Jay M. Jahangiri

Classes of uniformly starlike and convex functions

We note that for α> 1, if f ∈ SD(α, β) ,t henzf � (z)/f (z) lies in the region G ≡ G(α, β) ≡ {w :R e w> α|w −1|+β}, that is, part of the complex plane which contains w = 1a nd is bounded by the ellipse (u−(α 2 −β)/(α 2 −1)) 2 +(α 2 /(α 2 −1))v 2 = α 2 (1 − β) 2 /(α 2 − 1) 2 with vertices at the points ((α + β)/(α + 1), 0), ((α − β)/(α − 1), 0), ((α 2 − β)/(α 2 − 1), (β − 1)/ √ α 2 − 1), and ((α 2 − β)/(α 2 − 1), (1 − β)/ √ α 2 − 1) .S ince β β } and so SD(α, β) ⊂ S ∗ (β) .F orα = 1 if f ∈ SD(α, β) ,t henzf � (z)/f (z) belongs to the region which contains w = 2 and is bounded by parabola u = (v 2 + 1 − β 2 )/2(1 − β). Using the relation between convex and starlike functions, we define KD(α, β) as the class of functions f ∈ A if and only if zf � ∈ SD(α, β) .F orα = 1 and β = 0, we obtain the class KD(1, 0) of uniformly convex functions, first defined by Goodman [1]. Ronning [3] investigated the class KD(1 ,β )of uniformly convex functions of order β. For the class KD(α, 0) of α-uniformly convex function, see [2]. In this note, we study the coefficient bounds and Hadamard product or convolution properties of the classes SD(α, β) and KD(α, β). Using these results, we further show that the classes SD(α, β) and KD(α, β) are closed under certain integral operators. 2. Main results. First we give a sufficient coefficient bound for functions in SD(α, β). Theorem 2.1. If ∞=2[n(1 + α)− (α + β)]|an| < 1 − β, then f ∈ SD(α, β).

SUBORDINATION PROPERTIES OF p-VALENT FUNCTIONS DEFINED BY INTEGRAL OPERATORS

By applying certain integral operators to P-valent functions we define a comprehensive family of analytic functins. The subordinations properties of this family is studied, which in certain special cases yield some of the previously obtained results.

Construction of a Certain Class of Harmonic Close-To-Convex Functions Associated with the Alexander Integral Transform

The authors make use of the Alexander integral transforms of certain analytic functions (which are starlike or convex of positive order) with a view to investigating the construction of sense-preserving, univalent, and close-to-convex harmonic functions.

Coefficient Estimates for Certain Classes of Bi-Univalent Functions

A function analytic in the open unit disk is said to be bi-univalent in if both the function and its inverse map are univalent there. The bi-univalency condition imposed on the functions analytic in makes the behavior of their coefficients unpredictable. Not much is known about the behavior of the higher order coefficients of classes of bi-univalent functions. We use Faber polynomial expansions of bi-univalent functions to obtain estimates for their general coefficients subject to certain gap series as well as providing bounds for early coefficients of such functions.

Certain convex harmonic functions

We define and investigate a family of complex-valued harmonic convex univalent functions related to uniformly convex analytic functions. We obtain coefficient bounds, extreme points, distortion theorems, convolution and convex combinations for this family.

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.

Partial sums of functions of bounded turning

We determine conditions under which the partial sums of the Libera integral operator of functions of bounded turning are also of bounded turning.

Boundary fractional differential equation in a complex domain

We discuss univalent solutions of boundary fractional differential equations in a complex domain. The fractional operators are taken in the sense of the Srivastava-Owa calculus in the unit disk. The existence of subsolutions and supersolutions (maximal and minimal) is established. The existence of a unique univalent solution is imposed. Applications are constructed by making use of a transformation formula for fractional derivatives as well as generalized fractional derivatives.

Faber Polynomial Coefficient Estimates for Meromorphic Bi-Starlike Functions

We consider meromorphic starlike univalent functions that are also bi-starlike and find Faber polynomial coefficient estimates for these types of functions. A function is said to be bi-starlike if both the function and its inverse are starlike univalent.

Harmonic close-to-convex mappings

Sufficient coefficient conditions for complex functions to be close-to-convex harmonic or convex harmonic are given. Construction of close-to-convex harmonic functions is also studied by looking at transforms of convex analytic functions. Finally, a convolution property for harmonic functions is discussed.