Abstract
The main purpose of this paper is to establish a relationship between univalent harmonic mappings and Hardy spaces. The main result obtained in this paper improves previously published results. Moreover, we generalize some nice results in the analytic case to the harmonic case.
Highlights
Let Ω be a domain in the complex plane C and f be a complex-valued function of class C1 in Ω
It is well known that f is locally univalent if Jf (z) ̸= 0 in Ω and the converse is true if f is analytic
A harmonic mapping of the open unit disk U = {z ∈ C : |z| < 1} has the unique representation f = h+g, where h and g are analytic in U and g(0) = 0
Summary
Let Ω be a domain in the complex plane C and f be a complex-valued function of class C1 in Ω. A harmonic mapping of the open unit disk U = {z ∈ C : |z| < 1} has the unique representation f = h+g , where h and g are analytic in U and g(0) = 0 . Let H be the class of harmonic mappings f = h + g in the open unit disk U such that h(0) = g(0) = h′(0) − 1 = 0 . The class of functions f ∈ H that are sense-preserving and univalent in U is denoted by SH .
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