parts for multiplicative integrals, from $I_1$ we have

$$\begin{array}{cc}\hfill I_1=& {\left(\underset{0}{\stackrel{1}{\int }}{\left({\rho }^{*}{\left(\left(1-\varsigma \right)l+\varsigma \frac{l+k}{2}\right)}^{t^{\alpha }-\frac{1}{2}}\right)}^{dt}\right)}^{\frac{k-l}{4}}\hfill \\ \hfill =& \left(\underset{0}{\stackrel{1}{\int }}{\left({\rho }^{*}{\left(\left(1-t\right)l+t\frac{l+k}{2}\right)}^{\frac{k-l}{2}\left(\frac{1}{2}{\varsigma }^{\alpha }-\frac{1}{2}\right)}\right)}^{d\varsigma }\right)\hfill \\ \hfill =& \frac{{\left(\rho \left(\frac{l+k}{2}\right)\right)}^{\frac{1}{4}}}{{\left(\rho \left(l\right)\right)}^{-\frac{1}{4}}}.\frac{1}{\underset{0}{\stackrel{1}{\int }}{\left(\rho {\left(\left(\left(1-\varsigma \right)l+\varsigma \frac{l+k}{2}\right)\right)}^{\frac{\alpha }{2}{\varsigma }^{\alpha -1}}\right)}^{d\varsigma }}\hfill \\ \hfill =& {\left(\rho \left(l\right)\right)}^{\frac{1}{4}}{\left(\rho \left(\frac{l+k}{2}\right)\right)}^{\frac{1}{4}}\frac{1}{exp\left\{\underset{0}{\stackrel{1}{\int }}\frac{\alpha }{2}{\left(\varsigma \right)}^{\alpha -1}ln\left(\rho \left(\left(1-\varsigma \right)l+\varsigma \frac{l+k}{2}\right)\right)d\varsigma \right\}}\hfill \\ \hfill =& {\left(\rho \left(l\right)\right)}^{\frac{1}{4}}{\left(\rho \left(\frac{l+k}{2}\right)\right)}^{\frac{1}{4}}\frac{1}{exp\left\{\frac{2^{\alpha -1}\alpha }{{\left(k-l\right)}^{\alpha }}\underset{0}{\stackrel{1}{\int }}{\left(u-l\right)}^{\alpha -1}ln\left(\rho \left(u\right)\right)du\right\}}\hfill \\ \hfill =& \frac{{\left(\rho \left(l\right)\right)}^{\frac{1}{4}}{\left(\rho \left(\frac{l+k}{2}\right)\right)}^{\frac{1}{4}}}{{\left(exp\left\{\left(\frac{1}{\Gamma \left(\alpha \right)}\stackrel{\frac{l+k}{2}}{\underset{l}{\int }}{\left(u-l\right)}^{\alpha -1}ln\left(\rho \left(u\right)\right)du\right)\right\}\right)}^{\frac{2^{\alpha -1}\Gamma \left(\alpha +1\right)}{{\left(k-l\right)}^{\alpha }}}}\hfill \\ \hfill =& {\left(\rho \left(l\right)\right)}^{\frac{1}{4}}{\left(\rho \left(\frac{l+k}{2}\right)\right)}^{\frac{1}{4}}{\left(\left({}_l{I}_{*}^{\alpha }\rho \right)\left(\frac{l+k}{2}\right)\right)}^{-\frac{2^{\alpha -1}\Gamma \left(\alpha +1\right)}{{\left(k-l\right)}^{\alpha }}}.\hfill \end{array}$$

Similarly, we obtain

$$\begin{array}{cc}\hfill I_2=& {\left(\underset{0}{\stackrel{1}{\int }}{\left({\rho }^{*}{\left(\left(1-\varsigma \right)\frac{l+k}{2}+\varsigma k\right)}^{\frac{1}{2}-{\left(1-\varsigma \right)}^{\alpha }}\right)}^{d\varsigma }\right)}^{\frac{k-l}{4}}\hfill \\ \hfill =& \left(\underset{0}{\stackrel{1}{\int }}{\left({\rho }^{*}{\left(\left(1-\varsigma \right)\frac{l+k}{2}+\varsigma k\right)}^{\frac{k-l}{2}\left(\frac{1}{2}-\frac{1}{2}{\left(1-\varsigma \right)}^{\alpha }\right)}\right)}^{d\varsigma }\right)\hfill \\ \hfill =& \frac{{\left(\rho \left(k\right)\right)}^{\frac{1}{4}}}{{\left(\rho \left(\frac{l+k}{2}\right)\right)}^{-\frac{1}{4}}}.\frac{1}{\underset{0}{\stackrel{1}{\int }}{\left(\rho {\left(\left(1-\varsigma \right)\frac{l+k}{2}+\varsigma k\right)}^{\frac{\alpha }{2}{\left(1-\varsigma \right)}^{\alpha -1}}\right)}^{d\varsigma }}\hfill \\ \hfill =& \rho {\left(\frac{l+k}{2}\right)}^{\frac{1}{4}}\rho {\left(k\right)}^{\frac{1}{4}}.\frac{1}{exp\left\{\frac{\alpha }{2}\underset{0}{\stackrel{1}{\int }}{\left(1-\varsigma \right)}^{\alpha -1}ln\rho \left(\left(1-\varsigma \right)\frac{l+k}{2}+\varsigma k\right)d\varsigma \right\}}\hfill \\ \hfill =& {\left(\rho \left(\frac{l+k}{2}\right)\right)}^{\frac{1}{4}}{\left(\rho \left(k\right)\right)}^{\frac{1}{4}}.\frac{1}{exp\left\{\frac{2^{\alpha -1}}{{\left(k-l\right)}^{\alpha }}\alpha \underset{\frac{l+k}{2}}{\stackrel{k}{\int }}{\left(k-u\right)}^{\alpha -1}ln\rho \left(u\right)du\right\}}\hfill \\ \hfill =& {\left(\rho \left(\frac{l+k}{2}\right)\right)}^{\frac{n}{2\left(n+2\right)}}{\left(\rho \left(k\right)\right)}^{\frac{1}{n+2}}.\frac{1}{exp\left\{\frac{2^{\alpha -1}\Gamma \left(\alpha +1\right)}{{\left(k-l\right)}^{\alpha }}\left(\frac{1}{\Gamma \left(\alpha \right)}\underset{\frac{l+k}{2}}{\stackrel{k}{\int }}{\left(k-u\right)}^{\alpha -1}lnf\left(u\right)du\right)\right\}}\hfill \\ \hfill =& \frac{\rho {\left(\frac{l+k}{2}\right)}^{\frac{1}{4}}\rho {\left(k\right)}^{\frac{1}{4}}}{{\left(exp\left\{\left(\frac{1}{\Gamma \left(\alpha \right)}\underset{\frac{l+k}{2}}{\stackrel{k}{\int }}{\left(k-u\right)}^{\alpha -1}ln\rho \left(u\right)du\right)\right\}\right)}^{\frac{2^{\alpha -1}\Gamma \left(\alpha +1\right)}{{\left(k-l\right)}^{\alpha }}}}.\hfill \\ \hfill =& {\left(\rho \left(\frac{l+k}{2}\right)\right)}^{\frac{1}{4}}{\left(\rho \left(k\right)\right)}^{\frac{1}{4}}.{\left(\left({}_{*}{I}_{\frac{l+k}{2}}^{\alpha }\rho \right)\left(k\right)\right)}^{-\frac{2^{\alpha -1}\Gamma \left(\alpha +1\right)}{{\left(k-l\right)}^{\alpha }}}.\hfill \end{array}$$

Multiplying above equalities, we obtain

$$\begin{array}{cc}\hfill & I_1\times I_2={\left(\rho \left(l\right)\right)}^{\frac{1}{4}}{\left(\rho \left(\frac{l+k}{2}\right)\right)}^{\frac{1}{4}}{\left(\left({}_l{I}_{*}^{\alpha }\rho \right)\left(\frac{l+k}{2}\right)\right)}^{-\frac{2^{\alpha -1}\Gamma \left(\alpha +1\right)}{{\left(k-l\right)}^{\alpha }}}\hfill \\ \hfill & \times {\left(\rho \left(\frac{l+k}{2}\right)\right)}^{\frac{1}{4}}{\left(\rho \left(k\right)\right)}^{\frac{1}{4}}.{\left(\left({}_{*}{I}_{\frac{l+k}{2}}^{\alpha }\rho \right)\left(k\right)\right)}^{-\frac{2^{\alpha -1}\Gamma \left(\alpha +1\right)}{{\left(k-l\right)}^{\alpha }}}\hfill \\ \hfill =& {\left(\left(\rho \left(l\right)\right){\left(\rho \left(\frac{l+k}{2}\right)\right)}^2\left(\rho \left(k\right)\right)\right)}^{\frac{1}{4}}{\left(\left({}_l{I}_{*}^{\alpha }\rho \right)\left(\frac{l+k}{2}\right)\left({}_{*}{I}_{\frac{l+k}{2}}^{\alpha }\rho \right)\left(k\right)\right)}^{-\frac{2^{\alpha -1}\Gamma \left(\alpha +1\right)}{{\left(k-l\right)}^{\alpha }}},\hfill \end{array}$$

which is the result. The proof is completed. □

Theorem 4

Let $\rho :$$\left[l,k\right]$$\to {\mathbb{R}}^{+}$ be a multiplicatively differentiable mapping on $\left[l,k\right]$ with $l<k$. If $\left|ln{\rho }^{*}\right|\le ln\mathcal{M}$ on $[l,k]$, then we have

$$\begin{array}{cc}\hfill & \left|{\left(\left(\rho \left(l\right)\right){\left(\rho \left(\frac{l+k}{2}\right)\right)}^2\left(\rho \left(k\right)\right)\right)}^{\frac{1}{4}}{\left(\left({}_l{I}_{*}^{\alpha }\rho \right)\left(\frac{l+k}{2}\right)\left({}_{*}{I}_{\frac{l+k}{2}}^{\alpha }\rho \right)\left(k\right)\right)}^{-\frac{2^{\alpha -1}\Gamma \left(\alpha +1\right)}{{\left(k-l\right)}^{\alpha }}}\right|\hfill \\ \hfill \le & {\mathcal{M}}^{\frac{k-l}{4}\left(1+\frac{\alpha }{\alpha +1}{\left(\frac{1}{2}\right)}^{\frac{1}{\alpha }-1}\right)}.\hfill \end{array}$$

Proof. 

From Lemma 2, properties of multiplicative integral and using the fact that $\left|ln{f}^{*}\right|\le ln\mathcal{M}$, we have

$$\begin{array}{cc}\hfill & \left|{\left(\left(\rho \left(l\right)\right){\left(\rho \left(\frac{l+k}{2}\right)\right)}^2\left(\rho \left(k\right)\right)\right)}^{\frac{1}{4}}{\left(\left({}_l{I}_{*}^{\alpha }\rho \right)\left(\frac{l+k}{2}\right)\left({}_{*}{I}_{\frac{l+k}{2}}^{\alpha }\rho \right)\left(k\right)\right)}^{-\frac{2^{\alpha -1}\Gamma \left(\alpha +1\right)}{{\left(k-l\right)}^{\alpha }}}\right|\hfill \\ \hfill =& \left|{\left(\underset{0}{\stackrel{1}{\int }}{\left({\rho }^{*}{\left(\left(1-\varsigma \right)l+\varsigma \frac{l+k}{2}\right)}^{{\varsigma }^{\alpha }-\frac{1}{2}}\right)}^{d\varsigma }\right)}^{\frac{k-l}{4}}\right|\left|{\left(\underset{0}{\stackrel{1}{\int }}{\left({\rho }^{*}{\left(\left(1-\varsigma \right)\frac{l+k}{2}+\varsigma k\right)}^{\frac{1}{2}-{\left(1-\varsigma \right)}^{\alpha }}\right)}^{d\varsigma }\right)}^{\frac{k-l}{4}}\right|\hfill \\ \hfill =& \left|\left(\underset{0}{\stackrel{1}{\int }}{\left({\rho }^{*}{\left(\left(1-\varsigma \right)l+\varsigma \frac{l+k}{2}\right)}^{\frac{k-l}{4}\left({\varsigma }^{\alpha }-\frac{1}{2}\right)}\right)}^{dt}\right)\right|\left|\left(\underset{0}{\stackrel{1}{\int }}{\left({\rho }^{*}{\left(\left(1-\varsigma \right)\frac{l+k}{2}+\varsigma k\right)}^{\frac{k-l}{4}\left(\frac{1}{2}-{\left(1-\varsigma \right)}^{\alpha }\right)}\right)}^{d\varsigma }\right)\right|\hfill \\ \hfill \le & \left(exp\left\{\underset{0}{\stackrel{1}{\int }}\left|\frac{k-l}{4}\left({\varsigma }^{\alpha }-\frac{1}{2}\right)ln\left({\rho }^{*}\left(\left(1-\varsigma \right)l+\varsigma \frac{l+k}{2}\right)\right)\right|d\varsigma \right\}\right)\hfill \\ \hfill & \times \left(exp\left\{\underset{0}{\stackrel{1}{\int }}\left|\frac{k-l}{4}\left(\frac{1}{2}-{\left(1-\varsigma \right)}^{\alpha }\right)ln\left({\rho }^{*}\left(\left(1-\varsigma \right)\frac{l+k}{2}+\varsigma k\right)\right)\right|d\varsigma \right\}\right)\hfill \\ \hfill =& \left(exp\left\{\frac{k-l}{4}\underset{0}{\stackrel{1}{\int }}\left|{\varsigma }^{\alpha }-\frac{1}{2}\right|\left|ln\left({\rho }^{*}\left(\left(1-\varsigma \right)l+\varsigma \frac{l+k}{2}\right)\right)\right|d\varsigma \right\}\right)\hfill \\ \hfill & \times \left(exp\left\{\frac{k-l}{4}\underset{0}{\stackrel{1}{\int }}\left|\frac{1}{2}-{\left(1-\varsigma \right)}^{\alpha }\right|\left|ln\left({\rho }^{*}\left(\left(1-\varsigma \right)\frac{l+k}{2}+\varsigma k\right)\right)\right|d\varsigma \right\}\right)\hfill \\ \hfill \le & \left(exp\left\{\frac{k-l}{4}ln\mathcal{M}\underset{0}{\stackrel{1}{\int }}\left|{\varsigma }^{\alpha }-\frac{1}{2}\right|d\varsigma \right\}\right)\hfill \\ \hfill & \times \left(exp\left\{\frac{k-l}{4}ln\mathcal{M}\underset{0}{\stackrel{1}{\int }}\left|\frac{1}{2}-{\left(1-\varsigma \right)}^{\alpha }\right|d\varsigma \right\}\right)\hfill \\ \hfill =& \left(exp\left\{\frac{k-l}{4}ln\mathcal{M}\left(\underset{0}{\stackrel{{\left(\frac{1}{2}\right)}^{\frac{1}{\alpha }}}{\int }}\left(\frac{1}{2}-{\varsigma }^{\alpha }\right)d\varsigma +\underset{{\left(\frac{1}{2}\right)}^{\frac{1}{\alpha }}}{\stackrel{1}{\int }}\left({\varsigma }^{\alpha }-\frac{1}{2}\right)d\varsigma \right)\right\}\right)\hfill \\ \hfill & \times \left(exp\left\{\frac{k-l}{4}ln\mathcal{M}\left(\underset{0}{\stackrel{1-{\left(\frac{1}{2}\right)}^{\frac{1}{\alpha }}}{\int }}\left({\left(1-\varsigma \right)}^{\alpha }-\frac{1}{2}\right)d\varsigma +\underset{1-{\left(\frac{1}{2}\right)}^{\frac{1}{\alpha }}}{\stackrel{1}{\int }}\left(\frac{1}{2}-{\left(1-\varsigma \right)}^{\alpha }\right)d\varsigma \right)\right\}\right)\hfill \\ \hfill =& \left(exp\left\{\frac{k-l}{4}\left(\frac{2+2\alpha }{4\left(\alpha +1\right)}+\frac{2\alpha }{\alpha +1}{\left(\frac{1}{2}\right)}^{\frac{1}{\alpha }+1}\right)ln\mathcal{M}\right\}\right)\hfill \\ \hfill & \times \left(exp\left\{\frac{k-l}{4}\left(\frac{2+2\alpha }{4\left(\alpha +1\right)}+\frac{2\alpha }{\alpha +1}{\left(\frac{1}{2}\right)}^{1+\frac{1}{\alpha }}\right)ln\mathcal{M}\right\}\right)\hfill \\ \hfill =& {\mathcal{M}}^{\frac{k-l}{4}\left(1+\frac{\alpha }{\alpha +1}{\left(\frac{1}{2}\right)}^{\frac{1}{\alpha }-1}\right)}.\hfill \end{array}$$

The proof is completed. □

Corollary 1.

In Theorem 4, if we take $\alpha =1$, then we obtain

$$\left|{\left(\left(\rho \left(l\right)\right){\left(\rho \left(\frac{l+k}{2}\right)\right)}^2\left(\rho \left(k\right)\right)\right)}^{\frac{1}{4}}{\left(\underset{l}{\stackrel{k}{\int }}\rho {\left(u\right)}^{du}\right)}^{\frac{1}{l-k}}\right|\le {\mathcal{M}}^{\frac{3}{8}\left(k-l\right)}.$$

Theorem 5

Let $\rho :$$\left[l,k\right]$$\to {\mathbb{R}}^{+}$ be a multiplicative differentiable mapping on $\left[l,k\right]$ with $l<k$. If ${\rho }^{*}$ is multiplicatively convex function on $[l,k]$, then we have

$$\begin{array}{cc}\hfill & \left|{\left(\left(\rho \left(l\right)\right){\left(\rho \left(\frac{l+k}{2}\right)\right)}^2\left(\rho \left(k\right)\right)\right)}^{\frac{1}{4}}{\left(\left({}_l{I}_{*}^{\alpha }\rho \right)\left(\frac{l+k}{2}\right)\left({}_{*}{I}_{\frac{l+k}{2}}^{\alpha }\rho \right)\left(k\right)\right)}^{-\frac{2^{\alpha -1}\Gamma \left(\alpha +1\right)}{{\left(k-l\right)}^{\alpha }}}\right|\hfill \\ \hfill \le & {\left[\left({\rho }^{*}\left(l\right)\right)\left(f^{*}\left(k\right)\right)\right]}^{\frac{k-l}{2}\left(\frac{2-{\alpha }^2-3\alpha }{8\left(\alpha +1\right)\left(\alpha +2\right)}+\frac{\alpha }{\alpha +1}{\left(\frac{1}{2}\right)}^{1+\frac{1}{\alpha }}-\frac{\alpha }{2\left(\alpha +2\right)}{\left(\frac{1}{2}\right)}^{1+\frac{2}{\alpha }}\right)}\hfill \\ \hfill & \times {\left(f^{*}\left(\frac{l+k}{2}\right)\right)}^{\frac{k-l}{2}\left(\frac{2-\alpha }{4\left(\alpha +2\right)}+\frac{\alpha }{\alpha +2}{\left(\frac{1}{2}\right)}^{1+\frac{2}{\alpha }}\right)}.\hfill \end{array}$$

Proof. 

From Lemma 2, the properties of the multiplicative integral and the multiplicative convexity of ${\rho }^{*}$, we have

$$\begin{array}{cc}\hfill & \left|{\left(\left(\rho \left(l\right)\right){\left(\rho \left(\frac{l+k}{2}\right)\right)}^2\left(\rho \left(k\right)\right)\right)}^{\frac{1}{4}}{\left(\left({}_l{I}_{*}^{\alpha }\rho \right)\left(\frac{l+k}{2}\right)\left({}_{*}{I}_{\frac{l+k}{2}}^{\alpha }\rho \right)\left(k\right)\right)}^{-\frac{2^{\alpha -1}\Gamma \left(\alpha +1\right)}{{\left(k-l\right)}^{\alpha }}}\right|\hfill \\ \hfill =& \left|{\left(\underset{0}{\stackrel{1}{\int }}{\left({\rho }^{*}{\left(\left(1-\varsigma \right)l+\varsigma \frac{l+k}{2}\right)}^{{\varsigma }^{\alpha }-\frac{1}{2}}\right)}^{d\varsigma }\right)}^{\frac{k-l}{4}}\right|\hfill \\ \hfill & \times \left|{\left(\underset{0}{\stackrel{1}{\int }}{\left({\rho }^{*}{\left(\left(1-\varsigma \right)\frac{l+k}{2}+\varsigma k\right)}^{\frac{1}{2}-{\left(1-\varsigma \right)}^{\alpha }}\right)}^{d\varsigma }\right)}^{\frac{k-l}{4}}\right|\hfill \\ \hfill =& \underset{0}{\stackrel{1}{\int }}{\left|{\rho }^{*}{\left(\left(1-\varsigma \right)l+\varsigma \frac{l+k}{2}\right)}^{\frac{k-l}{2}\left(\frac{1}{2}{\varsigma }^{\alpha }-\frac{1}{4}\right)}\right|}^{d\varsigma }\hfill \\ \hfill & \times \underset{0}{\stackrel{1}{\int }}{\left|{\rho }^{*}{\left(\left(1-\varsigma \right)\frac{l+k}{2}+\varsigma k\right)}^{\frac{k-l}{2}\left(\frac{1}{4}-\frac{1}{2}{\left(1-\varsigma \right)}^{\alpha }\right)}\right|}^{d\varsigma }\hfill \\ \hfill \le & \left(exp\left\{\frac{k-l}{2}\underset{0}{\stackrel{1}{\int }}\left|\left(\frac{1}{2}{\varsigma }^{\alpha }-\frac{1}{4}\right)\right|\left|ln{\rho }^{*}\left(\left(1-\varsigma \right)l+\varsigma \frac{l+k}{2}\right)\right|d\varsigma \right\}\right)\hfill \\ \hfill & \times \left(exp\left\{\frac{k-l}{2}\underset{0}{\stackrel{1}{\int }}\left|\left(\frac{1}{4}-\frac{1}{2}{\left(1-\varsigma \right)}^{\alpha }\right)\right|\left|ln{\rho }^{*}\left(\left(1-\varsigma \right)\frac{l+k}{2}+\varsigma k\right)\right|d\varsigma \right\}\right)\hfill \\ \hfill \le & \left(exp\left\{\frac{k-l}{2}\underset{0}{\stackrel{1}{\int }}\left|\left(\frac{1}{2}{\varsigma }^{\alpha }-\frac{1}{4}\right)\right|\left|ln{\left({\rho }^{*}\left(l\right)\right)}^{\left(1-\varsigma \right)}{\left(f^{*}\left(\frac{l+k}{2}\right)\right)}^{\varsigma }\right|d\varsigma \right\}\right)\hfill \\ \hfill & \times \left(exp\left\{\frac{k-l}{2}\underset{0}{\stackrel{1}{\int }}\left|\left(\frac{1}{4}-\frac{1}{2}{\left(1-\varsigma \right)}^{\alpha }\right)\right|\left|ln{\left({\rho }^{*}\left(\frac{l+k}{2}\right)\right)}^{\left(1-\varsigma \right)}{\left(f^{*}\left(k\right)\right)}^{\varsigma }\right|d\varsigma \right\}\right)\hfill \\ \hfill =& exp\left\{\frac{k-l}{2}\left(\underset{0}{\stackrel{{\left(\frac{1}{2}\right)}^{\frac{1}{\alpha }}}{\int }}\left(\frac{1}{4}-\frac{1}{2}{\varsigma }^{\alpha }\right)\left|ln{\left({\rho }^{*}\left(l\right)\right)}^{\left(1-\varsigma \right)}{\left(f^{*}\left(\frac{l+k}{2}\right)\right)}^{\varsigma }\right|d\varsigma \right.\right.\hfill \\ \hfill & +\left.\left.\underset{{\left(\frac{1}{2}\right)}^{\frac{1}{\alpha }}}{\stackrel{1}{\int }}\left(\frac{1}{2}{\varsigma }^{\alpha }-\frac{1}{4}\right)\left|ln{\left({\rho }^{*}\left(l\right)\right)}^{\left(1-\varsigma \right)}{\left(f^{*}\left(\frac{l+k}{2}\right)\right)}^{\varsigma }\right|d\varsigma \right)\right\}\hfill \\ \hfill & \times exp\left\{\frac{k-l}{2}\left(\underset{0}{\stackrel{1-{\left(\frac{1}{2}\right)}^{\frac{1}{\alpha }}}{\int }}\left(\frac{1}{2}{\left(1-\varsigma \right)}^{\alpha }-\frac{1}{4}\right)\left|ln{\left({\rho }^{*}\left(\frac{l+k}{2}\right)\right)}^{\left(1-\varsigma \right)}{\left(f^{*}\left(k\right)\right)}^{\varsigma }\right|d\varsigma \right.\right.\hfill \\ \hfill & +\left.\left.\underset{1-{\left(\frac{1}{2}\right)}^{\frac{1}{\alpha }}}{\stackrel{1}{\int }}\left(\frac{1}{4}-\frac{1}{2}{\left(1-\varsigma \right)}^{\alpha }\right)\left|ln{\left({\rho }^{*}\left(\frac{l+k}{2}\right)\right)}^{\left(1-\varsigma \right)}{\left(f^{*}\left(k\right)\right)}^{\varsigma }\right|d\varsigma \right)\right\}\hfill \\ \hfill =& exp\left\{\frac{k-l}{2}\left(\underset{0}{\stackrel{{\left(\frac{1}{2}\right)}^{\frac{1}{\alpha }}}{\int }}\left(\frac{1}{4}-\frac{1}{2}{\varsigma }^{\alpha }\right)\left(\left(1-\varsigma \right)ln\left({\rho }^{*}\left(l\right)\right)+\varsigma ln\left(f^{*}\left(\frac{l+k}{2}\right)\right)\right)d\varsigma \right.\right.\hfill \\ \hfill & +\left.\left.\underset{{\left(\frac{1}{2}\right)}^{\frac{1}{\alpha }}}{\stackrel{1}{\int }}\left(\frac{1}{2}{\varsigma }^{\alpha }-\frac{1}{4}\right)\left(\left(1-\varsigma \right)ln\left({\rho }^{*}\left(l\right)\right)+\varsigma ln\left(f^{*}\left(\frac{l+k}{2}\right)\right)\right)d\varsigma \right)\right\}\hfill \\ \hfill & \times exp\left\{\frac{k-l}{2}\left(\underset{0}{\stackrel{1-{\left(\frac{1}{2}\right)}^{\frac{1}{\alpha }}}{\int }}\left(\frac{1}{2}{\left(1-\varsigma \right)}^{\alpha }-\frac{1}{4}\right)\left(\left(1-\varsigma \right)ln\left({\rho }^{*}\left(\frac{l+k}{2}\right)\right)+\varsigma ln\left(f^{*}\left(k\right)\right)\right)d\varsigma \right.\right.\hfill \\ \hfill & +\left.\left.\underset{1-{\left(\frac{1}{2}\right)}^{\frac{1}{\alpha }}}{\stackrel{1}{\int }}\left(\frac{1}{4}-\frac{1}{2}{\left(1-\varsigma \right)}^{\alpha }\right)\left(\left(1-\varsigma \right)ln\left({\rho }^{*}\left(\frac{l+k}{2}\right)\right)+\varsigma ln\left(f^{*}\left(k\right)\right)\right)d\varsigma \right)\right\}\hfill \\ \hfill =& exp\left\{\frac{k-l}{2}\left(ln\left({\rho }^{*}\left(l\right)\right)\underset{0}{\stackrel{{\left(\frac{1}{2}\right)}^{\frac{1}{\alpha }}}{\int }}\left(\frac{1}{4}-\frac{1}{2}{\varsigma }^{\alpha }\right)\left(1-\varsigma \right)d\varsigma \right.\right.\hfill \\ \hfill & +ln\left(f^{*}\left(\frac{l+k}{2}\right)\right)\underset{0}{\stackrel{{\left(\frac{1}{2}\right)}^{\frac{1}{\alpha }}}{\int }}\left(\frac{1}{4}-\frac{1}{2}{\varsigma }^{\alpha }\right)\varsigma d\varsigma +ln\left({\rho }^{*}\left(l\right)\right)\underset{{\left(\frac{1}{2}\right)}^{\frac{1}{\alpha }}}{\stackrel{1}{\int }}\left(\frac{1}{2}{\varsigma }^{\alpha }-\frac{1}{4}\right)\left(1-\varsigma \right)d\varsigma \hfill \\ \hfill & +\left.\left.ln\left(f^{*}\left(\frac{l+k}{2}\right)\right)\underset{{\left(\frac{1}{2}\right)}^{\frac{1}{\alpha }}}{\stackrel{1}{\int }}\left(\frac{1}{2}{\varsigma }^{\alpha }-\frac{1}{4}\right)\varsigma d\varsigma \right)\right\}\hfill \\ \hfill & \times exp\left\{\frac{k-l}{2}\left(ln\left({\rho }^{*}\left(\frac{l+k}{2}\right)\right)\underset{0}{\stackrel{1-{\left(\frac{1}{2}\right)}^{\frac{1}{\alpha }}}{\int }}\left(\frac{1}{2}{\left(1-\varsigma \right)}^{\alpha }-\frac{1}{4}\right)\left(1-\varsigma \right)d\varsigma \right.\right.\hfill \\ \hfill & +ln\left(f^{*}\left(k\right)\right)\underset{0}{\stackrel{1-{\left(\frac{1}{2}\right)}^{\frac{1}{\alpha }}}{\int }}\left(\frac{1}{2}{\left(1-\varsigma \right)}^{\alpha }-\frac{1}{4}\right)\varsigma d\varsigma +ln\left({\rho }^{*}\left(\frac{l+k}{2}\right)\right)\underset{1-{\left(\frac{1}{2}\right)}^{\frac{1}{\alpha }}}{\stackrel{1}{\int }}\left(\frac{1}{4}-\frac{1}{2}{\left(1-\varsigma \right)}^{\alpha }\right)\left(1-\varsigma \right)d\varsigma \hfill \\ \hfill & +\left.\left.ln\left(f^{*}\left(k\right)\right)\underset{1-{\left(\frac{1}{2}\right)}^{\frac{1}{\alpha }}}{\stackrel{1}{\int }}\left(\frac{1}{4}-\frac{1}{2}{\left(1-\varsigma \right)}^{\alpha }\right)\varsigma d\varsigma \right)\right\}\hfill \\ \hfill =& exp\left\{\frac{k-l}{2}\left(\left(\frac{\alpha }{4\left(\alpha +1\right)}{\left(\frac{1}{2}\right)}^{\frac{1}{\alpha }}-\frac{\alpha }{8\left(\alpha +2\right)8}{\left(\frac{1}{2}\right)}^{\frac{2}{\alpha }}\right)ln\left({\rho }^{*}\left(l\right)\right)\right.\right.\hfill \\ \hfill & +\frac{\alpha }{8\left(\alpha +2\right)}{\left(\frac{1}{2}\right)}^{\frac{2}{\alpha }}ln\left(f^{*}\left(\frac{l+k}{2}\right)\right)+\left(\frac{2-{\alpha }^2-3\alpha }{8\left(\alpha +1\right)\left(\alpha +2\right)}+\frac{\alpha }{4\left(\alpha +1\right)}{\left(\frac{1}{2}\right)}^{\frac{1}{\alpha }}-\frac{\alpha }{8\left(\alpha +2\right)}{\left(\frac{1}{2}\right)}^{\frac{2}{\alpha }}\right)ln\left({\rho }^{*}\left(l\right)\right)\hfill \\ \hfill & +\left.\left.\left(\frac{2-\alpha }{8\left(\alpha +2\right)}+\frac{\alpha }{8\left(\alpha +2\right)}{\left(\frac{1}{2}\right)}^{\frac{2}{\alpha }}\right)ln\left(f^{*}\left(\frac{l+k}{2}\right)\right)\right)\right\}\hfill \\ \hfill & \times exp\left\{\frac{k-l}{2}\left(\left(\frac{2-\alpha }{8\left(\alpha +2\right)}+\frac{\alpha }{8\left(\alpha +2\right)}{\left(\frac{1}{2}\right)}^{\frac{2}{\alpha }}\right)ln\left({\rho }^{*}\left(\frac{l+k}{2}\right)\right)\right.\right.\hfill \\ \hfill & +\left(\frac{2-{\alpha }^2-3\alpha }{8\left(\alpha +1\right)\left(\alpha +2\right)}+\frac{\alpha }{4\left(\alpha +1\right)}{\left(\frac{1}{2}\right)}^{\frac{1}{\alpha }}-\frac{\alpha }{8\left(\alpha +2\right)}{\left(\frac{1}{2}\right)}^{\frac{2}{\alpha }}\right)ln\left(f^{*}\left(k\right)\right)\hfill \\ \hfill & +\left.\left.\frac{\alpha }{8\left(\alpha +2\right)}{\left(\frac{1}{2}\right)}^{\frac{2}{\alpha }}ln\left({\rho }^{*}\left(\frac{l+k}{2}\right)\right)+\left(\frac{\alpha }{4\left(\alpha +1\right)}{\left(\frac{1}{2}\right)}^{\frac{1}{\alpha }}-\frac{\alpha }{8\left(\alpha +2\right)}{\left(\frac{1}{2}\right)}^{\frac{2}{\alpha }}\right)ln\left(f^{*}\left(k\right)\right)\right)\right\}\hfill \\ \hfill =& exp\left\{\frac{k-l}{2}\left(ln{\left({\rho }^{*}\left(l\right)\right)}^{\left(\frac{2-{\alpha }^2-3\alpha }{8\left(\alpha +1\right)\left(\alpha +2\right)}+\frac{\alpha }{\alpha +1}{\left(\frac{1}{2}\right)}^{1+\frac{1}{\alpha }}-\frac{\alpha }{2\left(\alpha +2\right)}{\left(\frac{1}{2}\right)}^{1+\frac{2}{\alpha }}\right)}\right.\right.\hfill \\ \hfill & +\left.\left.ln{\left(f^{*}\left(\frac{l+k}{2}\right)\right)}^{\frac{1}{2}\left(\frac{2-\alpha }{4\left(\alpha +2\right)}+\frac{\alpha }{\alpha +2}{\left(\frac{1}{2}\right)}^{1+\frac{2}{\alpha }}\right)}\right)\right\}exp\left\{\frac{k-l}{2}\left(ln{\left({\rho }^{*}\left(\frac{l+k}{2}\right)\right)}^{\frac{1}{2}\left(\frac{2-\alpha }{4\left(\alpha +2\right)}+\frac{\alpha }{\alpha +2}{\left(\frac{1}{2}\right)}^{1+\frac{2}{\alpha }}\right)}\right.\right.\hfill \\ \hfill & +\left.\left.ln{\left(f^{*}\left(k\right)\right)}^{\left(\frac{2-{\alpha }^2-3\alpha }{8\left(\alpha +1\right)\left(\alpha +2\right)}+\frac{\alpha }{\alpha +1}{\left(\frac{1}{2}\right)}^{1+\frac{1}{\alpha }}-\frac{\alpha }{2\left(\alpha +2\right)}{\left(\frac{1}{2}\right)}^{1+\frac{2}{\alpha }}\right)}\right)\right\}\hfill \\ \hfill =& {\left[\left({\rho }^{*}\left(l\right)\right)\left(f^{*}\left(k\right)\right)\right]}^{\frac{k-l}{2}\left(\frac{2-{\alpha }^2-3\alpha }{8\left(\alpha +1\right)\left(\alpha +2\right)}+\frac{\alpha }{\alpha +1}{\left(\frac{1}{2}\right)}^{1+\frac{1}{\alpha }}-\frac{\alpha }{2\left(\alpha +2\right)}{\left(\frac{1}{2}\right)}^{1+\frac{2}{\alpha }}\right)}{\left(f^{*}\left(\frac{l+k}{2}\right)\right)}^{\frac{k-l}{2}\left(\frac{2-\alpha }{4\left(\alpha +2\right)}+\frac{\alpha }{\alpha +2}{\left(\frac{1}{2}\right)}^{1+\frac{2}{\alpha }}\right)},\hfill \end{array}$$

where we have used

$$\begin{array}{cc}\hfill \underset{0}{\stackrel{{\left(\frac{1}{2}\right)}^{\frac{1}{\alpha }}}{\int }}\left(\frac{1}{4}-\frac{1}{2}{\varsigma }^{\alpha }\right)\left(1-\varsigma \right)d\varsigma =& \underset{1-{\left(\frac{1}{2}\right)}^{\frac{1}{\alpha }}}{\stackrel{1}{\int }}\left(\frac{1}{4}-\frac{1}{2}{\left(1-\varsigma \right)}^{\alpha }\right)\varsigma d\varsigma \hfill \\ \hfill =& \frac{\alpha }{4\left(\alpha +1\right)}{\left(\frac{1}{2}\right)}^{\frac{1}{\alpha }}-\frac{\alpha }{8\left(\alpha +2\right)}{\left(\frac{1}{2}\right)}^{\frac{2}{\alpha }},\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill \underset{0}{\stackrel{{\left(\frac{1}{2}\right)}^{\frac{1}{\alpha }}}{\int }}\left(\frac{1}{4}-\frac{1}{2}{\varsigma }^{\alpha }\right)\varsigma d\varsigma =& \underset{1-{\left(\frac{2}{n+2}\right)}^{\frac{1}{\alpha }}}{\stackrel{1}{\int }}\left(\frac{1}{4}-\frac{1}{2}{\left(1-\varsigma \right)}^{\alpha }\right)\left(1-\varsigma \right)d\varsigma \hfill \\ \hfill =& \frac{\alpha }{8\left(\alpha +2\right)}{\left(\frac{1}{2}\right)}^{\frac{2}{\alpha }},\hfill \end{array}$$

(3)

$$\begin{array}{cc}& \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\underset{{\left(\frac{1}{2}\right)}^{\frac{1}{\alpha }}}{\stackrel{1}{\int }}\left(\frac{1}{2}{\varsigma }^{\alpha }-\frac{1}{4}\right)\left(1-\varsigma \right)d\varsigma =\underset{0}{\stackrel{1-{\left(\frac{1}{2}\right)}^{\frac{1}{\alpha }}}{\int }}\left(\frac{1}{2}{\left(1-\varsigma \right)}^{\alpha }-\frac{1}{4}\right)\varsigma d\varsigma \hfill \\ & \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\frac{2-{\alpha }^2-3\alpha }{8\left(\alpha +1\right)\left(\alpha +2\right)}+\frac{\alpha }{4\left(\alpha +1\right)}{\left(\frac{1}{2}\right)}^{\frac{1}{\alpha }}-\frac{\alpha }{8\left(\alpha +2\right)}{\left(\frac{1}{2}\right)}^{\frac{2}{\alpha }},\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill \underset{{\left(\frac{1}{2}\right)}^{\frac{1}{\alpha }}}{\stackrel{1}{\int }}\left(\frac{1}{2}{\varsigma }^{\alpha }-\frac{1}{4}\right)\varsigma d\varsigma =& \underset{0}{\stackrel{1-{\left(\frac{1}{2}\right)}^{\frac{1}{\alpha }}}{\int }}\left(\frac{1}{2}{\left(1-\varsigma \right)}^{\alpha }-\frac{1}{4}\right)\left(1-\varsigma \right)d\varsigma \hfill \\ \hfill =& \frac{2-\alpha }{8\left(\alpha +2\right)}+\frac{\alpha }{8\left(\alpha +2\right)}{\left(\frac{1}{2}\right)}^{\frac{2}{\alpha }}.\hfill \end{array}$$

(5)

The proof is completed. □

Corollary 2.

In Theorem 5, using the multiplicative convexity of ${\rho }^{*}$ i.e., $f^{*}\left(\frac{l+k}{2}\right)\le \sqrt{{\rho }^{*}\left(l\right){\rho }^{*}\left(k\right)}$, we obtain

$$\begin{array}{cc}\hfill & \left|{\left(\left(\rho \left(l\right)\right){\left(\rho \left(\frac{l+k}{2}\right)\right)}^2\left(\rho \left(k\right)\right)\right)}^{\frac{1}{4}}{\left(\left({}_l{I}_{*}^{\alpha }\rho \right)\left(\frac{l+k}{2}\right)\left({}_{*}{I}_{\frac{l+k}{2}}^{\alpha }\rho \right)\left(k\right)\right)}^{-\frac{2^{\alpha -1}\Gamma \left(\alpha +1\right)}{{\left(k-l\right)}^{\alpha }}}\right|\hfill \\ \hfill \le & {\left[\left({\rho }^{*}\left(l\right)\right)\left(f^{*}\left(k\right)\right)\right]}^{\frac{k-l}{2}\left(\frac{1-\alpha }{4\left(\alpha +1\right)}+\frac{\alpha }{\alpha +1}{\left(\frac{1}{2}\right)}^{1+\frac{1}{\alpha }}\right)}.\hfill \end{array}$$

Corollary 3.

In Theorem 5, if we take $\alpha =1$, then we obtain

$$\begin{array}{cc}\hfill & \left|{\left(\left(\rho \left(l\right)\right){\left(\rho \left(\frac{l+k}{2}\right)\right)}^2\left(\rho \left(k\right)\right)\right)}^{\frac{1}{4}}{\left(\underset{l}{\stackrel{k}{\int }}\rho {\left(u\right)}^{du}\right)}^{\frac{1}{l-k}}\right|\hfill \\ \hfill \le & {\left[\left({\rho }^{*}\left(l\right)\right){\left(f^{*}\left(\frac{l+k}{2}\right)\right)}^2\left(f^{*}\left(k\right)\right)\right]}^{\frac{k-l}{32}}.\hfill \end{array}$$

Corollary 4.

In Corollary 3, using the multiplicative convexity of $f^{*}$, we obtain

$$\left|{\left(\left(\rho \left(l\right)\right){\left(\rho \left(\frac{l+k}{2}\right)\right)}^2\left(\rho \left(k\right)\right)\right)}^{\frac{1}{4}}{\left(\underset{l}{\stackrel{k}{\int }}\rho {\left(u\right)}^{du}\right)}^{\frac{1}{l-k}}\right|\le {\left[\left({\rho }^{*}\left(l\right)\right)\left(f^{*}\left(k\right)\right)\right]}^{\frac{k-l}{16}}.$$

3. Applications to Special Means

We shall consider the means for arbitrary real numbers $l,k$.

The Arithmetic mean: $A\left(l,k\right)=\frac{l+k}{2}$.

The Harmonic mean: $H\left(l,k\right)=\frac{2lk}{l+k}$

The logarithmic means: $L\left(l,k\right)=\frac{k-l}{lnk-lnl}$, $l,k>0$ and $l\ne k$.

The p-Logarithmic mean: $L_p\left(l,k\right)={\left(\frac{k^{p+1}-l^{p+1}}{\left(p+1\right)\left(k-l\right)}\right)}^{\frac{1}{p}}$, $l,k>0,l\ne k$ and $p\in \mathbb{R}\setminus \left\{-1,0\right\}$.

Proposition 2.

Let $l,k\in \mathbb{R}$ with $0<l<k$. Then, we have

$$\left|e^{\frac{1}{4}\left(2H^{-1}\left(l,k\right)-A^2\left(l,k\right)\right)-L^{-1}\left(l,k\right)}\right|\le e^{-\frac{3\left(k-l\right)}{8l^2}}.$$

Proof. 

The assertion follows from Corollary 1 applied to the function $\rho \left(\varsigma \right)=e^{\frac{1}{\varsigma }}$ whose ${\rho }^{*}\left(\varsigma \right)=e^{-\frac{1}{{\varsigma }^2}},$$\mathcal{M}=e^{-\frac{1}{l^2}}$ and ${\left(\underset{l}{\stackrel{k}{\int }}\rho {\left(u\right)}^{du}\right)}^{\frac{1}{l-k}}=exp\left\{-L^{-1}\left(l,k\right)\right\}$. □

Proposition 3.

Let $l,k\in \mathbb{R}$ with $0<l<k$. Then, we have

$$\left|e^{\frac{1}{4}\left(2A\left(l^p,k^p\right)+A^{2p}\left(l,k\right)\right)-L_p^p\left(l,k\right)}\right|\le e^{p\frac{\left(k-l\right)\left(l^{p-1}+k^{p-1}\right)}{16}}.$$

Proof. 

The assertion follows from Corollary 3, applied to the function $\rho \left(\varsigma \right)=e^{{\varsigma }^p}$ with $p\ge 2$ whose ${\rho }^{*}\left(\varsigma \right)=e^{p{\varsigma }^{p-1}}$ and ${\left(\underset{l}{\stackrel{k}{\int }}\rho {\left(u\right)}^{du}\right)}^{\frac{1}{l-k}}=exp\left\{-L_p^p\left(l,k\right)\right\}$. □

4. Conclusions

In this study, we have considered the fractional multiplicative Bullen-type integral inequalities, whose main results can be summarized as follows:

• A new fractional identity for multiplicatively integrals is proved.
• Some new fractional multiplicative Bullen-type inequalities for functions whose first multiplicative derivatives are multiplicative convex are established.
• Some special cases are derived.
• Applications of our findings are provided.