Herb Silverman

Univalent functions with negative coefficients

Coefficient, distortion, covering, and coefficient inequalities are determined for univalent functions with negative coefficients that are starlike of order a and convex of order a. Extreme points for these classes are also determined.

ON SANDWICH THEOREMS FOR SOME CLASSES OF ANALYTIC FUNCTIONS

The purpose of this present paper is to derive some subordination and superordination results for certain normalized analytic functions in the open unit disk. Relevant connections of the results, which are presented in the paper, with various known results are also considered.

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.

Coefficient Bounds for Certain Classes of Meromorphic Functions

Sharp bounds for are derived for certain classes and of meromorphic functions defined on the punctured open unit disk for which and , respectively, lie in a region starlike with respect to 1 and symmetric with respect to the real axis. Also, certain applications of the main results for a class of functions defined through Ruscheweyh derivatives are obtained.

Subordination and superordination results for a class of analytic multivalent functions.

We derive subordination and superordination results for a family of normalized analytic functions in the open unit disk defined by integral operators. We apply this to obtain sandwich results and generalizations of some known results.

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.

Classes of convex functions

We investigate a family that connects various subclasses of functions convex in the unit disk. We also look at generalized sequences for this family.

Coefficient bounds for inverses of classes of starlike functions

We find sharp coefficient bounds for the inverses of functions of the form that satisfy . We also solve the inverse coefficient problem in the cases n = 2, 3, 4 for such functions that are starlike of order α 0 ≤α≤1, and conjecture the correct bounds for all n.

Univalence for convolutions

The radius of univalence is found for the convolution f,# of functions f E S (normalized univalent functions) and g (5 C (close-to-convex functions). A lower bound for the radius ofunivalence is also determined when f and g range over all of S. Finally, a characterization ofC provides an inclusion relationship.

Classes of functions whose derivatives have positive real part

Abstract We investigate functions of the form f(z) = z − ∑n = 2x anzn, an ≥ 0, that are analytic in ¦z¦ and satisfy there the inequality Re {(1 − λ)f′ + λ(1 + zf″ f′ )} > 0 for some real λ. This family, denoted by M(λ), is shown for all real λ to contain only starlike functions whose derivatives have positive real parts. Relationships between M(λ) and other classes of univalent functions as well as coefficient and distortion bounds are found for λ ⩾ 0. A characterization for M(λ) when 0 ⩽ λ ⩽ 1 is also given.