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Certain Classes of Harmonic Multivalent Functions Based on Hadamard Product
Journal of Inequalities and Applications volume 2009, Article number: 759251 (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.
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 .
2. Main Results
Let be such that and are given by (1.1). Then if the inequality
is satisfied for some and
In view of (1.5), we need to prove that , where
Using the fact that , it sufficies to show that
Therefore, we obtain
By hypothesis, last expression is nonnegative. Thus the proof is complete.
The coeficient bounds (2.1) is sharp for the function
For , , if the inequality
holds, then .
For and , if the inequality
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
According to Corollaries 2.2 and 2.3, we must show that if the condition (2.9) does not hold, then that is, we must have
Choosing the values of on positive real axis where and using , the inequality (2.10) reduces to
where denotes (k+)p[−+], denotes (k+)p−+ and denotes
Letting , we obtain
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.
If , then for , and
These bounds are sharp.
Suppose . Let and . In view of (1.13), we get
Using Theorem 2.4(i), we obtain
The proofs of other cases are similar and so are omitted.
If , then
Suppose and . Then clco if and only if
In particular, the extreme points of are and .
Suppose . For functions of the form (2.17), we can write
On the other hand, for , we obtain
Thus , by Theorem 2.4.
Conversely, suppose that . Then, it follows Theorem 2.4 that
where , we obtain
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.
For let the functions given by
are in . Also suppose the given fixed harmonic functions are given by
For the convex combinations of can be expressed as
Thus the coefficient estimate given by Theorem 2.4 holds. Therefore, we obtain
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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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Ahuja, O.P., Güney, H.Ö. & Sakar, F.M. Certain Classes of Harmonic Multivalent Functions Based on Hadamard Product. J Inequal Appl 2009, 759251 (2009). https://doi.org/10.1155/2009/759251
- Harmonic Function
- Real Axis
- Coefficient Estimate
- Convex Combination
- Multivalent Function