Harmonic Functions Starlike in the Unit Disk

Abstract

DEDICATED TO KSU LATE PROFESSOR KENNETH B. CUMMINS 7r27r1911᎐5r13r1998Complex-valued harmonic functions that are univalent and sense-preserving in the unit disk ⌬ can be written in the form f s h q g, where h and g are analytic in ⌬. We give univalence criteria and sufficient coefficient conditions for normalized harmonic functions that are starlike of order ␣, 0 F ␣ -1. These coefficient conditions are also shown to be necessary when h has negative and g has positive coefficients. These lead to extreme points and distortion bounds. ᮊ 1999 Academic Press

A unified representation of some starlike and convex harmonic functions with negative coefficients

“…results obtained by Jahangiri [3]. (v) Putting k = 0 and γ = 0 in all the above results, we obtain the corresponding results obtained by Silverman [9], Jahangiri [4], and Silverman and Silvia [10]. (vi) Putting α = 0 in all the above results, we obtain the corresponding results for the class HG(k, γ).…”

“…Specializing the parameters α, γ and k, we obtain the following subclasses studied by various authors: [9], Jahangiri [4], and Silverman and Silvia [10]). …”

Pascu-Type Harmonic Functions with Positive Coefficients Involving Salagean Operator

“…By routine procedure (see [10][11][12][13]), we can easily prove the following results; hence we state the following theorems without proof for functions in V ℓ H ( , ).…”

“…Motivated by the earlier works of [11][12][13][14] on the subject of harmonic functions, in this paper an attempt has been made to study the class of functions ∈ V H associated with Salagean operator on harmonic functions. Further, we obtain a sufficient coefficient condition for functions ∈ H given by (3) and also show that this coefficient condition is necessary for functions ∈ V H , the class of harmonic functions with positive coefficients.…”

Quasiconformal Extensions of Starlike Harmonic Mappings in the Unit Disc

“…When f is harmonic, Avcı and Z lotkiewicz [2], Silverman [17] gave a sufficient coefficient condition for f to be univalent, sense-preserving and starlike when b 1 = 0. Jahangiri [13] generalized the result to the case that b 1 is not necessarily 0. He gave a sufficient coefficient condition for f to be univalent, sense-preserving and starlike of order α ∈ [0, 1) when b 1 is not necessarily 0 (Theorem 2.1).…”