Open Access

On Certain Multivalent Starlike or Convex Functions with Negative Coefficients

  • Neslihan Uyanik1,
  • Erhan Deniz2,
  • Ekrem Kadioǧlu2 and
  • Shigeyoshi Owa3Email author
Journal of Inequalities and Applications20102010:751721

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

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

Let denote the class of functions of the following form:
(1.1)

which are analytic and multivalent in the open unit dics

Let denote the th-order ordinary differential operator for a function   , that is,
(1.2)

where ;      

Next, we define the differential operator as
(1.3)
In view of (1.3), it is clear that
(1.4)

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

Finally, in terms of a differential operator defined by (1.3) above, let denote the subclass of consisting of functions which satisfy the following inequality:
(1.5)

where , , , ; ;

For , , and , we define the next subclasses of
(1.6)

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
(2.1)

(4)if , then ;

(5)if , then

Proof.

( ) Let We prove that
(2.2)
If has the series expansion
(2.3)
then
(2.4)
We use the fact that for and these imply
(2.5)

for

From (2.4) and (2.5), we deduce
(2.6)
By using the definition of from this last inequality we, obtain (2.2) which implies
(2.7)

hence

( ) Let be in Then (2.7) holds and, by using (2.3), this is equivalent to
(2.8)
For if , from (2.8) we obtain
(2.9)
which is equivalent to
(2.10)

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
(2.11)
and let We have
(2.12)
or
(2.13)

and then (see the definition of ).

Let now
(2.14)
Then, by a simple computation and by using the fact that
(2.15)
we obtain
(2.16)
where , , and
(2.17)
For we, have where is the disc with the center
(2.18)
and the radius
(2.19)

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), where and Then
(2.20)
which implies that
(2.21)
We have
(2.22)

where is given by (2.17).

From where and are given by (2.18) and (2.19), we obtain
(2.23)
If and , then
(2.24)
and if
(2.25)
then (2.24) also holds. By combining (2.24) with (2.23) and the definition of , we obtain that
(2.26)

Declarations

Authors’ Affiliations

(1)
Department of Mathematics, Kazim Karabekir Faculty of Education, Atatürk University
(2)
Department of Mathematics, Science and Art Faculty, Atatürk University
(3)
Department of Mathematics, Kinki University

References

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Copyright

© Neslihan Uyanik et al. 2010

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.