Abstract
Following Clunie and Sheil-Small, the class of normalized univalent harmonic mappings in the unit disk is denoted by $${\mathcal {S}}_{{\mathcal {H}}}$$ . The aim of the paper is to study the properties of a subclass of $${\mathcal {S}}_{{\mathcal {H}}}$$ , such that the analytic part is a convex function. We establish estimates of some functionals and bounds of the Bloch’s constant for co-analytic part.
Highlights
A complex-valued harmonic function f that is harmonic in a connected domain ⊂ C has the canonical representationCommunicated by V
Ruscheweyh [21] has obtained the best-possible estimates of higher order derivatives of bounded analytic functions on the disk
Bloch’s constant B f of a harmonic mapping f = h + gcan be expressed in terms of moduli of the derivatives of h and g
Summary
A complex-valued harmonic function f that is harmonic in a connected domain ⊂ C has the canonical representationCommunicated by V. Keywords Univalent harmonic mappings · Convex functions · Bloch’s constant · Normal family 1 Faculty of Mathematics and Natural Sciences, University of Rzeszow, ul.
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