Lobachevskii Journal of Mathematics

http://ljm.ksu.ru
ISSN 1818-9962

Vol. 21, 2006, 73–83 \begin{matrix}c \\ ◯ \end{matrix} Zhi-Gang Wang

Zhi-Gang Wang

CERTAIN SUBCLASSES OF CLOSE-TO-CONVEX AND

QUASI-CONVEX FUNCTIONS WITH RESPECT TO K-SYMMETRIC POINTS

(submitted by F. G. Avkhadiev)

Abstract. In the present paper, the author introduce two new subclasses C^{(k)}(α, β, γ) of close-to-convex functions and QC^{(k)}(α, β, γ) 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 A denote the class of functions of the form

f(z) = z + ∑_{n=2}^{∞} a_{n}z^{n},  (1.1)

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

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

2000 Mathematical Subject Classification. Primary 30C45.

Key words and phrases. Close-to-convex functions, quasi-convex functions, differential subordination, Hadamard product, k-symmetric points.

S^{∗} = \{f : f ∈ A and ℜ \{\frac{zf′(z)}{f(z)}\} > 0 (z ∈ U)\} ,
K = \{f : f ∈ A and ℜ \{1 + \frac{zf′′(z)}{f′(z)}\} > 0 (z ∈ U)\} ,
C = \{f : f ∈ A, ∃g ∈ S^{∗}, such that ℜ \{\frac{zf′(z)}{g(z)}\} > 0 (z ∈ U)\} ,

and

QC = \{f : f ∈ A, ∃g ∈ K, such that ℜ \{\frac{(zf′(z))′}{g′(z)} \} > 0 (z ∈ U)\} .

Let f(z) and F (z) be analytic in U. Then we say that the function f(z) is subordinate to F (z) in U, if there exists an analytic function ω(z) in U such that |ω(z)| ≤ |z| and f(z) = F (ω(z)), denoted by f ≺ F or f(z) ≺ F (z). If F (z) is univalent in U, then the subordination is equivalent to f(0) = F (0) and f(U) ⊂ F (U) (see [1]).

Sakaguchi [9] once introduced a class S_{s}^{∗} of functions starlike with respect to symmetric points, it consists of functions f(z) ∈ S satisfying

ℜ \{\frac{zf′(z)}{f(z) − f(−z)}\} > 0 (z ∈ U).

Following him, many authors discussed this class and its subclasses. And let C_{s}^{∗} denote the class of functions in S convex with respect to symmetric points, which satisfy the following inequality

ℜ \{\frac{(zf′(z))′}{f′(z) + f′(−z)}\} > 0 (z ∈ U).

Let S_{s}^{(k)}(α) denote the class of functions in S satisfying the following inequality

ℜ \{\frac{zf′(z)}{f_{k}(z)}\} > α (z ∈ U),

where 0 ≤ α < 1, k ≥ 1 is a fixed positive integer and f_{k}(z) is defined by the following equality

f_{k}(z) = \frac{1}{k} ∑_{ν=0}^{k−1} ε^{−ν}f(ε^{ν}z) (ε^{k} = 1; z ∈ U).  (1.2)

CLOSE-TO-CONVEX AND QUASI-CONVEX FUNCTIONS 75

And let C_{s}^{(k)}(α) denote the class of functions in S satisfying the following inequality

ℜ \{\frac{(zf′(z))′}{f′_{k}(z)} \} > α (z ∈ U),

where 0 ≤ α < 1, k ≥ 1 is a fixed positive integer and f_{k}(z) is defined by equality (1.2). The class S_{s}^{(k)}(α) of functions starlike with respect to ksymmetric points of order α was studied by Sokół [5, 6] and Stankiewicz [7].

Motivated by S_{s}^{(k)}(α) and C_{s}^{(k)}(α), we introduce and investigate the following two more generalized subclasses C^{(k)}(α, β, γ) and QC^{(k)}(α, β, γ) of S with respect to k-symmetric points, and obtain some interesting results.

Definition 1. Let C^{(k)}(α, β, γ) denote the class of functions in S satisfying the following inequality

|\frac{\frac{zf′(z)}{f_{k}(z)} − 1}{β \frac{zf′(z)}{f_{k}(z)} + (1 − γ)}| < 1 − α (z ∈ U),  (1.3)

where 0 ≤ α < 1, 0 ≤ β ≤ 1, 0 ≤ γ < 1, k ≥ 1 is a fixed positive integer and f_{k}(z) is defined by equality (1.2).

If β = 0 and γ = 0, then the class C^{(k)}(α, β, γ) reduces to the class S_{s}^{(k)}(α) [5, 6, 7]. If α = 0, β = 0, γ = 0 and k = 2, then the class C^{(k)}(α, β, γ) reduces to the class S_{s}^{∗} [9].

Definition 2. Let QC^{(k)}(α, β, γ) denote the class of functions in S satisfying the following inequality

|\frac{\frac{(zf′(z))′}{f′_{k}(z)} − 1}{β \frac{(zf′(z))′}{f′_{k}(z)} + (1 − γ)}| < 1 − α (z ∈ U),

where 0 ≤ α < 1, 0 ≤ β ≤ 1, 0 ≤ γ < 1, k ≥ 1 is a fixed positive integer and f_{k}(z) is defined by equality (1.2).

If β = 0 and γ = 0, then the class QC^{(k)}(α, β, γ) reduces to the class C_{s}^{(k)}(α). If α = 0, β = 0, γ = 0 and k = 2, then the class QC^{(k)}(α, β, γ) reduces to the class C_{s}^{∗}.

In our investigation of the classes C^{(k)}(α, β, γ) and QC^{(k)}(α, β, γ), we shall also make use of the following definition and lemmas.

Definition 3 (Hadamard Product or Convolution). Given two functions f, g ∈ A, where f(z) is given by (1.1) and g(z) is defined by

g(z) = z + ∑_{n=2}^{∞} b_{n}z^{n},

the Hadamard product (or convolution) f ∗ g is defined (as usual) by

(f ∗ g)(z) = z + ∑_{n=2}^{∞} a_{n}b_{n}z^{n} = (g ∗ f)(z).

Lemma 1 [2]. Let H(z) = 1 + ∑_{n=1}^{∞} h_{n}z^{n} be analytic in U, 0 ≤ α < 1, 0 ≤ β ≤ 1 and 0 ≤ γ < 1, then the condition

|\frac{H(z) − 1}{βH(z) + (1 − γ)}| < 1 − α (z ∈ U)

is equivalent to

H(z) ≺ \frac{1 + (1 − α)(1 − γ)z}{1 − (1 − α)βz} (z ∈ U).

Lemma 2. Let f(z) ∈ C^{(k)}(0, 0, 0), then we have f_{k}(z) ∈ S^{∗} ⊂ S.

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

Lemma 3. Let f(z) ∈ QC^{(k)}(0, 0, 0), then we have f_{k}(z) ∈ K ⊂ S^{∗}. This lemma is a special case of Theorem 2 in [13]. Lemma 4 [10]. Let −1 ≤ B_{2} ≤ B_{1} < A_{1} ≤ A_{2} ≤ 1, then we have

\frac{1 + A_{1}z}{1 + B_{1}z} ≺ \frac{1 + A_{2}z}{1 + B_{2}z} .

In the present paper, we shall provide the sufficient conditions and integral representations for functions belonging to the classes C^{(k)}(α, β, γ) and QC^{(k)}(α, β, γ), 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 C^{(k)}(α, β, γ) and QC^{(k)}(α, β, γ).

Theorem 1. Let 0 ≤ α < 1, 0 ≤ β ≤ 1 and 0 ≤ γ < 1, then we have

C^{(k)}(α, β, γ) ⊂ C^{(k)}(0, 0, 0) ⊂ C ⊂ S.

Proof. Suppose that f(z) ∈ C^{(k)}(α, β, γ), note that H(z) = zf′(z)/f_{k}(z) satisfying the condition of Lemma 1, from this we know that the condition (1.3) can be written as

\frac{zf′(z)}{f_{k}(z)} ≺ \frac{1 + (1 − α)(1 − γ)z}{1 − (1 − α)βz} (z ∈ U).  (2.1)

Note that

ℜ \{\frac{zf′(z)}{f_{k}(z)}\} = ℜ \{\frac{1 + (1 − α)(1 − γ)ω(z)}{1 − (1 − α)βω(z)}\} > \frac{1 + (1 − α)(1 − γ)}{1 − (1 − α)β} ≥ 0,

thus, by Lemma 2, we have C^{(k)}(α, β, γ) ⊂ C^{(k)}(0, 0, 0) ⊂ C ⊂ S.

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

Corollary 1. Let 0 ≤ α < 1, 0 ≤ β ≤ 1 and 0 ≤ γ < 1, then we have

QC^{(k)}(α, β, γ) ⊂ QC^{(k)}(0, 0, 0) ⊂ QC ⊂ C.

Theorem 2. Let 0 ≤ α_{1} ≤ α_{2} < 1, 0 ≤ β_{2} ≤ β_{1} ≤ 1, and 0 ≤ γ_{1} ≤ γ_{2} < 1, then we have

C^{(k)}(α_{2}, β_{2}, γ_{2}) ⊂ C^{(k)}(α_{1}, β_{1}, γ_{1}).

Proof. Suppose that f(z) ∈ C^{(k)}(α_{2}, β_{2}, γ_{2}), by (2.1), we have

\frac{zf′(z)}{f_{k}(z)} ≺ \frac{1 + (1 − α_{2})(1 − γ_{2})z}{1 − (1 − α_{2})β_{2}z} .

Since 0 ≤ α_{1} ≤ α_{2} < 1, 0 ≤ β_{2} ≤ β_{1} ≤ 1 and 0 ≤ γ_{1} ≤ γ_{2} < 1, then we have

−1 ≤ −(1−α_{1})β_{1} ≤ −(1−α_{2})β_{2} < (1−α_{2})(1−γ_{2}) ≤ (1−α_{1})(1−γ_{1}) ≤ 1.

Thus, by Lemma 4, we have

\frac{zf′(z)}{f_{k}(z)} ≺ \frac{1 + (1 − α_{2})(1 − γ_{2})z}{1 − (1 − α_{2})β_{2}z} ≺ \frac{1 + (1 − α_{1})(1 − γ_{1})z}{1 − (1 − α_{1})β_{1}z} ,

that is f(z) ∈ C^{(k)}(α_{1}, β_{1}, γ_{1}). This means that

C^{(k)}(α_{2}, β_{2}, γ_{2}) ⊂ C^{(k)}(α_{1}, β_{1}, γ_{1}),

and hence the proof is complete.

For the class QC^{(k)}(α, β, γ), we have

Corollary 2. Let 0 ≤ α_{1} ≤ α_{2} < 1, 0 ≤ β_{2} ≤ β_{1} ≤ 1, and 0 ≤ γ_{1} ≤ γ_{2} < 1, then we have

QC^{(k)}(α_{2}, β_{2}, γ_{2}) ⊂ QC^{(k)}(α_{1}, β_{1}, γ_{1}).

3. Sufficient Conditions

In this section, we give the sufficient conditions for functions belonging to the classes C^{(k)}(α, β, γ) and QC^{(k)}(α, β, γ).

Theorem 3. Let f(z) = z + ∑_{n=2}^{∞} a_{n}z^{n} be analytic in U, if for 0 ≤ α < 1, 0 ≤ β ≤ 1 and 0 ≤ γ < 1, we have

\begin{matrix}∑_{n=1}^{∞}\{nk+ & (1 − α)[(nk + 1)β + (1 − γ)]\} |a_{nk+1}| \\ + ∑_{n≠lk+1^{n=2}}^{∞} n[1 + (1 − α)β] |a_{n}| ≤ (1 − α)(1 + β − γ), \end{matrix} (3.1)

then f(z) ∈ C^{(k)}(α, β, γ).

Proof. Suppose that f(z) = z + ∑_{n=2}^{∞} a_{n}z^{n}, and f_{k}(z) is defined by equality (1.2). Then for z ∈ U, we have

M = |zf′(z) − f_{k}(z)| − (1 − α) |βzf′(z) + (1 − γ)f_{k}(z)|
= |z + ∑_{n=2}^{∞} na_{n}z^{n} − z − ∑_{n=2}^{∞} a_{n}c_{n}z^{n}|

−(1 − α) |β (z + ∑_{n=2}^{∞} na_{n}z^{n}) + (1 − γ) (z + ∑_{n=2}^{∞} a_{n}c_{n}z^{n})| ,

where

c_{n} = \frac{1}{k} ∑_{ν=0}^{k−1} ε^{(n−1)ν} (ε^{k} = 1).

Thus, for |z| = r < 1, we have

\begin{matrix}M ≤ & ∑_{n=2}^{∞}(n − c_{n}) |a_{n}| r^{n} − (1 − α)[(1 + β − γ)r \\ − ∑_{n=2}^{∞}[nβ + (1 − γ)c_{n}] |a_{n}| r^{n}] \\ < & ( ∑_{n=2}^{∞}\{(n − c_{n}) + (1 − α)[nβ \\ + (1 − γ)c_{n}]\} |a_{n}| − (1 − α)(1 + β − γ))r. \end{matrix}  (3.2)

From the definition of c_{n} we know

c_{n} = \{ \begin{matrix}1, & n = lk + 1, \\ 0, & n ≠ lk + 1. \end{matrix}  (3.3)

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

M < ∑_{n=1}^{∞}{nk + (1 − α)[(nk + 1)β + (1 − γ)]} |a_{nk+1}|

+ ∑_{n≠lk+1^{n=2}}^{∞} n[1 + (1 − α)β] |a_{n}| − (1 − α)(1 + β − γ).

From inequality (3.1) we know that M < 0, thus we can get inequality (1.3), that is f(z) ∈ C^{(k)}(α, β, γ). This completes the proof of Theorem 3.

For the class QC^{(k)}(α, β, γ), we have

Corollary 3. Let f(z) = z + ∑_{n=2}^{∞} a_{n}z^{n} be analytic in U, if for 0 ≤ α < 1, 0 ≤ β ≤ 1 and 0 ≤ γ < 1, we have

∑_{n=1}^{∞}(nk + 1){nk + (1 − α)[(nk + 1)β + (1 − γ)]} |a_{nk+1}|

+ ∑_{n≠lk+1^{n=2}}^{∞} n^{2}[1 + (1 − α)β] |a_{n}| ≤ (1 − α)(1 + β − γ),

then f(z) ∈ QC^{(k)}(α, β, γ).

4. Integral Representations

In this section, we give the integral representations of functions belonging to the classes C^{(k)}(α, β, γ) and QC^{(k)}(α, β, γ).

Theorem 4. Let f(z) ∈ C^{(k)}(α, β, γ), then we have

f_{k}(z) = z · exp \{\frac{1}{k} ∑_{μ=0}^{k−1} ∫_{0}^{ε^{μ}z} \frac{(1 − α)(1 + β − γ)ω(t)}{t[1 − (1 − α)βω(t)]}dt\} ,  (4.1)

where f_{k}(z) is defined by equality (1.2), ω(z) is analytic in U and ω(0) = 0, |ω(z)| < 1.

Proof. Suppose that f(z) ∈ C^{(k)}(α, β, γ), form (2.1) we can get

\frac{zf′(z)}{f_{k}(z)} = \frac{1 + (1 − α)(1 − γ)ω(z)}{1 − (1 − α)βω(z)},  (4.2)

where ω(z) is analytic in U and ω(0) = 0, |ω(z)| < 1. Substituting z by ε^{μ}z in (4.2) respectively (μ = 0, 1, 2, …, k − 1; ε^{k} = 1), we have

\frac{ε^{μ}zf′(ε^{μ}z)}{f_{k}(ε^{μ}z)} = \frac{1 + (1 − α)(1 − γ)ω(ε^{μ}z)}{1 − (1 − α)βω(ε^{μ}z)} (μ = 0, 1, 2, …, k − 1). (4.3)

Note that f_{k}(ε^{μ}z) = ε^{μ}f_{k}(z), summing (4.3), we can get

\frac{zf′_{k}(z)}{f_{k}(z)} = \frac{1}{k} ∑_{μ=0}^{k−1} \frac{1 + (1 − α)(1 − γ)ω(ε^{μ}z)}{1 − (1 − α)βω(ε^{μ}z)} (μ = 0, 1, 2, …, k − 1), (4.4)

from equality (4.4) we get

\frac{f′_{k}(z)}{f_{k}(z)} − \frac{1}{z} = \frac{1}{k} ∑_{μ=0}^{k−1} \frac{(1 − α)(1 + β − γ)ω(ε^{μ}z)}{z[1 − (1 − α)βω(ε^{μ}z)]}.  (4.5)

Integrating equality (4.5) we have

log \{\frac{f_{k}(z)}{z}\} = \frac{1}{k} ∑_{μ=0}^{k−1} ∫_{0}^{z} \frac{(1 − α)(1 + β − γ)ω(ε^{μ}ζ)}{ζ[1 − (1 − α)βω(ε^{μ}ζ)]}dζ

= \frac{1}{k} ∑_{μ=0}^{k−1} ∫_{0}^{ε^{μ}z} \frac{(1 − α)(1 + β − γ)ω(t)}{t[1 − (1 − α)βω(t)]}dt,

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

Theorem 5. Let f(z) ∈ C^{(k)}(α, β, γ), then we have

\begin{matrix}f(z) = ∫_{0}^{z} exp \{\frac{1}{k} ∑_{μ=0}^{k−1} ∫_{0}^{ε^{μ}ζ} \frac{(1 − α)(1 + β − γ)ω(t)}{t[1 − (1 − α)βω(t)]}dt\} \\ ·\frac{1 + (1 − α)(1 − γ)ω(ζ)}{1 − (1 − α)βω(ζ)}dζ, \end{matrix}  (4.6)

where ω(z) is analytic in U and ω(0) = 0, |ω(z)| < 1.

Proof. Suppose that f(z) ∈ C^{(k)}(α, β, γ), from equalities (4.1) and (4.2) we can get

f′(z) = \frac{f_{k}(z)}{z} · \frac{1 + (1 − α)(1 − γ)ω(z)}{1 − (1 − α)βω(z)}

= exp \{\frac{1}{k} ∑_{μ=0}^{k−1} ∫_{0}^{ε^{μ}z} \frac{(1 − α)(1 + β − γ)ω(t)}{t[1 − (1 − α)βω(t)]}dt\}

· \frac{1 + (1 − α)(1 − γ)ω(z)}{1 − (1 − α)βω(z)}.

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

For the class QC^{(k)}(α, β, γ), we have

Corollary 4. Let f(z) ∈ QC^{(k)}(α, β, γ), then we have

f_{k}(z) = ∫_{0}^{z} exp \{\frac{1}{k} ∑_{μ=0}^{k−1} ∫_{0}^{ε^{μ}ζ} \frac{(1 − α)(1 + β − γ)ω(t)}{t[1 − (1 − α)βω(t)]}dt\} dζ,

where f_{k}(z) is defined by equality (1.2), ω(z) is analytic in U and ω(0) = 0, |ω(z)| < 1.

Corollary 5. Let f(z) ∈ QC^{(k)}(α, β, γ), then we have

f(z) = ∫_{0}^{z} \frac{1}{ξ} ∫_{0}^{ξ} exp \{\frac{1}{k} ∑_{μ=0}^{k−1} ∫_{0}^{ε^{μ}ζ} \frac{(1 − α)(1 + β − γ)ω(t)}{t[1 − (1 − α)βω(t)]}dt\}

· \frac{1 + (1 − α)(1 − γ)ω(ζ)}{1 − (1 − α)βω(ζ)}dζdξ,

where ω(z) is analytic in U and ω(0) = 0, |ω(z)| < 1.

5. Convolution Conditions

At last, we provide the convolution conditions for the classes C^{(k)}(α, β, γ) and QC^{(k)}(α, β, γ).

Theorem 6. A function f(z) ∈ C^{(k)}(α, β, γ) if and only if

\begin{aligned}\frac{1}{z}\{f ∗ \{ & \frac{z}{(1 − z)^{2}} [1 − (1 − α)βe^{iθ}] \\ & − [1 + (1 − α)(1 − γ)e^{iθ}]h(z)\}\} ≠ 0 \end{aligned}  (5.1)

for all z ∈ U and 0 ≤ θ < 2π, where h(z) is given by (5.6).

Proof. Suppose that f(z) ∈ C^{(k)}(α, β, γ), by Theorem 1, we know that the condition (1.3) can be written as (2.1), since (2.1) is equivalent to

\frac{zf′(z)}{f_{k}(z)} ≠ \frac{1 + (1 − α)(1 − γ)e^{iθ}}{1 − (1 − α)βe^{iθ}}  (5.2)

for all z ∈ U and 0 ≤ θ < 2π. It is easy to know that the condition (5.2) can be written as

\frac{1}{z}\{zf′(z)[1 − (1 − α)βe^{iθ}] − f_{k}(z)[1 + (1 − α)(1 − γ)e^{iθ}]\} ≠ 0. (5.3)

On the other hand, it is well known that

zf′(z) = f(z) ∗ \frac{z}{(1 − z)^{2}} .  (5.4)

And from the definition of f_{k}(z) we know

f_{k}(z) = z + ∑_{n=2}^{∞} a_{n}c_{n}z^{n} = (f ∗ h)(z),  (5.5)

where

h(z) = z + ∑_{n=2}^{∞} c_{n}z^{n}  (5.6)

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.

82 ZHI-GANG WANG

For the class QC^{(k)}(α, β, γ), we have Corollary 6. A function f(z) ∈ QC^{(k)}(α, β, γ) if and only if

\begin{aligned}\frac{1}{z}\{f ∗ \{z\{\frac{z}{(1 − z)^{2}} [1 & − (1 − α)βe^{iθ}] \\ & − [1 + (1 − α)(1 − γ)e^{iθ}]h(z)\}′\}\} ≠ 0 \end{aligned}

for all z ∈ U and 0 ≤ θ < 2π, where h(z) is given by (5.6).

Acknowledgements

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.

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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