- Research Article
- Open Access

# On Certain Multivalent Starlike or Convex Functions with Negative Coefficients

- Neslihan Uyanik
^{1}, - Erhan Deniz
^{2}, - Ekrem Kadioǧlu
^{2}and - Shigeyoshi Owa
^{3}Email author

**2010**:751721

https://doi.org/10.1155/2010/751721

© Neslihan Uyanik et al. 2010

**Received:**2 April 2010**Accepted:**3 June 2010**Published:**15 June 2010

## Abstract

By means of a differential operator, we introduce and investigate some new subclasses of -valently analytic functions with negative coefficients, which are starlike or convex of complex order. Relevant connections of the definitions and results presented in this paper with those obtained in several earlier works on the subject are also pointed out.

## Keywords

- Differential Operator
- Convex Function
- Series Expansion
- Simple Computation
- Open Unit

## 1. Introduction

which are analytic and multivalent in the open unit dics

where ;

If we take and for , then become the differential operator defined by S l gean [1].

where , , , ; ;

where ; ; ;

Remark 1.1.

(
)
was studied by Nasr and Aouf [2] (also see Bulboac
et al*.* [3]).

(
)
and
,
were introduced by Srivastava et al*.* [4].

( ) and , were introduced by Silverman [5].

( ) and were introduced by Parvathan and Ponnusanny [6, pages 163-164].

( ) For and , the classes , , and are closely related with , , and which are defined by Owa and Sălăgean in [7].

In this paper we give relationships between the classes of , , and In the particular case when and , , and , we obtain the same results as in [8].

## 2. Main Results

Our main results are contained in

Theorem 2.1.

Let , and let ; then

(1)

(2)

(4)if , then ;

(5)if , then

Proof.

for

hence

Then multiplying the relation last inequality with we obtain

( ) if is a real positive number, then the definitions of and are equivalent, hence By using ( ) and ( ) from this theorem, we obtain ( ).

( ) We have the following two cases.

Case 1.

and then (see the definition of ).

We have where and we deduce that for all does not hold.

We have obtained that for but and in this case

Case 2.

We consider the function defined by (2.11) for In this case, the inequality (2.13) holds too and this implies that

We also obtain that like in Case 1.

where is given by (2.17).

## Declarations

## Authors’ Affiliations

## References

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- Nasr MA, Aouf MK: Starlike function of complex order.
*The Journal of Natural Sciences and Mathematics*1985, 25(1):1–12.MathSciNetMATHGoogle Scholar - Bulboacă T, Nasr MA, Sălăgean GŞ: Function with negative coefficients
-starlike of complex order.
*Universitatis Babeş-Bolyai. Studia Mathematica*1991, 36(2):7–12.MathSciNetMATHGoogle Scholar - Srivastava HM, Owa S, Chatterjea SK: A note on certain classes of starlike functions.
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*Proceedings of the American Mathematical Society*1975, 51: 109–116. 10.1090/S0002-9939-1975-0369678-0MathSciNetView ArticleMATHGoogle Scholar - R. Parvathan and S. Ponnusanny, Eds., “Open problems,” World Scientific, 1990, Conference on New Trends in Geometric Function Theory and Applications.Google Scholar
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## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.