Harmonic Close-to-convex Functions and Minimal Surfaces

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 $w(z)=\overline{f_{\stackrel{ˉ}{z}}}/f_z$ 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.

18 pages

References in corpus (1)

• Harmonic Shears of Slit and Polygonal Mappings