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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)
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.
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,
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 .
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 ; ; ;
() 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 .
2. Main Results
Our main results are contained in
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.
Let be defined by
and let We have
and then(see the definition of ).
Then, by a simple computation and by using the fact that
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
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
where is given by (2.17).
From where and are given by (2.18) and (2.19), we obtain
If and, then
then (2.24) also holds. By combining (2.24) with (2.23) and the definition of , we obtain that
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In this paper, we discuss the class of analytic functions with negative coefficients. Let us consider the functions given by
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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). https://doi.org/10.1155/2010/751721
- Differential Operator
- Convex Function
- Series Expansion
- Simple Computation
- Open Unit