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# On Certain Multivalent Starlike or Convex Functions with Negative Coefficients

*Journal of Inequalities and Applications*
**volume 2010**, Article number: 751721 (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

Let denote the class of functions of the following form:

which are analytic and multivalent in the open unit dics

Let denote the th-order ordinary differential operator for a function , that is,

where ;

Next, we define the differential operator as

In view of (1.3), it is clear that

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

Finally, in terms of a differential operator defined by (1.3) above, let denote the subclass ofconsisting of functionswhich satisfy the following inequality:

where , , , ; ;

For , , and , we define the next subclasses of

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)

(3)if , then

(4)if , then ;

(5)if , then

Proof.

() Let We prove that

If has the series expansion

then

We use the fact that for andthese imply

for

From (2.4) and (2.5), we deduce

By using the definition of from this last inequality we, obtain (2.2) which implies

hence

() Let be in Then (2.7) holds and, by using (2.3), this is equivalent to

For if , from (2.8) we obtain

which is equivalent to

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.

Let be defined by

and let We have

or

and then(see the definition of ).

Let now

Then, by a simple computation and by using the fact that

we obtain

where, , and

For we, have where is the disc with the center

and the radius

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.

() Let be given by (2.11), whereand Then

which implies that

We have

where is given by (2.17).

From where and are given by (2.18) and (2.19), we obtain

If and, then

and if

then (2.24) also holds. By combining (2.24) with (2.23) and the definition of , we obtain that

## 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.Nasr MA, Aouf MK: Starlike function of complex order.

*The Journal of Natural Sciences and Mathematics*1985, 25(1):1–12.Bulboacă T, Nasr MA, Sălăgean GŞ: Function with negative coefficients -starlike of complex order.

*Universitatis Babeş-Bolyai. Studia Mathematica*1991, 36(2):7–12.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.Silverman H: Univalent functions with negative coefficients.

*Proceedings of the American Mathematical Society*1975, 51: 109–116. 10.1090/S0002-9939-1975-0369678-0R. Parvathan and S. Ponnusanny, Eds., “Open problems,” World Scientific, 1990, Conference on New Trends in Geometric Function Theory and Applications.

Owa S, Sălăgean GS: Starlike or convex of complex order functions with negative coefficients.

*Sūrikaisekikenkyūsho Kōkyūroku*1998, (1062):77–83.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.1997.5808

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## Appendix

In this paper, we discuss the class of analytic functions with negative coefficients. Let us consider the functions given by

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### Cite this article

Uyanik, N., Deniz, E., Kadioǧlu, E. *et al.* On Certain Multivalent Starlike or Convex Functions with Negative Coefficients.
*J Inequal Appl* **2010, **751721 (2010). https://doi.org/10.1155/2010/751721

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

### Keywords

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