Weak subordination for convex univalent harmonic functions

Abstract

Keywords: de la Vallée Poussin means Subordination chain Convex univalent harmonic functions Hadamard product Convolution For two complex-valued harmonic functions f and F defined in the open unitwhenever t 1 , t 2 ∈ E and t 1 < t 2 . In this paper, we construct a weak subordination chain of convex univalent harmonic functions using a harmonic de la Vallée Poussin mean and a modified form of Pommerenke's criterion for a subordination chain of analytic functions.

Univalent Harmonic Functions Generated by Ruscheweyh Derivatives of Analytic Functions

“…For λ = 0 and p > 0, we have Proof. Suppose T λ,p [f ] = H + G, where H and G are given by (10). The necessary part is obviously true by Lemma 6.…”

“…The operator T 0 (z) = T 0,1 [I] was introduced and studied by Clunie and Sheil-Small [5]. In 2008, Muir [10]…”

“…[10] For f ∈ S, the function T p [f ] defined in Example 1 is locally univalent and sense-preserving in D if and only if f is convex analytic in D.Om P. Ahuja, Subzar Beig and V. Ravichandran -Univalent Harmonic FunctionsCorollary Let p > 0, λ ≥ 0 and the function f ∈ T be given by(6). Then the function T λ,p [f ] is locally univalent and sense-preserving in D if and only if∞ m=2 (λ + 1) m−1 (m − 1)!…”

Extreme points of subordination and weak subordination families of harmonic mappings

“…But this important property is not valid for harmonic mappings. In [10], Muir introduced the following concept for harmonic mappings. Definition 1.3.…”

Convolutions of harmonic half-plane mappings with harmonic vertical strip mappings

“…It has been proved in [20] that L c (z) map the unit disk U onto the generalized right half-plane, GR = {ω : Re(ω) > −1/(1 + c)} for each c > 0. Then if F is analytic in U and F(0) = 0, we have…”