- Research Article
- Open Access
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.
- Harmonic Function
- Real Axis
- Coefficient Estimate
- Convex Combination
- Multivalent Function
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.
For , denote by the set of all multivalent harmonic functions defined in the open unit disc , where and defined by
are analytic functions in .
Let be a fixed multivalent harmonic function given by
A function is said to be in the class if
where is a harmonic convolution of and . Note that . Using the fact
it follows that if and only if
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
Finally, denote by the subclass of functions in where
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 .
is satisfied for some and
By hypothesis, last expression is nonnegative. Thus the proof is complete.
The coeficient bounds (2.1) is sharp for the function
holds, then .
holds, then .
Let be such that and are given by (1.13). Also, suppose that and . Then
(i)for , if and only if
(ii)for , if and only if
where denotes (k + )p[ − + ], denotes (k+ )p − + and denotes
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.
In particular, the extreme points of are and .
Thus , by Theorem 2.4.
Thus can be expressed as (2.17). The proof for the case is similar and hence is omitted.
The class is closed under convex combinations.
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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