Lobachevskii Journal of Mathematics http://ljm.ksu.ru ISSN 1818-9962 Vol. 21, 2006, 73–83

© Zhi-Gang Wang

Abstract. In the present paper, the author introduce two new subclasses ${\mathcal{𝒞}}^{\left(k\right)}\left(\alpha ,\beta ,\gamma \right)$ of close-to-convex functions and ${\mathcal{𝒬}C}^{\left(k\right)}\left(\alpha ,\beta ,\gamma \right)$ of quasi-convex functions with respect to $k$-symmetric points. The sufficient conditions and integral representations for functions belonging to these classes are provided, the inclusion relationships and convolution conditions for these classes are also provided.

1. Introduction

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

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

We denote by ${\mathcal{𝒮}}^{\ast }$, $\mathcal{𝒦}$, $\mathcal{𝒞}$ and $\mathcal{𝒬}C$ the familiar subclasses of $\mathcal{𝒜}$ consisting of functions which are, respectively, starlike, convex, close-to-convex and quasi-convex in $\mathcal{𝒰}$. Thus, by definition, we have (see, for details, [4, 8]; see also [11, 12])

Let $f\left(z\right)$ and $F\left(z\right)$ be analytic in $\mathcal{𝒰}$. Then we say that the function $f\left(z\right)$ is subordinate to $F\left(z\right)$ in $\mathcal{𝒰}$, if there exists an analytic function $\omega \left(z\right)$ in $\mathcal{𝒰}$ such that $\left|\omega \left(z\right)\right|\le \left|z\right|$ and $f\left(z\right)=F\left(\omega \left(z\right)\right)$, denoted by $f\prec F$ or $f\left(z\right)\prec F\left(z\right)$. If $F\left(z\right)$ is univalent in $\mathcal{𝒰}$, then the subordination is equivalent to $f\left(0\right)=F\left(0\right)$ and $f\left(\mathcal{𝒰}\right)\subset F\left(\mathcal{𝒰}\right)$ (see [1]).

Sakaguchi [9] once introduced a class ${\mathcal{𝒮}}_s^{\ast }$ of functions starlike with respect to symmetric points, it consists of functions $f\left(z\right)\in \mathcal{𝒮}$ satisfying

Following him, many authors discussed this class and its subclasses. And let ${\mathcal{𝒞}}_s^{\ast }$ denote the class of functions in $\mathcal{𝒮}$ convex with respect to symmetric points, which satisfy the following inequality

Let ${\mathcal{𝒮}}_s^{\left(k\right)}\left(\alpha \right)$ denote the class of functions in $\mathcal{𝒮}$ satisfying the following inequality

where $0\le \alpha <1,k\ge 1$ is a fixed positive integer and $f_k\left(z\right)$ is defined by the following equality

And let ${\mathcal{𝒞}}_s^{\left(k\right)}\left(\alpha \right)$ denote the class of functions in $\mathcal{𝒮}$ satisfying the following inequality

where $0\le \alpha <1,k\ge 1$ is a fixed positive integer and $f_k\left(z\right)$ is defined by equality (1.2). The class ${\mathcal{𝒮}}_s^{\left(k\right)}\left(\alpha \right)$ of functions starlike with respect to $k$-symmetric points of order $\alpha$ was studied by Sokół [5, 6] and Stankiewicz [7].

Motivated by ${\mathcal{𝒮}}_s^{\left(k\right)}\left(\alpha \right)$ and ${\mathcal{𝒞}}_s^{\left(k\right)}\left(\alpha \right)$, we introduce and investigate the following two more generalized subclasses ${\mathcal{𝒞}}^{\left(k\right)}\left(\alpha ,\beta ,\gamma \right)$ and ${\mathcal{𝒬}C}^{\left(k\right)}\left(\alpha ,\beta ,\gamma \right)$ of $\mathcal{𝒮}$ with respect to $k$-symmetric points, and obtain some interesting results.

Definition 1. Let ${\mathcal{𝒞}}^{\left(k\right)}\left(\alpha ,\beta ,\gamma \right)$ denote the class of functions in $\mathcal{𝒮}$ satisfying the following inequality

where $0\le \alpha <1,0\le \beta \le 1,0\le \gamma <1,k\ge 1$ is a fixed positive integer and $f_k\left(z\right)$ is defined by equality (1.2).

If $\beta =0$ and $\gamma =0$, then the class ${\mathcal{𝒞}}^{\left(k\right)}\left(\alpha ,\beta ,\gamma \right)$ reduces to the class ${\mathcal{𝒮}}_s^{\left(k\right)}\left(\alpha \right)$ [5, 6, 7]. If $\alpha =0,\beta =0,\gamma =0$ and $k=2$, then the class ${\mathcal{𝒞}}^{\left(k\right)}\left(\alpha ,\beta ,\gamma \right)$ reduces to the class ${\mathcal{𝒮}}_s^{\ast }$ [9].

Definition 2. Let ${\mathcal{𝒬}C}^{\left(k\right)}\left(\alpha ,\beta ,\gamma \right)$ denote the class of functions in $\mathcal{𝒮}$ satisfying the following inequality

where $0\le \alpha <1,0\le \beta \le 1,0\le \gamma <1,k\ge 1$ is a fixed positive integer and $f_k\left(z\right)$ is defined by equality (1.2).

If $\beta =0$ and $\gamma =0$, then the class ${\mathcal{𝒬}C}^{\left(k\right)}\left(\alpha ,\beta ,\gamma \right)$ reduces to the class ${\mathcal{𝒞}}_s^{\left(k\right)}\left(\alpha \right)$. If $\alpha =0,\beta =0,\gamma =0$ and $k=2$, then the class ${\mathcal{𝒬}C}^{\left(k\right)}\left(\alpha ,\beta ,\gamma \right)$ reduces to the class ${\mathcal{𝒞}}_s^{\ast }$.

In our investigation of the classes ${\mathcal{𝒞}}^{\left(k\right)}\left(\alpha ,\beta ,\gamma \right)$ and ${\mathcal{𝒬}C}^{\left(k\right)}\left(\alpha ,\beta ,\gamma \right)$, we shall also make use of the following definition and lemmas.

Definition 3 (Hadamard Product or Convolution). Given two functions $f,g\in \mathcal{𝒜}$, where $f\left(z\right)$ is given by (1.1) and $g\left(z\right)$ is defined by

the Hadamard product (or convolution) $f\ast g$ is defined (as usual) by

Lemma 1 [2]. Let $H\left(z\right)=1+{\sum }_{n=1}^{\infty }{h}_n{z}^n$ be analytic in $\mathcal{𝒰}$, $0\le \alpha <1,0\le \beta \le 1$ and $0\le \gamma <1$, then the condition

is equivalent to

Lemma 2. Let $f\left(z\right)\in {\mathcal{𝒞}}^{\left(k\right)}\left(0,0,0\right)$, then we have $f_k\left(z\right)\in {\mathcal{𝒮}}^{\ast }\subset \mathcal{𝒮}.$

This lemma is a special case of Theorem 2 in [3].

Lemma 3. Let $f\left(z\right)\in {\mathcal{𝒬}C}^{\left(k\right)}\left(0,0,0\right)$, then we have $f_k\left(z\right)\in \mathcal{𝒦}\subset {\mathcal{𝒮}}^{\ast }.$

This lemma is a special case of Theorem 2 in [13].

Lemma 4 [10]. Let $-1\le B_2\le B_1<A_1\le A_2\le 1$, then we have

In the present paper, we shall provide the sufficient conditions and integral representations for functions belonging to the classes ${\mathcal{𝒞}}^{\left(k\right)}\left(\alpha ,\beta ,\gamma \right)$ and ${\mathcal{𝒬}C}^{\left(k\right)}\left(\alpha ,\beta ,\gamma \right)$, we shall also provide the inclusion relationships and convolution conditions for these classes.

2. Inclusion Relationships

First we give some inclusion relationships for the classes ${\mathcal{𝒞}}^{\left(k\right)}\left(\alpha ,\beta ,\gamma \right)$ and ${\mathcal{𝒬}C}^{\left(k\right)}\left(\alpha ,\beta ,\gamma \right)$.

Theorem 1. Let $0\le \alpha <1,0\le \beta \le 1$ and $0\le \gamma <1$, then we have

Proof. Suppose that $f\left(z\right)\in {\mathcal{𝒞}}^{\left(k\right)}\left(\alpha ,\beta ,\gamma \right)$, note that $H\left(z\right)=z{f}^{\prime }\left(z\right)∕f_k\left(z\right)$ satisfying the condition of Lemma 1, from this we know that the condition (1.3) can be written as

Note that

thus, by Lemma 2, we have ${\mathcal{𝒞}}^{\left(k\right)}\left(\alpha ,\beta ,\gamma \right)\subset {\mathcal{𝒞}}^{\left(k\right)}\left(0,0,0\right)\subset \mathcal{𝒞}\subset \mathcal{𝒮}.$

By means of Lemma 3, using the similar method as in Theorem 1, we have

Corollary 1. Let $0\le \alpha <1,0\le \beta \le 1$ and $0\le \gamma <1$, then we have

Theorem 2. Let $0\le {\alpha }_1\le {\alpha }_2<1,0\le {\beta }_2\le {\beta }_1\le 1,$ and $0\le {\gamma }_1\le {\gamma }_2<1$, then we have

Proof. Suppose that $f\left(z\right)\in {\mathcal{𝒞}}^{\left(k\right)}\left({\alpha }_2,{\beta }_2,{\gamma }_2\right)$, by (2.1), we have

Since $0\le {\alpha }_1\le {\alpha }_2<1,0\le {\beta }_2\le {\beta }_1\le 1$ and $0\le {\gamma }_1\le {\gamma }_2<1$, then we have

Thus, by Lemma 4, we have

that is $f\left(z\right)\in {\mathcal{𝒞}}^{\left(k\right)}\left({\alpha }_1,{\beta }_1,{\gamma }_1\right)$. This means that

and hence the proof is complete.

For the class ${\mathcal{𝒬}C}^{\left(k\right)}\left(\alpha ,\beta ,\gamma \right)$, we have

Corollary 2. Let $0\le {\alpha }_1\le {\alpha }_2<1,0\le {\beta }_2\le {\beta }_1\le 1,$ and $0\le {\gamma }_1\le {\gamma }_2<1$, then we have

3. Sufficient Conditions

In this section, we give the sufficient conditions for functions belonging to the classes ${\mathcal{𝒞}}^{\left(k\right)}\left(\alpha ,\beta ,\gamma \right)$ and ${\mathcal{𝒬}C}^{\left(k\right)}\left(\alpha ,\beta ,\gamma \right)$.

Theorem 3. Let $f\left(z\right)=z+{\sum }_{n=2}^{\infty }{a}_n{z}^n$ be analytic in $\mathcal{𝒰}$, if for $0\le \alpha <1,0\le \beta \le 1$ and $0\le \gamma <1$, we have

then $f\left(z\right)\in {\mathcal{𝒞}}^{\left(k\right)}\left(\alpha ,\beta ,\gamma \right)$.

Proof. Suppose that $f\left(z\right)=z+{\sum }_{n=2}^{\infty }{a}_n{z}^n$, and $f_k\left(z\right)$ is defined by equality (1.2). Then for $z\in \mathcal{𝒰}$, we have

where

Thus, for $\left|z\right|=r<1$, we have

From the definition of $c_n$ we know

Substituting (3.3) into inequality (3.2), we get

From inequality (3.1) we know that $M<0$, thus we can get inequality (1.3), that is $f\left(z\right)\in {\mathcal{𝒞}}^{\left(k\right)}\left(\alpha ,\beta ,\gamma \right)$. This completes the proof of Theorem 3.

For the class ${\mathcal{𝒬}C}^{\left(k\right)}\left(\alpha ,\beta ,\gamma \right)$, we have

Corollary 3. Let $f\left(z\right)=z+{\sum }_{n=2}^{\infty }{a}_n{z}^n$ be analytic in $\mathcal{𝒰}$, if for $0\le \alpha <1,0\le \beta \le 1$ and $0\le \gamma <1$, we have

then $f\left(z\right)\in {\mathcal{𝒬}C}^{\left(k\right)}\left(\alpha ,\beta ,\gamma \right)$.

4. Integral Representations

In this section, we give the integral representations of functions belonging to the classes ${\mathcal{𝒞}}^{\left(k\right)}\left(\alpha ,\beta ,\gamma \right)$ and ${\mathcal{𝒬}C}^{\left(k\right)}\left(\alpha ,\beta ,\gamma \right)$.

Theorem 4. Let $f\left(z\right)\in {\mathcal{𝒞}}^{\left(k\right)}\left(\alpha ,\beta ,\gamma \right)$, then we have

where $f_k\left(z\right)$ is defined by equality (1.2), $\omega \left(z\right)$ is analytic in $\mathcal{𝒰}$ and $\omega \left(0\right)=0$, $\left|\omega \left(z\right)\right|<1$.

Proof. Suppose that $f\left(z\right)\in {\mathcal{𝒞}}^{\left(k\right)}\left(\alpha ,\beta ,\gamma \right)$, form (2.1) we can get

where $\omega \left(z\right)$ is analytic in $\mathcal{𝒰}$ and $\omega \left(0\right)=0$, $\left|\omega \left(z\right)\right|<1$. Substituting $z$ by ${\varepsilon }^{\mu }z$ in (4.2) respectively $\left(\mu =0,1,2,\dots ,k-1;{\varepsilon }^k=1\right)$, we have

Note that $f_k\left({\varepsilon }^{\mu }z\right)={\varepsilon }^{\mu }{f}_k\left(z\right)$, summing (4.3), we can get

from equality (4.4) we get

Integrating equality (4.5) we have

from the above equality, we can get equality (4.1) easily. Hence the proof is complete.

Theorem 5. Let $f\left(z\right)\in {\mathcal{𝒞}}^{\left(k\right)}\left(\alpha ,\beta ,\gamma \right)$, then we have

where $\omega \left(z\right)$ is analytic in $\mathcal{𝒰}$ and $\omega \left(0\right)=0$, $\left|\omega \left(z\right)\right|<1$.

Proof. Suppose that $f\left(z\right)\in {\mathcal{𝒞}}^{\left(k\right)}\left(\alpha ,\beta ,\gamma \right)$, from equalities (4.1) and (4.2) we can get

Integrating the above equality, we can get equality (4.6) easily. Hence the proof is complete.

For the class ${\mathcal{𝒬}C}^{\left(k\right)}\left(\alpha ,\beta ,\gamma \right)$, we have

Corollary 4. Let $f\left(z\right)\in {\mathcal{𝒬}C}^{\left(k\right)}\left(\alpha ,\beta ,\gamma \right)$, then we have

where $f_k\left(z\right)$ is defined by equality (1.2), $\omega \left(z\right)$ is analytic in $\mathcal{𝒰}$ and $\omega \left(0\right)=0$, $\left|\omega \left(z\right)\right|<1$.

Corollary 5. Let $f\left(z\right)\in {\mathcal{𝒬}C}^{\left(k\right)}\left(\alpha ,\beta ,\gamma \right)$, then we have

where $\omega \left(z\right)$ is analytic in $\mathcal{𝒰}$ and $\omega \left(0\right)=0$, $\left|\omega \left(z\right)\right|<1$.

5. Convolution Conditions

At last, we provide the convolution conditions for the classes ${\mathcal{𝒞}}^{\left(k\right)}\left(\alpha ,\beta ,\gamma \right)$ and ${\mathcal{𝒬}C}^{\left(k\right)}\left(\alpha ,\beta ,\gamma \right)$.

Theorem 6. A function $f\left(z\right)\in {\mathcal{𝒞}}^{\left(k\right)}\left(\alpha ,\beta ,\gamma \right)$ if and only if

for all $z\in \mathcal{𝒰}$ and $0\le \theta <2\pi$, where $h\left(z\right)$ is given by (5.6).

Proof. Suppose that $f\left(z\right)\in {\mathcal{𝒞}}^{\left(k\right)}\left(\alpha ,\beta ,\gamma \right)$, by Theorem 1, we know that the condition (1.3) can be written as (2.1), since (2.1) is equivalent to

for all $z\in \mathcal{𝒰}$ and $0\le \theta <2\pi$. It is easy to know that the condition (5.2) can be written as

On the other hand, it is well known that

And from the definition of $f_k\left(z\right)$ we know

where

for $c_n$ satisfy equality (3.3). Substituting (5.4) and (5.5) into (5.3), we can get (5.1). This completes the proof of Theorem 6.

For the class ${\mathcal{𝒬}C}^{\left(k\right)}\left(\alpha ,\beta ,\gamma \right)$, we have

Corollary 6. A function $f\left(z\right)\in {\mathcal{𝒬}C}^{\left(k\right)}\left(\alpha ,\beta ,\gamma \right)$ if and only if

for all $z\in \mathcal{𝒰}$ and $0\le \theta <2\pi$, where $h\left(z\right)$ is given by (5.6).

This work was supported by the Scientific Research Fund of Hunan Provincial Education Department and the Hunan Provincial Natural Science Foundation (No. 05JJ30013) of People’s Republic of China.

References

College of Mathematics and Computing Science, Changsha University of Science and Technology, Changsha, Hunan 410076, People’s Republic of China

E-mail address: zhigwang@163.com

Received 2006, March 16