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On Certain Multivalent Starlike or Convex Functions with Negative Coefficients


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



(3)if , then


(4)if , then ;

(5)if , then


() Let We prove that


If has the series expansion




We use the fact that for andthese 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), 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



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Correspondence to Shigeyoshi Owa.


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

which are analytic in . For such a function , we say that if it satisfies
for some complex number with .If we define the function for by
then we know that is analytic in , , and . Thus is the Carathéodory function. Since the extremal function for the Carathéodory function is given by
we can write
This shows us that
Noting that
we see that
that is,
It follows from the above that
Calculating the above integrations, we have that
Therefore, we obtain that
that is,
Consequently, the function defined by the above is the extremal function for the class . But our class is defined with analytic functions with negative coefficients. Thus we do not know how we can consider the extremal function for this class.

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Uyanik, N., Deniz, E., Kadioǧlu, E. et al. On Certain Multivalent Starlike or Convex Functions with Negative Coefficients. J Inequal Appl 2010, 751721 (2010).

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