Om P. Ahuja

Convolutions for special classes of harmonic univalent functions

Abstract Ruscheweyh and Sheil-Small proved the PolyarSchoenberg conjecture that the class of convex analytic functions is closed under convolution or Hadamard product. They also showed that close-to-convexity is preserved under convolution with convex analytic functions. In this note, we investigate harmonic analogs. Beginning with convex analytic functions, we form certain harmonic functions which preserve close-to-convexity under convolution. An auxiliary function enables us to obtain necessary and sufficient convolution conditions for convex and starlike harmonic functions, which lead to sufficient coefficient bounds for inclusion in these classes.

Planar harmonic convolution operators generated by hypergeometric functions

Let S Ĥ be the family of all planar harmonic, univalent and sense-preserving mappings f=h+g¯ where h and g are analytic funtions in the open unit disk. The purpose of this article is to investigate connections between the theory of harmonic mappings in the plane and hypergeometric functions by applying certain planar harmonic convolution operators on various subclasses of S Ĥ . Dedicated to Professor Eveyln M. Silvia, late Professor of Mathematics, UC Davis February 8, 1948–January 21, 2006

Connections between various subclasses of planar harmonic mappings involving hypergeometric functions

The main purpose of this paper is to establish connections between various subclasses of harmonic mappings in the plane by applying certain convolution operators involving hypergeometric functions. To be more precise, we investigate such connections with harmonic k-uniformly starlike and harmonic k-uniformly convex mappings in the plane.

Harmonic starlikeness and convexity of integral operators generated by hypergeometric series

The object of this article is to study harmonic starlikeness and harmonic convexity of integral operators generated by hypergeometeric series. We also consider special cases of harmonic starlikeness and convexity of integral operators generated by incomplete beta functions.

Certain multipliers of univalent harmonic functions

Abstract Inequalities involving multipliers using the sequences { c n } and { d n } of positive real numbers are introduced for complex-valued harmonic univalent functions. By specializing { c n } and { d n } , we determine representation theorems, distortion bounds, convolutions, convex combinations, and neighbourhoods for such functions. The theorems presented, in many cases, confirm or generalize various well-known results for corresponding classes of harmonic functions.

Radius Constants for Functions with the Prescribed Coefficient Bounds

For an analytic univalent function in the unit disk, it is well-known that for . But the inequality does not imply the univalence of . This motivated several authors to determine various radii constants associated with the analytic functions having prescribed coefficient bounds. In this paper, a survey of the related work is presented for analytic and harmonic mappings. In addition, we establish a coefficient inequality for sense-preserving harmonic functions to compute the bounds for the radius of univalence, radius of full starlikeness/convexity of order () for functions with prescribed coefficient bound on the analytic part.

Neighborhoods of Starlike and Convex Functions Associated with Parabola

Let be a normalized analytic function defined on the unit disk and for . For , a function if lies in the parabolic region . Let be the corresponding class consisting of functions such that lies in the region . For an appropriate , the -neighbourhood of a function is shown to consist of functions in the class .

Certain Classes of Harmonic Multivalent Functions Based on Hadamard Product

We define and investigate two special subclasses of the class of complex-valued harmonic multivalent functions based on Hadamard product.

Convolutions of Harmonic Functions with Certain Dilatations

The convolution of harmonic functions, unlike the analytic case, proved to be very challenging. In this paper, we introduce dilatation conditions that guarantee the convolution of two harmonic functions to be locally one-to-one, sense-preserving, and close-to-convex harmonic in the unit disk.

Differential subordinations and argument inequalities

Abstract The main object of the present paper is to investigate certain properties of multivalent functions associated with a linear operator I p λ ( a , c ) .