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Harmonic close-to-convex functions and minimal surfaces

Saminathan Ponnusamy
Organization: Indian Statistical Institute (ISI), Chennai Centre, SETS (Society for Electronic Transactions and Security)
Antti Rasila
Organization: Department of Mathematics and Systems Analysis, Aalto University
A. Sairam Kaliraj
Organization: bDepartment of Mathematics, Indian Institute of Technology Madras
Journal / Anthology

Complex Variables and Elliptic Equations: An International Journal
Year: 2014
Volume: 59
Issue: 7
Page range: 9861002

In this paper, we study the family C0 H of sense-preserving complex-valued harmonic functions f that are normalized close-to-convex functions on the open unit disk D with fz (0) = 0. We derive a sufficient condition for f to belong to the class C0 H. We take the analytic part of f to be zF(a, b; c; z) or zF(a, b; c; z2) and for a suitable choice of co-analytic part of f , the second complex dilatation ω(z) = fz/ fz 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 ofminimal surfaces are given.Graphs generated by using Mathematica are used for illustration.

*Mathematics > Algebra

coefficient inequality, univalence, close-to-convex, convex in vertical, direction, univalent harmonic functions, Gaussian hypergeometric functions, minimal surfaces