On the Fekete-Szeg\"o problem associated with Libera type close-to-convex functions

TL;DR

This paper introduces a new subclass of close-to-convex functions and derives Fekete-Szeg"o inequalities for them, expanding the understanding of their geometric properties and relationships to known classes.

Contribution

It presents a novel subclass of close-to-convex functions and establishes new Fekete-Szeg"o inequalities using an innovative approach.

Findings

Derived new Fekete-Szeg"o inequalities for the subclass

Unified several known results as special cases

Enhanced understanding of close-to-convex function properties

Abstract

The main purpose of this paper is to introduce a new comprehensive subclass of analytic close-to-convex functions and derive Fekete-Szeg\"o inequalities for functions belonging to this new class by using a different way. Various known special cases of our results are also pointed out.

Equations

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

$$\left|a_{3}-\mu a_{2}^{2}\right|\leq\left\{\begin{array}[]{lll}3-4\mu&,&\mu\leq 0\\ 1+2\exp\left(\frac{-2\mu}{1-\mu}\right)&,&0\leq\mu\leq 1\\ 4\mu-3&,&\mu\geq 1\end{array}\right.$$

$${\varphi }_{\mu }\left(f\right)=a_3-\mu a_2^2=\frac{1}{6}\left(f^{\prime \prime \prime }\left(0\right)-\frac{3\mu }{2}{\left(f^{\prime \prime }\left(0\right)\right)}^2\right)$$

$${\varphi }_{\mu }\left(e^{-i\theta }f\left(e^{i\theta }z\right)\right)=e^{2i\theta }{\varphi }_{\mu }\left(f\right)\left(\theta \in \mathbb{R}\right).$$

$${\varphi }_0\left(f\right)=a_3,$$

$${\varphi }_1\left(f\right)=a_3-a_2^2$$

$$S_f\left(z\right)={\left(\frac{f^{\prime \prime }\left(z\right)}{f^{\prime }\left(z\right)}\right)}^{\prime }-\frac{1}{2}{\left(\frac{f^{\prime \prime }\left(z\right)}{f^{\prime }\left(z\right)}\right)}^2.$$

$$\Re \left(\frac{z{f}^{\prime }(z)}{f(z)}\right)>\beta \left(z\in \mathbb{U}\right).$$

$$\Re \left(\frac{z{f}^{\prime }(z)}{g(z)}\right)>\alpha \left(z\in \mathbb{U}\right)$$

$$\left|a_3-\mu a_2^2\right|\le \left(1-\beta \right)\max \left\{1,\left|3-2\beta -4\mu \left(1-\beta \right)\right|\right\}.$$

$$\left|a_{3}-\mu a_{2}^{2}\right|\leq\left\{\begin{array}[]{lll}3-4\mu&,&\mu\leq\frac{1}{3}\\ &&\\ \frac{1}{3}-\frac{4}{9\mu}&,&\frac{1}{3}\leq\mu\leq\frac{2}{3}\\ &&\\ 1&,&\frac{2}{3}\leq\mu\leq 1\\ &&\\ 4\mu-3&,&\mu\geq 1\end{array}\right..$$

$$3\left|a_{3}-\mu a_{2}^{2}\right|\leq\left\{\begin{array}[]{lll}\left(3-2\beta\right)\left(3-2\alpha-\beta\right)-3\mu\left(2-\alpha-\beta\right)^{2}&,&\mu\leq\frac{2\left(1-\beta\right)}{3\left(2-\alpha-\beta\right)}\\ &&\\ 1-2\alpha+\beta\left(3-2\beta\right)+\frac{4}{3\mu}\left(1-\beta\right)^{2}&,&\frac{2\left(1-\beta\right)}{3\left(2-\alpha-\beta\right)}\leq\mu\leq\frac{2}{3}\\ &&\\ 3-2\alpha-\beta&,&\frac{2}{3}\leq\mu\leq\frac{2\left(2-\beta\right)\left(3-2\alpha-\beta\right)}{3\left(2-\alpha-\beta\right)^{2}}\\ &&\\ \left(2\beta-3\right)\left(3-2\alpha-\beta\right)+3\mu\left(2-\alpha-\beta\right)^{2}&,&\mu\geq\frac{2\left(2-\beta\right)\left(3-2\alpha-\beta\right)}{3\left(2-\alpha-\beta\right)^{2}}\end{array}\right..$$

$$\Re \left(\frac{z{f}^{\prime }\left(z\right)+\lambda z^2{f}^{\prime \prime }\left(z\right)}{g\left(z\right)}\right)>\alpha \left(z\in \mathbb{U}\right).$$

$$\Re \left(\frac{z{f}^{\prime }\left(z\right)+\left(\lambda -\delta +2\lambda \delta \right)z^2{f}^{\prime \prime }\left(z\right)+\lambda \delta z^3{f}^{\prime \prime \prime }(z)}{g\left(z\right)}\right)>\alpha \left(z\in \mathbb{U}\right),$$

$$\Re \left(\frac{z{f}^{\prime }\left(z\right)+\lambda z^2{f}^{\prime \prime }\left(z\right)}{g\left(z\right)}\right)>0\left(z\in \mathbb{U}\right),$$

$$p(0)=1\text{and}\Re \left(p\left(z\right)\right)>0.$$

$$\left|c_n\right|\le 2\left(n\ge 1\right).$$

$$\left|c_2-\nu c_1^2\right|\le 2\max \left\{1,\left|2\nu -1\right|\right\},$$

$$p\left(z\right)=\frac{1+z^2}{1-z^2}\text{and}p\left(z\right)=\frac{1+z}{1-z}.$$

$$g(z)=z+\sum _{k=2}^{\infty }{b}_k{z}^k\left(z\in \mathbb{U}\right)$$

$$\left|b_2\right|\le 2\left(1-\beta \right)$$

$$\left|b_3\right|\le \left(1-\beta \right)\left(3-2\beta \right).$$

$$\left(1+\lambda -\delta +2\lambda \delta \right)\left|a_2\right|\le 2-\alpha -\beta$$

$$3\left(1+2\lambda -2\delta +6\lambda \delta \right)\left|a_3\right|\le \left(3-2\alpha -\beta \right)\left(3-2\beta \right).$$

$$\Re \left(\frac{z{f}^{\prime }\left(z\right)+\left(\lambda -\delta +2\lambda \delta \right)z^2{f}^{\prime \prime }\left(z\right)+\lambda \delta z^3{f}^{\prime \prime \prime }(z)}{g\left(z\right)}\right)>\alpha \left(z\in \mathbb{U}\right).$$

$$\frac{z{f}^{\prime }\left(z\right)+\left(\lambda -\delta +2\lambda \delta \right)z^2{f}^{\prime \prime }\left(z\right)+\lambda \delta z^3{f}^{\prime \prime \prime }(z)}{g\left(z\right)}=\alpha +\left(1-\alpha \right)p(z),$$

$$p(z)=1+c_{1}z+c_2{z}^2+\cdots \in \mathcal{P}.$$

$$\frac{z+\sum _{k=2}^{\infty }k\left[\left(1-\lambda +\delta \right)+k\left(\lambda -\delta \right)+k\left(k-1\right)\lambda \delta \right]a_k{z}^k}{z+\sum _{k=2}^{\infty }{b}_k{z}^k}=1+\left(1-\alpha \right)\sum _{k=1}^{\infty }{c}_k{z}^k.$$

$$2\left(1+\lambda -\delta +2\lambda \delta \right)a_2=b_2+\left(1-\alpha \right)c_1$$

$$3\left(1+2\lambda -2\delta +6\lambda \delta \right)a_3=b_3+\left(1-\alpha \right)b_2{c}_1+\left(1-\alpha \right)c_2.$$

Full text

1. Introduction

Let $\mathcal{A}$ denote the class of functions of the form

[TABLE]

which are analytic in the open unit disk $\mathbb{U}=\left\{z\in\mathbb{C}:\left|z\right|<1\right\}.$ Also let $\mathcal{S}$ denote the subclass of $\mathcal{A}$ consisting of univalent functions in $\mathbb{U}.$

For $f\in\mathcal{S}$ given by $\left(\ref{1.1}\right)$, Fekete and Szegö [12] proved a noticeable result that

[TABLE]

holds. The result is sharp in the sense that for each $\mu$ there is a function in the class under consideration for which equality holds.

The coefficient functional

[TABLE]

on $f\in\mathcal{A}$ represents various geometric quantities as well as in the sense that this behaves well with respect to the rotation, namely

[TABLE]

In fact, other than the simplest case when

[TABLE]

we have several important ones. For example,

[TABLE]

represents $S_{f}\left(0\right)/6,$ where $S_{f}$ denotes the Schwarzian derivative

[TABLE]

Thus it is quite natural to ask about inequalities for $\phi_{\mu}$ corresponding to subclasses of $\mathcal{S}.$ This is called Fekete-Szegö problem. Actually, many authors have considered this problem for typical classes of univalent functions (see, for instance [1, 4, 5, 6, 7, 8, 9, 10, 14, 15, 16, 17]).

A function $f\in\mathcal{A}$ is said to be starlike of order $\beta$ $\left(0\leq\beta<1\right)$ if it satisfies the inequality

[TABLE]

We denote the class which consists of all functions $f\in\mathcal{A}$ that are starlike of order $\beta$ by $\mathcal{S}^{\ast}(\beta)$. It is well-known that $\mathcal{S}^{\ast}(\beta)\subset\mathcal{S}^{\ast}(0)=\mathcal{S}^{\ast}\subset\mathcal{S}.$

Let $0\leq\alpha,\beta<1.$ A function $f\in\mathcal{A}$ is said to be close-to-convex of order $\alpha$ and type $\beta$ if there exists a function $g\in\mathcal{S}^{\ast}\left(\beta\right)$ such that the inequality

[TABLE]

holds. We denote the class which consists of all functions $f\in\mathcal{A}$ that are close-to-convex of order $\alpha$ and type $\beta$ by $\mathcal{C}(\alpha,\beta)$. This class is introduced by Libera [16].

In particular, when $\beta=0$ we have $\mathcal{C}(\alpha,0)=\mathcal{C}(\alpha)$ of close-to-convex functions of order $\alpha$, and also we get $\mathcal{C}(0,0)=\mathcal{C}$ of close-to-convex functions introduced by Kaplan [13]. It is well-known that $\mathcal{S}^{\ast}\subset\mathcal{C}\subset\mathcal{S}$.

Keogh and Merkes [14] stated the Fekete-Szegö inequalities for functions in the classes $\mathcal{S}^{\ast}\left(\beta\right)$ and $\mathcal{C}$, respectively, as follows:

Theorem 1**.**

For $0\leq\beta<1$, let $f\left(z\right)$ given by $\left(\ref{1.1}\right)$ belongs to the function class $\mathcal{S}^{\ast}(\beta).$ Then for any real number $\mu,$

[TABLE]

Theorem 2**.**

If $f\in\mathcal{C}$ and if $\mu$ is real, then

[TABLE]

For each $\mu$, there is a function in $\mathcal{C}$ such that equality holds.

Recently, Darus and Thomas [9] generalized the results of Theorem 2 for functions $f\in\mathcal{C}(\alpha,\beta)$ as the following.

Theorem 3**.**

Let $f\in\mathcal{C}(\alpha,\beta)$ and be given by $\left(\ref{1.1}\right)$. Then for $0\leq\alpha,\beta<1$,

[TABLE]

For each $\mu$, there is a function in $\mathcal{C}(\alpha,\beta)$ such that equality holds.

Al-Abbadi and Darus [3] introduced a general subclass $\mathcal{U}_{\lambda,\alpha}^{\beta}$ of close-to-convex functions as follows:

Definition 1**.**

For $0\leq\lambda<1$, $0\leq\alpha<1$ and $0\leq\beta<1$, let the function $f\in\mathcal{A}$ be given by $\left(\ref{1.1}\right)$. Then the function $f\in\mathcal{U}_{\lambda,\alpha}^{\beta}$ if and only if there exist $g\in\mathcal{S}^{\ast}(\beta)$ such that

[TABLE]

Now we define a more comprehensive class of close-to-convex functions of order $\alpha$ and type $\beta$:

Definition 2**.**

Let $0\leq\alpha<1$ and $0\leq\delta\leq\lambda\leq 1.$We denote by $\mathcal{K}_{\lambda,\delta}\left(\alpha,\beta\right)$ the class of functions $f\in\mathcal{A}$ satisfying

[TABLE]

where $g\in\mathcal{S}^{\ast}(\beta);\;0\leq\beta<1.$

Remark 1**.**

$(i)$* For $\delta=0,$ the class $\mathcal{K}_{\lambda,\delta}\left(\alpha,\beta\right)$ reduces to the class $\mathcal{U}_{\lambda,\alpha}^{\beta}$ defined in Definition 1.*

$(ii)$* For $\delta=0$ and $\alpha=0,$ the class $\mathcal{K}_{\lambda,\delta}\left(\alpha,\beta\right)$ reduces to the class $\mathcal{U}_{\lambda}^{\beta}$ which consists of functions $f\in\mathcal{A}$ satisfying*

[TABLE]

where $g\in\mathcal{S}^{\ast}(\beta);\;0\leq\beta<1;\;0\leq\lambda\leq 1.$ This class introduced and studied by Al-Abbadi and Darus **[2]**.

$(iii)$* For $\delta=0$ and $\lambda=0,$ the class $\mathcal{K}_{\lambda,\delta}\left(\alpha,\beta\right)$ reduces to the class $\mathcal{C}(\alpha,\beta).$*

2. Preliminary Results

We denote by $\mathcal{P}$ a class of the analytic functions in $\mathbb{U}$ with

[TABLE]

We shall require the following lemmas.

Lemma 1**.**

[11]** Let $p\in\mathcal{P}$ with $p\left(z\right)=1+c_{1}z+c_{2}z^{2}+\cdots.$ Then

[TABLE]

Lemma 2**.**

[17]** Let $p\in\mathcal{P}$ with $p\left(z\right)=1+c_{1}z+c_{2}z^{2}+\cdots.$ Then for any complex number $\nu$

[TABLE]

and the result is sharp for the functions given by

[TABLE]

Lemma 3**.**

[18]** Let the function $g$ defined by

[TABLE]

belongs to the function class $\mathcal{S}^{\ast}(\beta)\,\left(0\leq\beta<1\right)$. Then we have

[TABLE]

and

[TABLE]

Lemma 4**.**

Let the function $f$ given by $\left(\ref{1.1}\right)$ belongs to the function class $\mathcal{K}_{\lambda,\delta}\left(\alpha,\beta\right).$ Then

[TABLE]

and

[TABLE]

Proof.

Let the function $f\in\mathcal{K}_{\lambda,\delta}\left(\alpha,\beta\right)$ be of the form $\left(\ref{1.1}\right)$. Therefore, there exists a function $g\in\mathcal{S}^{\ast}(\beta)$, defined in $\left(\ref{1.2}\right)$, so that

[TABLE]

It follows from the above inequality that

[TABLE]

with

[TABLE]

The equality $\left(\ref{2.2}\right)$ implies the equality

[TABLE]

Equating coefficients of both sides, we have

[TABLE]

and

[TABLE]

Using Lemma 1 and Lemma 3 in $\left(\ref{2.4}\right)$ and $\left(\ref{2.5}\right)$, we easily get $\left(\ref{2.1}\right)$ and $\left(\ref{2.a}\right)$, respectively. ∎

Remark 2**.**

In Lemma 4, letting $\delta=0$; $\delta=0,\alpha=0$; or $\delta=\lambda=0$, we have [3, Lemma 3], [2, Lemma 2.3] and [9, Lemma 3], respectively.

3. Main Results

In this section, we begin by solving the Fekete-Szegö problem for functions belonging to the class $\mathcal{K}_{\lambda,\delta}\left(\alpha,\beta\right)$ when $\mu\in\mathbb{C}.$

Theorem 4**.**

Let $f\left(z\right)$ given by $\left(\ref{1.1}\right)$ belongs to the function class $\mathcal{K}_{\lambda,\delta}\left(\alpha,\beta\right).$ Then, for any complex number $\mu,$

[TABLE]

where

[TABLE]

Proof.

Let the function $f\in\mathcal{K}_{\lambda,\delta}\left(\alpha,\beta\right)$ be of the form $\left(\ref{1.1}\right)$. For the simplicity, we set

[TABLE]

From $\left(\ref{2.4}\right)$ and $\left(\ref{2.5}\right)$, we obtain

[TABLE]

So we have

[TABLE]

Hence, by means of Theorem 1 and Lemmas 1-3, we get desired result. ∎

Letting $\delta=0$ in Theorem 4, we get following consequence.

Corollary 1**.**

Let $f\left(z\right)$ given by $\left(\ref{1.1}\right)$ belongs to the function class $\mathcal{U}_{\lambda,\alpha}^{\beta}.$ Then, for any complex number $\mu,$

[TABLE]

where

[TABLE]

Letting $\delta=0$ and $\alpha=0$ in Theorem 4, we get following consequence.

Corollary 2**.**

Let $f\left(z\right)$ given by $\left(\ref{1.1}\right)$ belongs to the function class $\mathcal{U}_{\lambda}^{\beta}.$ Then, for any complex number $\mu,$

[TABLE]

where $\Psi_{\lambda}$ is defined by $\left(\ref{psi}\right)$.

Letting $\delta=0$ and $\lambda=0$ in Theorem 4, we get following consequence.

Corollary 3**.**

Let $f\left(z\right)$ given by $\left(\ref{1.1}\right)$ belongs to the function class $\mathcal{C}(\alpha,\beta).$ Then, for any complex number $\mu,$

[TABLE]

Now we prove our main result when $\mu$ is real.

Theorem 5**.**

Let $f\left(z\right)$ given by $\left(\ref{1.1}\right)$ belongs to the function class $\mathcal{K}_{\lambda,\delta}\left(\alpha,\beta\right).$ Then

[TABLE]

For each $\mu$, there is a function in $\mathcal{K}_{\lambda,\delta}\left(\alpha,\beta\right)$ such that equality holds.

Proof.

Let $f\in\mathcal{K}_{\lambda,\delta}\left(\alpha,\beta\right)$ be given by $\left(\ref{1.1}\right)$ and let us define the function

[TABLE]

Then it is worthy to note that the condition $\left(\ref{def.1}\right)$ is equal to

[TABLE]

Since $g\in\mathcal{S}^{\ast}(\beta)$, by the definition of close-to-convex function of order $\alpha$ and type $\beta$, we deduce that $F_{\lambda,\delta}\in\mathcal{C}(\alpha,\beta).$ By the definition of close-to-convex function class $\mathcal{C}(\alpha,\beta)$, there exists two functions $p,q\in\mathcal{P}$ with

[TABLE]

and

[TABLE]

such that

[TABLE]

We assume that the function $F_{\lambda,\delta}$ is of the form

[TABLE]

By Theorem 3, $F_{\lambda,\delta}\in\mathcal{C}(\alpha,\beta)$ implies that

[TABLE]

Now equating the coefficients of $\left(\ref{4.a}\right)$ and $\left(\ref{4.b}\right)$, we obtain

[TABLE]

where $\tau$ and $\sigma$ defined by $\left(\ref{3.2}\right)$. Hence we get from the above equalities that

[TABLE]

Taking $\rho=\mu\sigma/\tau^{2}$ in $\left(\ref{4.1}\right)$, we get desired estimate $\left(\ref{4.x}\right)$.

Finally, sharpness of the results in $\left(\ref{4.x}\right)$ is getting by

(i) in Case 1: upon choosing

[TABLE]

(ii) in Case 2: upon choosing

[TABLE]

(iii) in Case 3: upon choosing

[TABLE]

(iv) in Case 4: upon choosing

[TABLE]

∎

Remark 3**.**

Note that our method proves easily the theorems given by [3, Theorem 1] and [2, Theorem 3.1] with a different way.

Corollary 4**.**

In Theorem 5, letting $\delta=0$; $\delta=0,\alpha=0$; $\delta=\lambda=0$; or $\delta=\lambda=0,\alpha=\beta=0$, we have [3, Theorem 1], [2, Theorem 3.1], Theorem 3 and Theorem 2, respectively.