# Neighborhoods and partial sums of certain subclass of starlike functions

## 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.

## 1 Introduction

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

$f\left(z\right)=z+\underset{k=m+1}{\overset{\mathrm{âˆž}}{âˆ‘}}{a}_{k}{z}^{k}\phantom{\rule{1em}{0ex}}\left(mâˆˆ\mathbb{N}:=\left\{1,2,3,â€¦\right\}\right),$
(1.1)

which are analytic in the open unit disk

A function $fâˆˆ{\mathcal{A}}_{m}$ is said to be in the class ${\mathcal{S}}_{m}^{âˆ—}\left(\mathrm{Î²}\right)$ of starlike functions of order Î² if it satisfies the inequality

$\mathrm{â„œ}\left(\frac{z{f}^{â€²}\left(z\right)}{f\left(z\right)}\right)>\mathrm{Î²}\phantom{\rule{1em}{0ex}}\left(zâˆˆ\mathbb{U};0â‰¦\mathrm{Î²}<1\right).$
(1.2)

Assuming that $\mathrm{Î±}â‰§0$, $0â‰¦\mathrm{Î²}<1$ and $fâˆˆ{\mathcal{A}}_{m}$, we say that a function $fâˆˆ{\mathcal{H}}_{m}\left(\mathrm{Î±},\mathrm{Î²}\right)$ if it satisfies the condition

$\mathrm{â„œ}\left(\frac{z{f}^{â€²}\left(z\right)}{f\left(z\right)}+\mathrm{Î±}\frac{{z}^{2}{f}^{â€³}\left(z\right)}{f\left(z\right)}\right)>\mathrm{Î±}\mathrm{Î²}\left(\mathrm{Î²}+\frac{m}{2}âˆ’1\right)+\mathrm{Î²}âˆ’\frac{m\mathrm{Î±}}{2}\phantom{\rule{1em}{0ex}}\left(zâˆˆ\mathbb{U}\right).$
(1.3)

For convenience, throughout this paper, we write

${\mathrm{Î³}}_{m}:=\mathrm{Î±}\mathrm{Î²}\left(\mathrm{Î²}+\frac{m}{2}âˆ’1\right)+\mathrm{Î²}âˆ’\frac{m\mathrm{Î±}}{2}.$
(1.4)

Recently, Ravichandran et al. [1] proved that ${\mathcal{H}}_{m}\left(\mathrm{Î±},\mathrm{Î²}\right)âŠ‚{\mathcal{S}}_{m}^{âˆ—}\left(\mathrm{Î²}\right)$. 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 ${\mathcal{H}}_{m}\left(\mathrm{Î±},\mathrm{Î²}\right)$ and a subclass of ${\mathcal{H}}_{m}\left(\mathrm{Î±},\mathrm{Î²}\right)$ with negative coefficients. Furthermore, Singh et al. [3] generalized the class ${\mathcal{H}}_{m}\left(\mathrm{Î±},\mathrm{Î²}\right)$ and found several sufficient conditions for starlikeness. In the present paper, we aim at proving the neighborhoods and partial sums of the class ${\mathcal{H}}_{m}\left(\mathrm{Î±},\mathrm{Î²}\right)$.

## 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 AltintaÅŸ et al. [6â€“9], CÇŽtaÅŸ [10], Frasin [11], Keerthi et al. [12] and Srivastava et al. [13], we begin by introducing here the Î´-neighborhood of a function $fâˆˆ{\mathcal{A}}_{m}$ of the form (1.1) by means of the definition

(2.1)

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

Theorem 1 If $fâˆˆ{\mathcal{A}}_{m}$ satisfies the condition

$\frac{f\left(z\right)+\mathrm{Îµ}z}{1+\mathrm{Îµ}}âˆˆ{\mathcal{H}}_{m}\left(\mathrm{Î±},\mathrm{Î²}\right)\phantom{\rule{1em}{0ex}}\left(\mathrm{Îµ}âˆˆ\mathbb{C};|\mathrm{Îµ}|<\mathrm{Î´};\mathrm{Î´}>0\right),$
(2.2)

then

${\mathcal{N}}_{\mathrm{Î´}}\left(f\right)âŠ‚{\mathcal{H}}_{m}\left(\mathrm{Î±},\mathrm{Î²}\right).$
(2.3)

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

$|\frac{\frac{z{f}^{â€²}\left(z\right)}{f\left(z\right)}+\mathrm{Î±}\frac{{z}^{2}{f}^{â€³}\left(z\right)}{f\left(z\right)}âˆ’1}{\frac{z{f}^{â€²}\left(z\right)}{f\left(z\right)}+\mathrm{Î±}\frac{{z}^{2}{f}^{â€³}\left(z\right)}{f\left(z\right)}âˆ’\left(2{\mathrm{Î³}}_{m}âˆ’1\right)}|<1\phantom{\rule{1em}{0ex}}\left(zâˆˆ\mathbb{U}\right),$
(2.4)

we easily find from (2.4) that a function $gâˆˆ{\mathcal{H}}_{m}\left(\mathrm{Î±},\mathrm{Î²}\right)$ if and only if

which is equivalent to

(2.5)

where

$h\left(z\right)=z+\underset{k=m+1}{\overset{\mathrm{âˆž}}{âˆ‘}}{c}_{k}{z}^{k}\phantom{\rule{1em}{0ex}}\left({c}_{k}:=\frac{k+\mathrm{Î±}k\left(kâˆ’1\right)âˆ’1âˆ’\left[k+\mathrm{Î±}k\left(kâˆ’1\right)âˆ’\left(2{\mathrm{Î³}}_{m}âˆ’1\right)\right]\mathrm{Ïƒ}}{2\left({\mathrm{Î³}}_{m}âˆ’1\right)\mathrm{Ïƒ}}\right).$
(2.6)

It follows from (2.6) that

$\begin{array}{rcl}|{c}_{k}|& =& |\frac{k+\mathrm{Î±}k\left(kâˆ’1\right)âˆ’1âˆ’\left[k+\mathrm{Î±}k\left(kâˆ’1\right)âˆ’\left(2{\mathrm{Î³}}_{m}âˆ’1\right)\right]\mathrm{Ïƒ}}{2\left({\mathrm{Î³}}_{m}âˆ’1\right)\mathrm{Ïƒ}}|\\ â‰¦& \frac{k+\mathrm{Î±}k\left(kâˆ’1\right)âˆ’1+\left[k+\mathrm{Î±}k\left(kâˆ’1\right)âˆ’\left(2{\mathrm{Î³}}_{m}âˆ’1\right)\right]|\mathrm{Ïƒ}|}{2\left(1âˆ’{\mathrm{Î³}}_{m}\right)|\mathrm{Ïƒ}|}\\ =& \frac{k\left(1+k\mathrm{Î±}âˆ’\mathrm{Î±}\right)âˆ’{\mathrm{Î³}}_{m}}{1âˆ’{\mathrm{Î³}}_{m}}\phantom{\rule{1em}{0ex}}\left(|\mathrm{Ïƒ}|=1\right).\end{array}$

If $fâˆˆ{\mathcal{A}}_{m}$ satisfies the condition (2.2), we deduce from (2.5) that

or, equivalently,

$|\frac{\left(fâˆ—h\right)\left(z\right)}{z}|â‰§\mathrm{Î´}\phantom{\rule{1em}{0ex}}\left(zâˆˆ\mathbb{U};\mathrm{Î´}>0\right).$
(2.7)

We now suppose that

$q\left(z\right)=z+\underset{k=m+1}{\overset{\mathrm{âˆž}}{âˆ‘}}{d}_{k}{z}^{k}âˆˆ{\mathcal{N}}_{\mathrm{Î´}}\left(f\right).$

It follows from (2.1) that

$\begin{array}{rcl}|\frac{\left(\left(qâˆ’f\right)âˆ—h\right)\left(z\right)}{z}|& =& |\underset{k=m+1}{\overset{\mathrm{âˆž}}{âˆ‘}}\left({d}_{k}âˆ’{a}_{k}\right){c}_{k}{z}^{kâˆ’1}|\\ â‰¦& |z|\underset{k=m+1}{\overset{\mathrm{âˆž}}{âˆ‘}}\frac{k\left(1+k\mathrm{Î±}âˆ’\mathrm{Î±}\right)âˆ’{\mathrm{Î³}}_{m}}{1âˆ’{\mathrm{Î³}}_{m}}|{d}_{k}âˆ’{a}_{k}|<\mathrm{Î´}.\end{array}$
(2.8)

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

$|\frac{\left(qâˆ—h\right)\left(z\right)}{z}|=|\frac{\left(\left[f+\left(qâˆ’f\right)\right]âˆ—h\right)\left(z\right)}{z}|â‰§|\frac{\left(fâˆ—h\right)\left(z\right)}{z}|âˆ’|\frac{\left(\left(qâˆ’f\right)âˆ—h\right)\left(z\right)}{z}|>0,$

which implies that

Therefore, we conclude that

$q\left(z\right)âˆˆ{\mathcal{N}}_{\mathrm{Î´}}\left(f\right)âŠ‚{\mathcal{H}}_{m}\left(\mathrm{Î±},\mathrm{Î²}\right).$

We thus complete the proof of Theorem 1.â€ƒâ–¡

Next, we derive the partial sums of the class ${\mathcal{H}}_{m}\left(\mathrm{Î±},\mathrm{Î²}\right)$. For some recent investigations involving the partial sums in analytic function theory, one can refer to [14â€“16] and the references cited therein.

Theorem 2 Let $fâˆˆ{\mathcal{A}}_{m}$ be given by (1.1) and define the partial sums ${f}_{n}\left(z\right)$ of f by

${f}_{n}\left(z\right)=z+\underset{k=m+1}{\overset{n}{âˆ‘}}{a}_{k}{z}^{k}\phantom{\rule{1em}{0ex}}\left(nâˆˆ\mathbb{N};nâ‰§m+1\right).$
(2.9)

If

$\underset{k=m+1}{\overset{\mathrm{âˆž}}{âˆ‘}}\frac{k\left(1+k\mathrm{Î±}âˆ’\mathrm{Î±}\right)âˆ’{\mathrm{Î³}}_{m}}{1âˆ’{\mathrm{Î³}}_{m}}|{a}_{k}|â‰¦1\phantom{\rule{1em}{0ex}}\left(\mathrm{Î±}â‰§0;0â‰¦\mathrm{Î²}<1;{\mathrm{Î³}}_{m}<1\right),$
(2.10)

then

(1) $fâˆˆ{\mathcal{H}}_{m}\left(\mathrm{Î±},\mathrm{Î²}\right)$;

(2)

$\mathrm{â„œ}\left(\frac{f\left(z\right)}{{f}_{n}\left(z\right)}\right)â‰§\frac{n\left(1+\mathrm{Î±}+n\mathrm{Î±}\right)}{\left(n+1\right)\left(1+n\mathrm{Î±}\right)âˆ’{\mathrm{Î³}}_{m}}\phantom{\rule{1em}{0ex}}\left(nâˆˆ\mathbb{N};nâ‰§m+1;zâˆˆ\mathbb{U}\right)$
(2.11)

and

$\mathrm{â„œ}\left(\frac{{f}_{n}\left(z\right)}{f\left(z\right)}\right)â‰§\frac{\left(n+1\right)\left(1+n\mathrm{Î±}\right)âˆ’{\mathrm{Î³}}_{m}}{\left(n+1\right)\left(1+n\mathrm{Î±}\right)+1âˆ’2{\mathrm{Î³}}_{m}}\phantom{\rule{1em}{0ex}}\left(nâˆˆ\mathbb{N};nâ‰§m+1;zâˆˆ\mathbb{U}\right).$
(2.12)

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

Proof (1) Suppose that ${f}_{1}\left(z\right)=z$. We know that $zâˆˆ{\mathcal{H}}_{m}\left(\mathrm{Î±},\mathrm{Î²}\right)$, which implies that

$\frac{{f}_{1}\left(z\right)+\mathrm{Îµ}z}{1+\mathrm{Îµ}}=zâˆˆ{\mathcal{H}}_{m}\left(\mathrm{Î±},\mathrm{Î²}\right).$

From (2.10), we easily find that

$\underset{k=m+1}{\overset{\mathrm{âˆž}}{âˆ‘}}\frac{k\left(1+k\mathrm{Î±}âˆ’\mathrm{Î±}\right)âˆ’{\mathrm{Î³}}_{m}}{1âˆ’{\mathrm{Î³}}_{m}}|{a}_{k}âˆ’0|â‰¦1,$

which implies that $fâˆˆ{\mathcal{N}}_{1}\left(z\right)$. In view of Theorem 1, we deduce that

$fâˆˆ{\mathcal{N}}_{1}\left(z\right)âŠ‚{\mathcal{H}}_{m}\left(\mathrm{Î±},\mathrm{Î²}\right).$

(2) It is easy to verify that

$\begin{array}{rcl}\frac{\left(n+1\right)\left[1+\left(n+1\right)\mathrm{Î±}âˆ’\mathrm{Î±}\right]âˆ’{\mathrm{Î³}}_{m}}{1âˆ’{\mathrm{Î³}}_{m}}& =& \frac{\left(n+1\right)\left(1+n\mathrm{Î±}\right)âˆ’{\mathrm{Î³}}_{m}}{1âˆ’{\mathrm{Î³}}_{m}}\\ >& \frac{n\left(1+n\mathrm{Î±}âˆ’\mathrm{Î±}\right)âˆ’{\mathrm{Î³}}_{m}}{1âˆ’{\mathrm{Î³}}_{m}}>1\phantom{\rule{1em}{0ex}}\left(nâˆˆ\mathbb{N}\right).\end{array}$

Therefore, we have

$\underset{k=m+1}{\overset{n}{âˆ‘}}|{a}_{k}|+\frac{\left(n+1\right)\left(1+n\mathrm{Î±}\right)âˆ’{\mathrm{Î³}}_{m}}{1âˆ’{\mathrm{Î³}}_{m}}\underset{k=n+1}{\overset{\mathrm{âˆž}}{âˆ‘}}|{a}_{k}|â‰¦\underset{k=m+1}{\overset{\mathrm{âˆž}}{âˆ‘}}\frac{k\left(1+k\mathrm{Î±}âˆ’\mathrm{Î±}\right)âˆ’{\mathrm{Î³}}_{m}}{1âˆ’{\mathrm{Î³}}_{m}}|{a}_{k}|â‰¦1.$
(2.13)

We now suppose that

$\begin{array}{rcl}\mathrm{Ïˆ}\left(z\right)& =& \frac{\left(n+1\right)\left(1+n\mathrm{Î±}\right)âˆ’{\mathrm{Î³}}_{m}}{1âˆ’{\mathrm{Î³}}_{m}}\left(\frac{f\left(z\right)}{{f}_{n}\left(z\right)}âˆ’\frac{n\left(1+\mathrm{Î±}+n\mathrm{Î±}\right)}{\left(n+1\right)\left(1+n\mathrm{Î±}\right)âˆ’{\mathrm{Î³}}_{m}}\right)\\ =& 1+\frac{\frac{\left(n+1\right)\left(1+n\mathrm{Î±}\right)âˆ’{\mathrm{Î³}}_{m}}{1âˆ’{\mathrm{Î³}}_{m}}{âˆ‘}_{k=n+1}^{\mathrm{âˆž}}{a}_{k}{z}^{kâˆ’1}}{1+{âˆ‘}_{k=m+1}^{n}{a}_{k}{z}^{kâˆ’1}}.\end{array}$
(2.14)

It follows from (2.13) and (2.14) that

$|\frac{\mathrm{Ïˆ}\left(z\right)âˆ’1}{\mathrm{Ïˆ}\left(z\right)+1}|â‰¦\frac{\frac{\left(n+1\right)\left(1+n\mathrm{Î±}\right)âˆ’{\mathrm{Î³}}_{m}}{1âˆ’{\mathrm{Î³}}_{m}}{âˆ‘}_{k=n+1}^{\mathrm{âˆž}}|{a}_{k}|}{2âˆ’2{âˆ‘}_{k=m+1}^{n}|{a}_{k}|âˆ’\frac{\left(n+1\right)\left(1+n\mathrm{Î±}\right)âˆ’{\mathrm{Î³}}_{m}}{1âˆ’{\mathrm{Î³}}_{m}}{âˆ‘}_{k=n+1}^{\mathrm{âˆž}}|{a}_{k}|}â‰¦1\phantom{\rule{1em}{0ex}}\left(zâˆˆ\mathbb{U}\right),$

which shows that

$\mathrm{â„œ}\left(\mathrm{Ïˆ}\left(z\right)\right)â‰§0\phantom{\rule{1em}{0ex}}\left(zâˆˆ\mathbb{U}\right).$
(2.15)

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

Moreover, if we put

$f\left(z\right)=z+\frac{1âˆ’{\mathrm{Î³}}_{m}}{\left(n+1\right)\left(1+n\mathrm{Î±}\right)âˆ’{\mathrm{Î³}}_{m}}{z}^{n+1}\phantom{\rule{1em}{0ex}}\left(nâˆˆ\mathbb{N}âˆ–\left\{1,2,â€¦,mâˆ’1\right\};mâˆˆ\mathbb{N}\right),$
(2.16)

then for $z=r{e}^{i\mathrm{Ï€}/n}$, we have

$\frac{f\left(z\right)}{{f}_{n}\left(z\right)}=1+\frac{1âˆ’{\mathrm{Î³}}_{m}}{\left(n+1\right)\left(1+n\mathrm{Î±}\right)âˆ’{\mathrm{Î³}}_{m}}{z}^{n}â†’\frac{n\left(1+\mathrm{Î±}+n\mathrm{Î±}\right)}{\left(n+1\right)\left(1+n\mathrm{Î±}\right)âˆ’{\mathrm{Î³}}_{m}}\phantom{\rule{1em}{0ex}}\left(râ†’{1}^{âˆ’}\right),$

which implies that the bound in (2.11) is the best possible for each $nâˆˆ\mathbb{N}âˆ–\left\{1,2,â€¦,mâˆ’1\right\}$.

Similarly, we suppose that

$\begin{array}{rcl}\mathrm{Ï†}\left(z\right)& =& \frac{\left(n+1\right)\left(1+n\mathrm{Î±}\right)+1âˆ’2{\mathrm{Î³}}_{m}}{1âˆ’{\mathrm{Î³}}_{m}}\left(\frac{{f}_{n}\left(z\right)}{f\left(z\right)}âˆ’\frac{\left(n+1\right)\left(1+n\mathrm{Î±}\right)âˆ’{\mathrm{Î³}}_{m}}{\left(n+1\right)\left(1+n\mathrm{Î±}\right)+1âˆ’2{\mathrm{Î³}}_{m}}\right)\\ =& 1âˆ’\frac{\frac{\left(n+1\right)\left(1+n\mathrm{Î±}\right)+1âˆ’2{\mathrm{Î³}}_{m}}{1âˆ’{\mathrm{Î³}}_{m}}{âˆ‘}_{k=n+1}^{\mathrm{âˆž}}{a}_{k}{z}^{kâˆ’1}}{1+{âˆ‘}_{k=m+1}^{\mathrm{âˆž}}{a}_{k}{z}^{kâˆ’1}}.\end{array}$
(2.17)

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

$|\frac{\mathrm{Ï†}\left(z\right)âˆ’1}{\mathrm{Ï†}\left(z\right)+1}|â‰¦\frac{\frac{\left(n+1\right)\left(1+n\mathrm{Î±}\right)+1âˆ’2{\mathrm{Î³}}_{m}}{1âˆ’{\mathrm{Î³}}_{m}}{âˆ‘}_{k=n+1}^{\mathrm{âˆž}}|{a}_{k}|}{2âˆ’2{âˆ‘}_{k=m+1}^{n}|{a}_{k}|âˆ’\frac{n\left(1+\mathrm{Î±}+n\mathrm{Î±}\right)}{1âˆ’{\mathrm{Î³}}_{m}}{âˆ‘}_{k=n+1}^{\mathrm{âˆž}}|{a}_{k}|}â‰¦1\phantom{\rule{1em}{0ex}}\left(zâˆˆ\mathbb{U}\right),$

which implies that

$\mathrm{â„œ}\left(\mathrm{Ï†}\left(z\right)\right)â‰§0\phantom{\rule{1em}{0ex}}\left(zâˆˆ\mathbb{U}\right).$
(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.â€ƒâ–¡

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 $fâˆˆ{\mathcal{A}}_{m}$ be given by (1.1) and define the partial sums ${f}_{n}\left(z\right)$ of f by (2.9). If the condition (2.10) holds, then

$\mathrm{â„œ}\left(\frac{{f}^{â€²}\left(z\right)}{{f}_{n}^{\mathrm{â€²}}\left(z\right)}\right)â‰§\frac{\left(n+1\right)\left(n\mathrm{Î±}+{\mathrm{Î³}}_{m}\right)âˆ’{\mathrm{Î³}}_{m}}{\left(n+1\right)\left(1+n\mathrm{Î±}\right)âˆ’{\mathrm{Î³}}_{m}}\phantom{\rule{1em}{0ex}}\left(nâˆˆ\mathbb{N};nâ‰§m+1;zâˆˆ\mathbb{U}\right)$
(2.19)

and

$\mathrm{â„œ}\left(\frac{{f}_{n}^{\mathrm{â€²}}\left(z\right)}{{f}^{â€²}\left(z\right)}\right)â‰§\frac{\left(n+1\right)\left(1+n\mathrm{Î±}\right)âˆ’{\mathrm{Î³}}_{m}}{\left(n+1\right)\left(2+n\mathrm{Î±}âˆ’{\mathrm{Î³}}_{m}\right)âˆ’{\mathrm{Î³}}_{m}}\phantom{\rule{1em}{0ex}}\left(nâˆˆ\mathbb{N};nâ‰§m+1;zâˆˆ\mathbb{U}\right).$
(2.20)

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

Remark By setting $\mathrm{Î±}=0$ and $m=1$ in Theorems 2 and 3, we get the corresponding results obtained by Silverman [16].

## References

1. Ravichandran V, Selvaraj C, Rajalaksmi R: Sufficient conditions for starlike functions of order Î± . J. Inequal. Pure Appl. Math. 2002., 3: Article ID 81 (electronic)

2. Liu M-S, Zhu Y-C, Srivastava HM: Properties and characteristics of certain subclasses of starlike functions of order Î² . Math. Comput. Model. 2008, 48: 402â€“419. 10.1016/j.mcm.2006.09.026

3. Singh S, Gupta S, Singh S: Starlikeness of analytic maps satisfying a differential inequality. Gen. Math. 2010, 18: 51â€“58.

4. Goodman AW: Univalent functions and nonanalytic curves. Proc. Am. Math. Soc. 1957, 8: 598â€“601. 10.1090/S0002-9939-1957-0086879-9

5. Ruscheweyh S: Neighborhoods of univalent functions. Proc. Am. Math. Soc. 1981, 81: 521â€“527. 10.1090/S0002-9939-1981-0601721-6

6. AltintaÅŸ O: Neighborhoods of certain p -valently analytic functions with negative coefficients. Appl. Math. Comput. 2007, 187: 47â€“53. 10.1016/j.amc.2006.08.101

7. AltintaÅŸ O, Owa S: Neighborhoods of certain analytic functions with negative coefficients. Int. J. Math. Math. Sci. 1996, 19: 797â€“800. 10.1155/S016117129600110X

8. AltintaÅŸ O, Ã–zkan Ã–, Srivastava HM: Neighborhoods of a class of analytic functions with negative coefficients. Appl. Math. Lett. 2000, 13: 63â€“67.

9. AltintaÅŸ O, Ã–zkan Ã–, Srivastava HM: Neighborhoods of a certain family of multivalent functions with negative coefficients. Comput. Math. Appl. 2004, 47: 1667â€“1672. 10.1016/j.camwa.2004.06.014

10. CÇŽtaÅŸ A: Neighborhoods of a certain class of analytic functions with negative coefficients. Banach J. Math. Anal. 2009, 3: 111â€“121.

11. Frasin BA: Neighborhoods of certain multivalent functions with negative coefficients. Appl. Math. Comput. 2007, 193: 1â€“6. 10.1016/j.amc.2007.03.026

12. Keerthi BS, Gangadharan A, Srivastava HM: Neighborhoods of certain subclasses of analytic functions of complex order with negative coefficients. Math. Comput. Model. 2008, 47: 271â€“277. 10.1016/j.mcm.2007.04.004

13. Srivastava HM, Eker SS, Seker B: Inclusion and neighborhood properties for certain classes of multivalently analytic functions of complex order associated with the convolution structure. Appl. Math. Comput. 2009, 212: 66â€“71. 10.1016/j.amc.2009.01.077

14. Frasin BA: Partial sums of certain analytic and univalent functions. Acta Math. Acad. Paedagog. NyhÃ¡zi. 2005, 21: 135â€“145. (electronic)

15. Frasin BA: Generalization of partial sums of certain analytic and univalent functions. Appl. Math. Lett. 2008, 21: 735â€“741. 10.1016/j.aml.2007.08.002

16. Silverman H: Partial sums of starlike and convex functions. J. Math. Anal. Appl. 1997, 209: 221â€“227. 10.1006/jmaa.1997.5361

## 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 Peopleâ€™s Republic of China.

## Author information

Authors

### Corresponding author

Correspondence to Zhi-Gang Wang.

### Competing interests

The authors declare that they have no competing interests.

### Authorsâ€™ contributions

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

## Rights and permissions

Reprints and Permissions

Wang, ZG., Yuan, XS. & Shi, L. Neighborhoods and partial sums of certain subclass of starlike functions. J Inequal Appl 2013, 163 (2013). https://doi.org/10.1186/1029-242X-2013-163