Coefficient-Related Studies and Fekete-Szegö Type Inequalities for New Classes of Bi-Starlike and Bi-Convex Functions

Abstract

In this paper, we define certain families ${\mathcal{S}}_E^{*}\left(\vartheta \right)$ and ${\mathcal{C}}_E\left(\vartheta \right)$ of holomorphic and bi-univalent functions which are defined in the open unit disk U. We establish upper bounds for the initial Taylor–Maclaurin coefficients and Fekete–Szegö type inequalities for functions in these families.

1. Introduction and Preliminaries

We indicate by $\mathcal{A}$ the family of all holomorphic functions of the type

$$f\left(z\right)=z+\sum _{k=2}^{\infty }{a}_k{z}^k$$

(1)

in the open unit disk $U=\{z\in C:|z|<1\}$.

Also denote by $\mathcal{W}$ the subfamily of $\mathcal{A}$ consisting of functions which are also univalent in U.

A function $f\in \mathcal{W}$ is called starlike of order $\gamma (0\le \gamma <1)$ if

$$\Re \left(\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}\right)>\gamma ,(z\in U)$$

and a function $f\in \mathcal{W}$ is called convex of order $\gamma (0\le \gamma <1)$ if

$$\Re \left(\frac{z{f}^{″}\left(z\right)}{f^{\prime }\left(z\right)}+1\right)>\gamma ,(z\in U).$$

We denote by ${\mathcal{S}}^{*}\left(\gamma \right)$ and $\mathcal{C}\left(\gamma \right)$ the families of functions that are starlike of order $\gamma$ and convex of order $\gamma$ in U, respectively.

The famous Koebe one-quarter theorem [1] ensures that the image of U under each univalent function $f\in \mathcal{A}$ contains a disk of radius $\frac{1}{4}$. Furthermore, each function $f\in \mathcal{W}$ has an inverse $f^{-1}$ defined by $f^{-1}\left(f\left(z\right)\right)=z$ and

$$f\left(f^{-1}\left(w\right)\right)=w,\left(\left|w\right|<r_0\left(f\right),r_0\left(f\right)\ge \frac{1}{4}\right)$$

where

$$g\left(w\right)=f^{-1}\left(w\right)=w-a_2{w}^2+(2a_2^2-a_3)w^3-(5a_2^3-5a_2{a}_3+a_4)w^4+\cdots .$$

We say that the function $f\in \mathcal{A}$ is bi-univalent in U if both f and $f^{-1}$ are univalent in U. The family of all bi-univalent functions in U is denoted by E.

A very large number of investigations related to bi-univalent functions have been published (see, for example, [2,3,4,5,6,7,8,9,10,11,12,13]). Following Srivastava et al. [14], we choose to recall the following examples of functions in the family $E:$

$$\frac{z}{1-z},-log(1-z)\mathrm{and}\frac{1}{2}log\left(\frac{1+z}{1-z}\right).$$

Note that the family E is not empty. However, the Koebe function is not a member of $E.$

The problem of how to obtain the general coefficient bounds on the Taylor–Maclaurin coefficients

$$|a_n|(n\in \mathbb{N};n\geqq 3)$$

for functions $f\in E$ has still not been completely addressed for many of the subfamilies of E. The Fekete–Szegö functional $\left|a_3-\eta a_2^2\right|$ for $f\in \mathcal{W}$ is well-known for its rich history in the field of geometric function theory. Its origin was in the disproof by Fekete and Szegö [1] of the Littlewood–Paley conjecture that the coefficients of odd univalent functions are bounded by unity. Many authors have obtained Fekete–Szegö inequalities for different families of functions. This topic has become of considerable interest to researchers of geometric function theory (see, for example, [2,3,5,6,11,14,15,16,17,18,19,20,21,22,23,24,25]).

With a view to recalling the principle of subordination between holomorphic functions, let the functions f and g be holomorphic in U. The function f is subordinate to g, if there exists a Schwarz function $\omega$, which is analytic in U with

$$\omega \left(0\right)=0\mathrm{and}\left|\omega \right(z\left)\right|<1(z\in U),$$

such that

$$f\left(z\right)=g\left(\omega \left(z\right)\right).$$

We denote this subordination is denoted by

$$f\prec g\mathrm{or}f\left(z\right)\prec g\left(z\right)(z\in U).$$

It is well known that (see [26]), if the function g is univalent in U, then

$$f\prec g(z\in U)⟺f\left(0\right)=g\left(0\right)\mathrm{and}f\left(U\right)\subseteq g\left(U\right).$$

Now, we study the function

$$\vartheta \left(z\right)=e^{\left(z+\tau \frac{z^2}{2}\right)}=1+z+\frac{z^2}{2}+\frac{1+\tau }{6}{z}^3+\frac{1+3\tau }{24}{z}^4+\cdots$$

(2)

with its domain of definition as the open unit disk U. We note that $\vartheta \left(z\right)$ is holomorphic function with a positive real part in U such that $\vartheta \left(0\right)=1,{\vartheta }^{\prime }\left(0\right)>0$ and $\vartheta$ maps U onto a region starlike with respect to 1 and symmetric with respect to the real axis.

Lemma 1

([27], p. 41). Let the function $x\in P$ be given by the following series:

$$x\left(z\right)=1+x_{1}z+x_2{z}^2+\cdots (z\in U).$$

The sharp estimate given by

$$|x_n|\leqq 2(n\in \mathbb{N})$$

holds true.

2. Set of Main Results

We now provide the following subfamilies of holomorphic bi-starlike and bi-convex functions.

Definition 1.

A function $f\in E$ is said to be in the family ${\mathcal{S}}_E^{*}\left(\vartheta \right)$ if it fulfils the subordinations:

$$\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}\prec e^{\left(z+\tau \frac{z^2}{2}\right)}=:\vartheta \left(z\right)$$

and

$$\frac{w{g}^{\prime }\left(w\right)}{g\left(w\right)}\prec e^{\left(w+\tau \frac{w^2}{2}\right)}=:\vartheta \left(w\right),$$

where $1\leqq \tau <2$ and $g\left(w\right)=f^{-1}\left(w\right)$.

Definition 2.

A function $f\in E$ is said to be in the family ${\mathcal{C}}_E\left(\vartheta \right)$ if it fulfils the subordinations:

$$1+\frac{z{f}^{″}\left(z\right)}{f^{\prime }\left(z\right)}\prec e^{\left(z+\tau \frac{z^2}{2}\right)}=:\vartheta \left(z\right)$$

and

$$1+\frac{w{g}^{″}\left(w\right)}{g^{\prime }\left(w\right)}\prec e^{\left(w+\tau \frac{w^2}{2}\right)}=:\vartheta \left(w\right),$$

where $1\leqq \tau <2$ and $g\left(w\right)=f^{-1}\left(w\right)$.

Theorem 1.

Let f given by (1) be in the family ${\mathcal{S}}_E^{*}\left(\vartheta \right)$. Then

$$|a_2|\leqq min\left\{1,\sqrt{\frac{2}{\left|3-\tau \right|}}\right\}$$

and

$$|a_3|\leqq min\left\{\frac{\tau +2}{2},\frac{3}{2}\right\}.$$

Proof. 

Let $f\in {\mathcal{S}}_E^{*}\left(\vartheta \right)$ and $g=f^{-1}$. Then there are holomorphic functions $\mathfrak{F},\mathfrak{G}:U⟶U$ with $\mathfrak{F}\left(0\right)=\mathfrak{G}\left(0\right)=0$, fulfils the following conditions:

$$\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}=\vartheta \left(\mathfrak{F}\left(z\right)\right),z\in U$$

(3)

and

$$\frac{w{g}^{\prime }\left(w\right)}{g\left(w\right)}=\vartheta \left(\mathfrak{G}\left(w\right)\right),w\in U.$$

(4)

We defined the functions x and y by

$$x\left(z\right)=\frac{1+\mathfrak{F}\left(z\right)}{1-\mathfrak{F}\left(z\right)}=1+x_{1}z+x_2{z}^2+\cdots$$

and

$$y\left(z\right)=\frac{1+\mathfrak{G}\left(z\right)}{1-\mathfrak{G}\left(z\right)}=1+y_{1}z+y_2{z}^2+\cdots .$$

Then the functions x and y are analytic in U with $x\left(0\right)=y\left(0\right)=1$. Since we have $\mathfrak{F},\mathfrak{G}:U⟶U$, each of the functions x and y has a positive real part in U.

For $\mathfrak{F}\left(z\right)$ and $\mathfrak{G}\left(z\right)$, we have

$$\mathfrak{F}\left(z\right)=\frac{x\left(z\right)-1}{x\left(z\right)+1}=\frac{1}{2}\left[x_{1}z+\left(x_2-\frac{x_1^2}{2}\right)z^2\right]+\cdots (z\in U)$$

(5)

and

$$\mathfrak{G}\left(z\right)=\frac{y\left(z\right)-1}{y\left(z\right)+1}=\frac{1}{2}\left[y_{1}z+\left(y_2-\frac{y_1^2}{2}\right)z^2\right]+\cdots (z\in U).$$

(6)

Substituting (5) and (6) into (3) and (4) and applying (2), we get

$$\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}=\vartheta \left(\mathfrak{F}\left(z\right)\right)=e^{\left(\frac{x\left(z\right)-1}{x\left(z\right)+1}+\tau \frac{{\left(\frac{x\left(z\right)-1}{x\left(z\right)+1}\right)}^2}{2}\right)}=1+\frac{1}{2}{x}_{1}z+\left[\frac{x_2}{2}+\frac{(\tau -1)x_1^2}{8}\right]z^2+\cdots$$

(7)

and

$$\frac{w{g}^{\prime }\left(w\right)}{g\left(w\right)}=\vartheta \left(\mathfrak{G}\left(w\right)\right)=e^{\left(\frac{y\left(w\right)-1}{y\left(w\right)+1}+\tau \frac{{\left(\frac{y\left(w\right)-1}{y\left(w\right)+1}\right)}^2}{2}\right)}=1+\frac{1}{2}{y}_{1}w+\left[\frac{y_2}{2}+\frac{(\tau -1)y_1^2}{8}\right]w^2+\cdots$$

(8)

From equating the coefficients in (7) and (8), yields

$$a_2=\frac{1}{2}{x}_1,$$

(9)

$$2a_3-a_2^2=\frac{x_2}{2}+\frac{(\tau -1)x_1^2}{8},$$

(10)

$$-a_2=\frac{1}{2}{y}_1$$

(11)

and

$$3a_2^2-2a_3=\frac{y_2}{2}+\frac{(\tau -1)y_1^2}{8}.$$

(12)

From the relations (9) and (11), we have

$$x_1=-y_1$$

(13)

and

$$2a_2^2=\frac{1}{4}(x_1^2+y_1^2).$$

(14)

Now we add (10) to (12) and we obtain

$$2a_2^2=\frac{1}{2}(x_2+y_2)+\frac{1}{8}(\tau -1)(x_1^2+y_1^2).$$

(15)

Substituting the value of $x_1^2+y_1^2$ from (14) in the right hand side of (15), we deduce that

$$a_2^2=\frac{x_2+y_2}{2(3-\tau )}.$$

(16)

Applying Lemma 1 for the coefficients $x_1,x_2,y_1,y_2$ in (14) and (16), we get

$$|a_2|\leqq 1,|a_2|\leqq \sqrt{\frac{2}{\left|3-\tau \right|}},$$

which gives the estimates of $|a_2|$.

Furthermore, in order to find the bound on $|a_3|$, we subtract (12) from (10) and also applying (13), we obtain $x_1^2=y_1^2$, hence

$$4(a_3-a_2^2)=\frac{1}{2}(x_2-y_2),$$

(17)

then by substituting the value of $a_2^2$ from (14) into (17), gives

$$a_3=\frac{x_1^2+y_1^2+x_2-y_2}{8}.$$

We have

$$|a_3|\leqq \frac{3}{2}.$$

In addition, substituting the value of $a_2^2$ from (15) into (17), we get

$$a_3=\frac{x_2-y_2}{8}+\frac{x_2+y_2}{4}+\frac{(\tau -1)(x_1^2+y_1^2)}{16}$$

and we have

$$|a_3|\leqq \frac{\tau +2}{2},$$

which gives us the desired estimates for the coefficient $|a_3|$. □

Theorem 2.

Let f given by (1) be in the family ${\mathcal{C}}_E\left(\vartheta \right)$. Then

$$|a_2|\leqq min\left\{\frac{1}{4},\sqrt{\frac{1}{2\left|2-\tau \right|}}\right\}$$

and

$$|a_3|\leqq min\left\{\frac{3\tau +5}{12},\frac{5}{12}\right\}.$$

Proof. 

Let $f\in {\mathcal{C}}_E\left(\vartheta \right)$ and $g=f^{-1}$. Then there are holomorphic functions $\mathfrak{F},\mathfrak{G}:U⟶U$ such that

$$1+\frac{z{f}^{″}\left(z\right)}{f^{\prime }\left(z\right)}=\vartheta \left(\mathfrak{F}\left(z\right)\right),z\in U$$

(18)

and

$$1+\frac{w{g}^{″}\left(w\right)}{g^{\prime }\left(w\right)}=\vartheta \left(\mathfrak{G}\left(w\right)\right),w\in U,$$

(19)

where $\mathfrak{F}\left(z\right)$ and $\mathfrak{G}\left(z\right)$ have the forms (5) and (6). From (18), (19) and (2), we deduce that

$$1+\frac{z{f}^{″}\left(z\right)}{f^{\prime }\left(z\right)}=\vartheta \left(\mathfrak{F}\left(z\right)\right)=e^{\left(\frac{x\left(z\right)-1}{x\left(z\right)+1}+\tau \frac{{\left(\frac{x\left(z\right)-1}{x\left(z\right)+1}\right)}^2}{2}\right)}=1+\frac{1}{2}{x}_{1}z+\left[\frac{x_2}{2}+\frac{(\tau -1)x_1^2}{8}\right]z^2+\cdots$$

(20)

and

$$1+\frac{w{g}^{″}\left(w\right)}{g^{\prime }\left(w\right)}=\vartheta \left(\mathfrak{G}\left(w\right)\right)=e^{\left(\frac{y\left(w\right)-1}{y\left(w\right)+1}+\tau \frac{{\left(\frac{y\left(w\right)-1}{y\left(w\right)+1}\right)}^2}{2}\right)}=1+\frac{1}{2}{y}_{1}w+\left[\frac{y_2}{2}+\frac{(\tau -1)y_1^2}{8}\right]w^2+\cdots$$

(21)

Equating the coefficients in (20) and (21) yields

$$2a_2=\frac{1}{2}{x}_1,$$

(22)

$$6a_3-4a_2^2=\frac{x_2}{2}+\frac{(\tau -1)x_1^2}{8},$$

(23)

$$-2a_2=\frac{1}{2}{y}_1$$

(24)

and

$$8a_2^2-6a_3=\frac{y_2}{2}+\frac{(\tau -1)y_1^2}{8}.$$

(25)

By the relations (22) and (24), we have

$$x_1=-y_1$$

(26)

and

$$8a_2^2=\frac{1}{4}(x_1^2+y_1^2).$$

(27)

If we add (23) to (25), we obtain

$$4a_2^2=\frac{1}{2}(x_2+y_2)+\frac{1}{8}(\tau -1)(x_1^2+y_1^2).$$

(28)

Substituting the value of $x_1^2+y_1^2$ from (27) in the right hand side of (28), we deduce that

$$a_2^2=\frac{x_2+y_2}{8(2-\tau )}.$$

(29)

Applying Lemma 1 for the coefficients $x_1,x_2,y_1,y_2$ in (27) and (29), we get

$$|a_2|\leqq \frac{1}{4},\left|a_2\right|\leqq \sqrt{\frac{1}{2(2-\tau )}},$$

which gives the estimates of $|a_2|$.

Furthermore, in order to find the bound on $|a_3|$, we subtract (25) from (23) and also applying (26), we obtain $x_1^2=y_1^2$, hence

$$12(a_3-a_2^2)=\frac{1}{2}(x_2-y_2),$$

(30)

then by substituting the value of $a_2^2$ from (27) into (30), gives

$$a_3=\frac{x_2-y_2}{24}+\frac{x_1^2+y_1^2}{32}.$$

We obtain

$$|a_3|\leqq \frac{5}{12}.$$

Substituting the value of $a_2^2$ from (28) into (30), we get

$$a_3=\frac{x_2-y_2}{24}+\frac{x_2+y_2}{8}+\frac{(\tau -1)(x_1^2+y_1^2)}{32}$$

and we have

$$|a_3|\leqq \frac{3\tau +5}{12},$$

which gives us the desired estimates for the coefficient $|a_3|$. □

In the next theorems, we provide the Fekete–Szegö type inequalities for the functions of the families ${\mathcal{S}}_E^{*}\left(\vartheta \right)$ and ${\mathcal{C}}_E\left(\vartheta \right)$.

Theorem 3.

For $\eta \in \mathbb{R}$, let $f\in {\mathcal{S}}_E^{*}\left(\vartheta \right)$ be of the form (1). Then

$$\left|a_3-\eta a_2^2\right|\leqq \left\{\begin{array}{c}\frac{1}{2};\left|\eta -1\right|\leqq \frac{\left|3-\tau \right|}{4},\hfill \\ \\ \frac{2\left|\eta -1\right|}{\left|3-\tau \right|};\left|\eta -1\right|\geqq \frac{\left|3-\tau \right|}{4}.\hfill \end{array}\right.$$

Proof. 

It follows from (16) and (17) that

$$\begin{array}{cc}\hfill a_3-\eta a_2^2& =\frac{x_2-y_2}{8}+\left(1-\eta \right)a_2^2\hfill \\ \hfill & =\frac{x_2-y_2}{8}+\frac{(x_2+y_2)\left(1-\eta \right)}{2(3-\tau )}\hfill \\ \hfill & =\frac{1}{2}\left[\left(\frac{1-\eta }{3-\tau }+\frac{1}{4}\right)x_2+\left(\frac{1-\eta }{3-\tau }-\frac{1}{4}\right)y_2\right].\hfill \end{array}$$

According to Lemma 1 and (2), we find that

$$\left|a_3-\eta a_2^2\right|\leqq \left\{\begin{array}{c}\frac{1}{2},0\leqq \left|\frac{1-\eta }{3-\tau }\right|\leqq \frac{1}{4},\hfill \\ \\ \left|\frac{2(1-\eta )}{3-\tau }\right|,\left|\frac{1-\eta }{3-\tau }\right|\geqq \frac{1}{4}.\hfill \end{array}\right.$$

After some computations, we obtain

$$\left|a_3-\eta a_2^2\right|\leqq \left\{\begin{array}{c}\frac{1}{2};\left|\eta -1\right|\leqq \frac{\left|3-\tau \right|}{4},\hfill \\ \\ \frac{2\left|\eta -1\right|}{\left|3-\tau \right|};\left|\eta -1\right|\geqq \frac{\left|3-\tau \right|}{4}.\hfill \end{array}\right.$$

□

Putting $\eta =1$ in Theorem 3, we obtain the following result:

Corollary 1.

If $f\in {\mathcal{S}}_E^{*}\left(\vartheta \right)$ be of the form (1), then

$$\left|a_3-a_2^2\right|\leqq \frac{1}{2}.$$

Theorem 4.

For $\eta \in \mathbb{R}$, let $f\in {\mathcal{C}}_E\left(\vartheta \right)$ be of the form (1). Then

$$\left|a_3-\eta a_2^2\right|\leqq \left\{\begin{array}{c}\frac{1}{6};\left|\eta -1\right|\leqq \frac{\left|2-\tau \right|}{3},\hfill \\ \\ \frac{\left|\eta -1\right|}{2\left|2-\tau \right|};\left|\eta -1\right|\geqq \frac{\left|2-\tau \right|}{3}.\hfill \end{array}\right.$$

Proof. 

It follows from (29) and (30) that

$$\begin{array}{cc}\hfill a_3-\eta a_2^2& =\frac{x_2-y_2}{24}+\left(1-\eta \right)a_2^2\hfill \\ \hfill & =\frac{x_2-y_2}{24}+\frac{(x_2+y_2)\left(1-\eta \right)}{8(2-\tau )}\hfill \\ \hfill & =\frac{1}{8}\left[\left(\frac{1-\eta }{2-\tau }+\frac{1}{3}\right)x_2+\left(\frac{1-\eta }{2-\tau }-\frac{1}{3}\right)y_2\right].\hfill \end{array}$$

According to Lemma 1 and (2), we find that

$$\left|a_3-\eta a_2^2\right|\leqq \left\{\begin{array}{c}\frac{1}{6},0\leqq \left|\frac{1-\eta }{2-\tau }\right|\leqq \frac{1}{3},\hfill \\ \\ \left|\frac{1-\eta }{2(2-\tau )}\right|,\left|\frac{1-\eta }{2-\tau }\right|\geqq \frac{1}{3}.\hfill \end{array}\right.$$

After some computations, we obtain

$$\left|a_3-\eta a_2^2\right|\leqq \left\{\begin{array}{c}\frac{1}{6};\left|\eta -1\right|\leqq \frac{\left|2-\tau \right|}{3},\hfill \\ \\ \frac{\left|\eta -1\right|}{2\left|2-\tau \right|};\left|\eta -1\right|\geqq \frac{\left|2-\tau \right|}{3}.\hfill \end{array}\right.$$

□

Putting $\eta =1$ in Theorem 4, we obtain the following result:

Corollary 2.

If the function $f\in {\mathcal{C}}_E\left(\vartheta \right)$ is of the form (1), then

$$\left|a_3-a_2^2\right|\leqq \frac{1}{6}.$$

3. Conclusions

The fact that we can find many unique and effective uses of a large variety of specific polynomials in geometric function theory provided the primary inspiration for the analysis in this article. The primary objective was to create certain families of holomorphic and bi-univalent functions. We generated Taylor–Maclaurin coefficient inequalities for functions in these families and considered the famous Fekete–Szegö type inequalities. In the future, the contents of the paper may be generalized to other operators and stimulate further research related to other families.