Close-to-Convexity of Convolutions of Classes of Harmonic Functions

Raj Kumar Garg,Jay M Jahangiri

doi:10.1155/2018/3808513

Abstract

For j=1,2 and for positive integers m and n, we consider classes of harmonic functions fj=hj+gj¯, where g1(z)=znh1(z) and g2′(z)=znh2′(z) or g1′(z)=znh1′(z) and g2′(z)=zmh2′(z), and we prove that their convolution f1⁎f2=h1⁎h2+g1⁎g2¯ is locally one-to-one, sense-preserving, and close-to-convex harmonic in z<1.

Highlights

Let A denote the class of functions that are analytic in the open unit disk E fl {z : |z| < 1} and let A󸀠 be the subclass of A consisting of functions h with the normalization h(0) = h󸀠(0) − 1 = 0
By Lewy’s Theorem, a necessary and sufficient condition for the harmonic function f = h + g to be locally one-to-one and sense-preserving in E is that its Jacobian Jf = |h󸀠|2 − |g󸀠|2 should be positive or equivalently if and only if h󸀠 ≠ 0 in E and the second complex dilatation ω of f satisfies |ω| = |g󸀠/h󸀠| < 1 in E
Without loss of generality, we consider those locally one-toone and sense-preserving harmonic functions f = h + g that are normalized by f(0) = h(0) = 0 and fz(0) = 1 and have the representation

Summary

Introduction

By Lewy’s Theorem (see [1, 2] or [3]), a necessary and sufficient condition for the harmonic function f = h + g to be locally one-to-one and sense-preserving in E is that its Jacobian Jf = |h󸀠|2 − |g󸀠|2 should be positive or equivalently if and only if h󸀠 ≠ 0 in E and the second complex dilatation ω of f satisfies |ω| = |g󸀠/h󸀠| < 1 in E. Without loss of generality, we consider those locally one-toone and sense-preserving harmonic functions f = h + g that are normalized by f(0) = h(0) = 0 and fz(0) = 1 and have the representation A󸀠 be ≥ −1/2 in or g1󸀠 = znh1󸀠 , g2󸀠 = znh2󸀠 , α ≥ 0, F = h1∗h2+g1 ∗ g2 is locally one-to-one, sense-preserving, and close-to-convex harmonic in

Results

Conclusion