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Neighborhoods and partial sums of certain subclass of starlike functions

verfasst von: Zhi-Gang Wang, Xin-Sheng Yuan, Lei Shi

Erschienen in: Journal of Inequalities and Applications | Ausgabe 1/2013

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Abstract

The main purpose of the present paper is to derive the neighborhoods and partial sums of a certain subclass of starlike functions.

MSC: 30C45.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors completed the paper together. They also read and approved the final manuscript.

1 Introduction

Let

A_{m}

denote the class of functions f of the form

f (z) = z + \sum_{k = m + 1}^{\infty} a_{k} z^{k} (m \in N : = {1, 2, 3, \dots}),

(1.1)

which are analytic in the open unit disk

U : = {z : z \in C and | z | < 1} .

A function

f \in A_{m}

is said to be in the class

S_{m}^{*} (β)

of starlike functions of order β if it satisfies the inequality

ℜ (\frac{z f^{'} (z)}{f (z)}) > β (z \in U; 0 ≦ β < 1) .

(1.2)

Assuming that

α ≧ 0

0 ≦ β < 1

and

f \in A_{m}

, we say that a function

f \in H_{m} (α, β)

if it satisfies the condition

ℜ (\frac{z f^{'} (z)}{f (z)} + α \frac{z^{2} f^{″} (z)}{f (z)}) > α β (β + \frac{m}{2} - 1) + β - \frac{m α}{2} (z \in U) .

(1.3)

For convenience, throughout this paper, we write

γ_{m} : = α β (β + \frac{m}{2} - 1) + β - \frac{m α}{2} .

(1.4)

Recently, Ravichandran et al. [1] proved that

H_{m} (α, β) \subset S_{m}^{*} (β)

. Subsequently, Liu et al. [2] derived various properties and characteristics such as inclusion relationships, Hadamard products, coefficient estimates, distortion theorems and cover theorems for the class

H_{m} (α, β)

and a subclass of

H_{m} (α, β)

with negative coefficients. Furthermore, Singh et al. [3] generalized the class

H_{m} (α, β)

and found several sufficient conditions for starlikeness. In the present paper, we aim at proving the neighborhoods and partial sums of the class

H_{m} (α, β)

2 Main results

Following the earlier works (based upon the familiar concept of a neighborhood of analytic functions) by Goodman [4] and Ruscheweyh [5], and (more recently) by Altintaş et al. [6‐9], Cǎtaş [10], Frasin [11], Keerthi et al. [12] and Srivastava et al. [13], we begin by introducing here the δ-neighborhood of a function

f \in A_{m}

of the form (1.1) by means of the definition

\begin{array}{rcl} N_{δ} (f) & : = & {g \in A_{m} : g (z) = z + \sum_{k = m + 1}^{\infty} b_{k} z^{k} and \\ \sum_{k = m + 1}^{\infty} \frac{k (1 + k α - α) - γ_{m}}{1 - γ_{m}} | a_{k} - b_{k} | ≦ δ (δ, α ≧ 0; 0 ≦ β < 1; γ_{m} < 1)} . \end{array}

(2.1)

By making use of the definition (2.1), we now derive the following result.

Theorem 1 If

f \in A_{m}

satisfies the condition

\frac{f (z) + ε z}{1 + ε} \in H_{m} (α, β) (ε \in C; | ε | < δ; δ > 0),

(2.2)

then

N_{δ} (f) \subset H_{m} (α, β) .

(2.3)

Proof By noting that the condition (1.3) can be rewritten as follows:

| \frac{\frac{z f^{'} (z)}{f (z)} + α \frac{z^{2} f^{″} (z)}{f (z)} - 1}{\frac{z f^{'} (z)}{f (z)} + α \frac{z^{2} f^{″} (z)}{f (z)} - (2 γ_{m} - 1)} | < 1 (z \in U),

(2.4)

we easily find from (2.4) that a function

g \in H_{m} (α, β)

if and only if

\frac{z g^{'} (z) + α z^{2} g^{″} (z) - g (z)}{z g^{'} (z) + α z^{2} g^{″} (z) - (2 γ_{m} - 1) g (z)} \neq σ (z \in U; σ \in C; | σ | = 1),

which is equivalent to

\frac{(g * h) (z)}{z} \neq 0 (z \in U),

(2.5)

where

h (z) = z + \sum_{k = m + 1}^{\infty} c_{k} z^{k} (c_{k} : = \frac{k + α k (k - 1) - 1 - [k + α k (k - 1) - (2 γ_{m} - 1)] σ}{2 (γ_{m} - 1) σ}) .

(2.6)

It follows from (2.6) that

\begin{array}{rcl} | c_{k} | & = & | \frac{k + α k (k - 1) - 1 - [k + α k (k - 1) - (2 γ_{m} - 1)] σ}{2 (γ_{m} - 1) σ} | \\ ≦ & \frac{k + α k (k - 1) - 1 + [k + α k (k - 1) - (2 γ_{m} - 1)] | σ |}{2 (1 - γ_{m}) | σ |} \\ = & \frac{k (1 + k α - α) - γ_{m}}{1 - γ_{m}} (| σ | = 1) . \end{array}

f \in A_{m}

satisfies the condition (2.2), we deduce from (2.5) that

\frac{(f * h) (z)}{z} \neq - ε (| ε | < δ; δ > 0),

or, equivalently,

| \frac{(f * h) (z)}{z} | ≧ δ (z \in U; δ > 0) .

(2.7)

We now suppose that

q (z) = z + \sum_{k = m + 1}^{\infty} d_{k} z^{k} \in N_{δ} (f) .

It follows from (2.1) that

\begin{array}{rcl} | \frac{((q - f) * h) (z)}{z} | & = & | \sum_{k = m + 1}^{\infty} (d_{k} - a_{k}) c_{k} z^{k - 1} | \\ ≦ & | z | \sum_{k = m + 1}^{\infty} \frac{k (1 + k α - α) - γ_{m}}{1 - γ_{m}} | d_{k} - a_{k} | < δ . \end{array}

(2.8)

Combining (2.7) and (2.8), we easily find that

| \frac{(q * h) (z)}{z} | = | \frac{([f + (q - f)] * h) (z)}{z} | ≧ | \frac{(f * h) (z)}{z} | - | \frac{((q - f) * h) (z)}{z} | > 0,

which implies that

\frac{(q * h) (z)}{z} \neq 0 (z \in U) .

Therefore, we conclude that

q (z) \in N_{δ} (f) \subset H_{m} (α, β) .

We thus complete the proof of Theorem 1. □

Next, we derive the partial sums of the class

H_{m} (α, β)

. For some recent investigations involving the partial sums in analytic function theory, one can refer to [14‐16] and the references cited therein.

Theorem 2 Let

f \in A_{m}

be given by (1.1) and define the partial sums

f_{n} (z)

of f by

f_{n} (z) = z + \sum_{k = m + 1}^{n} a_{k} z^{k} (n \in N; n ≧ m + 1) .

(2.9)

\sum_{k = m + 1}^{\infty} \frac{k (1 + k α - α) - γ_{m}}{1 - γ_{m}} | a_{k} | ≦ 1 (α ≧ 0; 0 ≦ β < 1; γ_{m} < 1),

(2.10)

then

(1)

f \in H_{m} (α, β)

;

(2)

ℜ (\frac{f (z)}{f_{n} (z)}) ≧ \frac{n (1 + α + n α)}{(n + 1) (1 + n α) - γ_{m}} (n \in N; n ≧ m + 1; z \in U)

(2.11)

and

ℜ (\frac{f_{n} (z)}{f (z)}) ≧ \frac{(n + 1) (1 + n α) - γ_{m}}{(n + 1) (1 + n α) + 1 - 2 γ_{m}} (n \in N; n ≧ m + 1; z \in U) .

(2.12)

The bounds in (2.11) and (2.12) are sharp.

Proof (1) Suppose that

f_{1} (z) = z

. We know that

z \in H_{m} (α, β)

, which implies that

\frac{f_{1} (z) + ε z}{1 + ε} = z \in H_{m} (α, β) .

From (2.10), we easily find that

\sum_{k = m + 1}^{\infty} \frac{k (1 + k α - α) - γ_{m}}{1 - γ_{m}} | a_{k} - 0 | ≦ 1,

which implies that

f \in N_{1} (z)

. In view of Theorem 1, we deduce that

f \in N_{1} (z) \subset H_{m} (α, β) .

(2) It is easy to verify that

\begin{array}{rcl} \frac{(n + 1) [1 + (n + 1) α - α] - γ_{m}}{1 - γ_{m}} & = & \frac{(n + 1) (1 + n α) - γ_{m}}{1 - γ_{m}} \\ > & \frac{n (1 + n α - α) - γ_{m}}{1 - γ_{m}} > 1 (n \in N) . \end{array}

Therefore, we have

\sum_{k = m + 1}^{n} | a_{k} | + \frac{(n + 1) (1 + n α) - γ_{m}}{1 - γ_{m}} \sum_{k = n + 1}^{\infty} | a_{k} | ≦ \sum_{k = m + 1}^{\infty} \frac{k (1 + k α - α) - γ_{m}}{1 - γ_{m}} | a_{k} | ≦ 1 .

(2.13)

We now suppose that

\begin{array}{rcl} ψ (z) & = & \frac{(n + 1) (1 + n α) - γ_{m}}{1 - γ_{m}} (\frac{f (z)}{f_{n} (z)} - \frac{n (1 + α + n α)}{(n + 1) (1 + n α) - γ_{m}}) \\ = & 1 + \frac{\frac{(n + 1) (1 + n α) - γ_{m}}{1 - γ_{m}} \sum_{k = n + 1}^{\infty} a_{k} z^{k - 1}}{1 + \sum_{k = m + 1}^{n} a_{k} z^{k - 1}} . \end{array}

(2.14)

It follows from (2.13) and (2.14) that

| \frac{ψ (z) - 1}{ψ (z) + 1} | ≦ \frac{\frac{(n + 1) (1 + n α) - γ_{m}}{1 - γ_{m}} \sum_{k = n + 1}^{\infty} | a_{k} |}{2 - 2 \sum_{k = m + 1}^{n} | a_{k} | - \frac{(n + 1) (1 + n α) - γ_{m}}{1 - γ_{m}} \sum_{k = n + 1}^{\infty} | a_{k} |} ≦ 1 (z \in U),

which shows that

ℜ (ψ (z)) ≧ 0 (z \in U) .

(2.15)

Combining (2.14) and (2.15), we deduce that the assertion (2.11) holds true.

Moreover, if we put

f (z) = z + \frac{1 - γ_{m}}{(n + 1) (1 + n α) - γ_{m}} z^{n + 1} (n \in N ∖ {1, 2, \dots, m - 1}; m \in N),

(2.16)

then for

z = r e^{i π / n}

, we have

\frac{f (z)}{f_{n} (z)} = 1 + \frac{1 - γ_{m}}{(n + 1) (1 + n α) - γ_{m}} z^{n} \to \frac{n (1 + α + n α)}{(n + 1) (1 + n α) - γ_{m}} (r \to 1^{-}),

which implies that the bound in (2.11) is the best possible for each

n \in N ∖ {1, 2, \dots, m - 1}

Similarly, we suppose that

\begin{array}{rcl} φ (z) & = & \frac{(n + 1) (1 + n α) + 1 - 2 γ_{m}}{1 - γ_{m}} (\frac{f_{n} (z)}{f (z)} - \frac{(n + 1) (1 + n α) - γ_{m}}{(n + 1) (1 + n α) + 1 - 2 γ_{m}}) \\ = & 1 - \frac{\frac{(n + 1) (1 + n α) + 1 - 2 γ_{m}}{1 - γ_{m}} \sum_{k = n + 1}^{\infty} a_{k} z^{k - 1}}{1 + \sum_{k = m + 1}^{\infty} a_{k} z^{k - 1}} . \end{array}

(2.17)

In view of (2.13) and (2.17), we conclude that

| \frac{φ (z) - 1}{φ (z) + 1} | ≦ \frac{\frac{(n + 1) (1 + n α) + 1 - 2 γ_{m}}{1 - γ_{m}} \sum_{k = n + 1}^{\infty} | a_{k} |}{2 - 2 \sum_{k = m + 1}^{n} | a_{k} | - \frac{n (1 + α + n α)}{1 - γ_{m}} \sum_{k = n + 1}^{\infty} | a_{k} |} ≦ 1 (z \in U),

which implies that

ℜ (φ (z)) ≧ 0 (z \in U) .

(2.18)

Combining (2.17) and (2.18), we readily get the assertion (2.12) of Theorem 2. The bound in (2.12) is sharp with the extremal function f given by (2.16).

The proof of Theorem 2 is thus completed. □

Finally, we turn to ratios involving derivatives. The proof of Theorem 3 below is much akin to that of Theorem 2, we here choose to omit the analogous details.

Theorem 3 Let

f \in A_{m}

be given by (1.1) and define the partial sums

f_{n} (z)

of f by (2.9). If the condition (2.10) holds, then

ℜ (\frac{f^{'} (z)}{f_{n}^{'} (z)}) ≧ \frac{(n + 1) (n α + γ_{m}) - γ_{m}}{(n + 1) (1 + n α) - γ_{m}} (n \in N; n ≧ m + 1; z \in U)

(2.19)

and

ℜ (\frac{f_{n}^{'} (z)}{f^{'} (z)}) ≧ \frac{(n + 1) (1 + n α) - γ_{m}}{(n + 1) (2 + n α - γ_{m}) - γ_{m}} (n \in N; n ≧ m + 1; z \in U) .

(2.20)

The bounds in (2.19) and (2.20) are sharp with the extremal function given by (2.16).

Remark By setting

α = 0

and

m = 1

in Theorems 2 and 3, we get the corresponding results obtained by Silverman [16].

Acknowledgements

Dedicated to Professor Hari M Srivastava.

The present investigation was supported by the National Natural Science Foundation under Grants 11226088 and 11101053, the Key Project of Chinese Ministry of Education under Grant 211118, the Excellent Youth Foundation of Educational Committee of Hunan Province under Grant 10B002, the Open Fund Project of Key Research Institute of Philosophies and Social Sciences in Hunan Universities under Grants 11FEFM02 and 12FEFM02, and the Key Project of Natural Science Foundation of Educational Committee of Henan Province under Grant 12A110002 of the People’s Republic of China.

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors completed the paper together. They also read and approved the final manuscript.

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Titel: Neighborhoods and partial sums of certain subclass of starlike functions
verfasst von: Zhi-Gang Wang
Xin-Sheng Yuan
Lei Shi
Publikationsdatum: 01.12.2013
Verlag: Springer International Publishing
Erschienen in: Journal of Inequalities and Applications / Ausgabe 1/2013
Elektronische ISSN: 1029-242X
DOI: https://doi.org/10.1186/1029-242X-2013-163

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Neighborhoods and partial sums of certain subclass of starlike functions

Abstract

Competing interests

Authors’ contributions

1 Introduction

2 Main results

Acknowledgements

Competing interests

Authors’ contributions

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Abstract

Competing interests

Authors’ contributions

1 Introduction

2 Main results

Acknowledgements

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Authors’ contributions

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