Harmonic Close-to-convex Functions and Minimal Surfaces

TL;DR

This paper characterizes a class of harmonic close-to-convex functions, providing conditions for their inclusion, and explores their connection to minimal surfaces with explicit representations and visualizations.

Contribution

It introduces a sufficient condition for harmonic close-to-convex functions to belong to a specific class and links these functions to minimal surfaces via explicit formulas.

Findings

Derived a sufficient condition for class inclusion.

Connected harmonic functions to minimal surface representations.

Provided explicit formulas and visualizations for the 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 $w(z)=\bar{f_{\bar{z}}}/f_z$ 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.