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  • Research Article
  • Open Access

On Certain Multivalent Starlike or Convex Functions with Negative Coefficients

  • 1,
  • 2,
  • 2 and
  • 3Email author
Journal of Inequalities and Applications20102010:751721

  • Received: 2 April 2010
  • Accepted: 3 June 2010
  • Published:


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.


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

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 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:

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



(3)if , then

(4)if , then ;

(5)if , then


( ) Let We prove that
If has the series expansion
We use the fact that for and these imply


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


( ) 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

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), where and 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


Authors’ Affiliations

Department of Mathematics, Kazim Karabekir Faculty of Education, Atatürk University, Erzurum, 25240, Turkey
Department of Mathematics, Science and Art Faculty, Atatürk University, Erzurum, 25240, Turkey
Department of Mathematics, Kinki University, Higashi-Osaka, Osaka 577-8502, Japan


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© Neslihan Uyanik et al. 2010

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