Abstract
We introduced a new subclass of univalent harmonic functions defined by the shear construction in the present paper. First, we showed that the convolutions of two special subclass harmonic mappings are convex in the horizontal direction. Secondly, we proved a necessary and sufficient condition for the above subclass of harmonic mappings to be convex in the horizontal direction. We also presented some basic examples of univalent harmonic functions explaining the behavior of the image domains.
Highlights
Let f = u + iV be a continuous complex-valued harmonic mapping in the open unit disc D = {z ∈ C : |z| < 1}, where u and V are real-valued harmonic functions in D. Such functions can be expressed as f = h + g; here h is known as the analytic part and g the coanalytic part of f, respectively
The Jacobian of the mapping f = h + g is given by Jf = |h|2 − |g|2
We denote by SH the class of harmonic, sense-preserving, and univalent mappings in D, normalized by the conditions f(0) = 0 and fz(0) = 1
Summary
Dorff et al [5] obtained some results involving convolutions of f0 with right half-plane mappings and vertical strip mappings We present some basic examples of harmonic mappings satisfying the conditions of the theorems and illustrate them graphically with the help of the Mathematica software With these examples we explain the behavior of the image domains. If |a0,k| ≥ |an−k,k|, by Lemma 5 we know that not all roots of pk(z) = 0 are inside the unit disc D; we would not need to construct the function pk+1(z). The following lemma is a necessary and sufficient condition of all zeros of (24) lying inside the unit disc. The images of concentric circles inside D under the harmonic mapping P(z) and concentric circles under the convolution map P(z) ∗ P(z) are shown in Figures 1 and 2, respectively
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