- Research Article
- Open Access
On Certain Multivalent Starlike or Convex Functions with Negative Coefficients
© Neslihan Uyanik et al. 2010
- Received: 2 April 2010
- Accepted: 3 June 2010
- Published: 15 June 2010
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
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 .
where , , , ; ;
where ; ; ;
( ) and , were introduced by Srivastava et al. .
( ) and , were introduced by Silverman .
( ) 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 .
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 .
Our main results are contained in
Let , and let ; then
(4)if , then ;
(5)if , then
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.
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
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).
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