Injectivity of sections of univalent harmonic mappings

Abstract

If S denotes the class of functions h(z)=z+∑n=2∞anzn which are analytic and univalent in the unit disk |z|<1, then the classical result of Szegö shows that every section sn(h)(z)=∑k=1nakzk of h is univalent in |z|<1/4. Exact (largest) radius of univalence rn of sn(h) remains an open problem, although the corresponding results for sections of various geometric subclasses of S have been obtained which include those h∈S for which Reh′(z)>0 holds in the unit disk. However, no attempt has been made to derive harmonic analog of these results. The central object in this case is the class SH0 of sense-preserving harmonic univalent mappings f=h+g¯ defined on the unit disk, normalized so that h(0)=g(0)=h′(0)−1=g′(0)=0. Our primary objective in this paper is to solve the univalency of every section of a harmonic function in the class PH0(α), where PH0(α) denotes the class of normalized univalent harmonic mapping f=h+g¯ in the unit disk |z|<1 satisfying the condition Re(h′(z)−α)>|g′(z)| for |z|<1, where g′(0)=0 and 0≤α<1. In this paper, we first present sharp bounds for the moduli of the coefficients an, bn of f∈PH0(α) and then determine the value of r so that the partial sums of f∈PH0 are close-to-convex in |z|<r.