Certain Classes of Harmonic Multivalent Functions Based on Hadamard Product
© Om P. Ahuja et al. 2009
Received: 25 February 2009
Accepted: 12 September 2009
Published: 29 September 2009
We define and investigate two special subclasses of the class of complex-valued harmonic multivalent functions based on Hadamard product.
A continuous function is a complex-valued harmonic function in a complex domain if both and are real harmonic in . In any simply connected domain , we can write , where and are analytic in . We call the analytic part and the co-analytic part of . Note that reduces to if the coanalytic part is zero.
A function in is called -uniformly multivalent harmonic starlike function associated with a fixed multivalent harmonic function . The set is a comprehensive family that contains several previously studied subclasses of ; for example, if we let
In this paper, we investigate coefficient conditions, extreme points, and distortion bounds for functions in the family . We observe that the results so obtained for this main family can be viewed as extensions and generalizations for various subclasses of and .
2. Main Results
By hypothesis, last expression is nonnegative. Thus the proof is complete.
The coeficient bounds (2.1) is sharp for the function
If the condition (2.10) does not hold, then the numerator in (2.12) is negative for sufficiently close to 1. Hence there exists in for which (2.12) is negative.Therefore, it follows that and so the proof is complete.
These bounds are sharp.
The proofs of other cases are similar and so are omitted.
This present investigation is supported with the Project no. DÜBAP-07-02-21 by Dicle University, The committee of the Scientific Research Projects.
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