- 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.
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 .
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.
- Ahuja OP, Jahangiri JM: Multivalent harmonic starlike functions with missing coefficients. Mathematical Sciences Research Journal 2003,7(9):347–352.MathSciNetMATHGoogle Scholar
- Güney HÖ, Ahuja OP: Inequalities involving multipliers for multivalent harmonic functions. Journal of Inequalities in Pure and Applied Mathematics 2006,7(5, article 190):1–9.MathSciNetMATHGoogle Scholar
- Ahuja OP, Jahangiri JM: Multivalent harmonic starlike functions. Annales Universitatis Mariae Curie-Skłodowska. Sectio A 2001,55(1):1–13.MathSciNetMATHGoogle Scholar
- Jahangiri JM: Harmonic functions starlike in the unit disk. Journal of Mathematical Analysis and Applications 1999,235(2):470–477. 10.1006/jmaa.1999.6377MathSciNetView ArticleMATHGoogle Scholar
- Silverman H: Harmonic univalent functions with negative coefficients. Journal of Mathematical Analysis and Applications 1998,220(1):283–289. 10.1006/jmaa.1997.5882MathSciNetView ArticleMATHGoogle Scholar
- Silverman H, Silvia EM: Subclasses of harmonic univalent functions. New Zealand Journal of Mathematics 1999,28(2):275–284.MathSciNetMATHGoogle Scholar
- Ahuja OP, Aghalary R, Joshi SB: Harmonic univalent functions associated with k-uniformly starlike functions. Mathematical Sciences Research Journal 2005,9(1):9–17.MathSciNetMATHGoogle Scholar
- Jahangiri JM, Murugusundaramoorthy G, Vijaya K: On starlikeness of certain multivalent harmonic functions. Journal of Natural Geometry 2003,24(1–2):1–10.MathSciNetMATHGoogle Scholar
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