# 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.

## 1. Introduction

which are analytic and multivalent in the open unit dics

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

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.

Proof.

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

## Declarations

## Authors’ Affiliations

## References

- Sălăgean GŠ: Subclasses of univalent functions. In Complex Analysis—Fifth Romanian-Finnish Seminar Part 1 (Bucharest, 1981), 1983, Lecture Notes in Mathematics. Volume 1013. Springer; 362–372.Google Scholar
- 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.
*Rendiconti del Seminario Matematico della Università di Padova*1987, 77: 115–124.MathSciNetMATHGoogle Scholar - Silverman H: Univalent functions with negative coefficients.
*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
- Owa S, Sălăgean GS: Starlike or convex of complex order functions with negative coefficients.
*Sūrikaisekikenkyūsho Kōkyūroku*1998, (1062):77–83.Google Scholar - Owa S, Sălăgean GS: On an open problem of S. Owa.
*Journal of Mathematical Analysis and Applications*1998, 218(2):453–457. 10.1006/jmaa.1997.5808MathSciNetView ArticleMATHGoogle Scholar

## 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.