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arXiv · 1209.0202

Harmonic Close-to-convex Functions and Minimal Surfaces

S. Ponnusamy·A. Rasila·A. Sairam Kaliraj
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Abstract

In this paper, we study the family ${\mathcal C}_{H}^0$ of sense-preserving complex-valued harmonic functions $f$ that are normalized close-to-convex functions on the open unit disk $\mathbb{D}$ with $f_{\bar{z}}(0)=0$. We derive a sufficient condition for $f$ to belong to the class $\CC_{H}^0$. We take the analytic part of $f$ to be $zF(a,b;c;z)$ or $zF(a,b;c;z^2)$ and for a suitable choice of co-analytic part of $f$, the second complex dilatation $w(z)=\bar{f_{\bar{z}}}/f_z$ turns out to be a square of an analytic function. Hence $f$ is lifted to a minimal surface expressed by an isothermal parameter. Explicit representation for classes of minimal surfaces are given. Graphs generated by using Mathematica are used for illustration.

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S. Ponnusamy, A. Rasila, A. Sairam Kaliraj. 2012-09-02. Harmonic Close-to-convex Functions and Minimal Surfaces. https://arxiv.org/abs/1209.0202

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