A convolution property of univalent harmonic right half-plane mappings
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A convolution property of univalent harmonic right half-plane mappings Md Firoz Ali1 · Vasudevarao Allu2
· Nirupam Ghosh3
Received: 19 April 2020 / Accepted: 19 June 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract We consider the convolution of right half-plane harmonic mappings in the unit disk D := {z ∈ C : |z| < 1} with respective dilatations eiα (z + a)/(1 + az) and −z, where −1 < a < 1 and α ∈ R. We prove that such convolutions are locally univalent and convex in the horizontal direction under certain condition. Keywords Analytic · Univalent · Starlike · Convex · Close-to-convex functions · Close-to-convex harmonic mappings · Convolution · Right half-plane mappings Mathematics Subject Classification Primary 30C45 · 30C50
1 Introduction and notation Harmonic mapping techniques have been used to study and solve fluid flow problems (see [1]). In 2012, Aleman and Constantin [1] provided a nice connection between geometric function theory and fluid dynamics. They provided a method of solving
Communicated by Adrian Constantin.
B
Vasudevarao Allu [email protected] Md Firoz Ali [email protected] Nirupam Ghosh [email protected]
1
Department of Mathematics, National Institute of Technology Durgapur, Durgapur, West Bengal 713209, India
2
School of Basic Sciences, Indian Institute of Technology Bhubaneswar, Argul, Bhubaneswar, Odisha 752050, India
3
Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, Bangalore Centre, 8th Mile Mysore Road, Bangalore 560059, India
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M. F. Ali et al.
the incompressible two dimensional Euler equations by means of univalent harmonic maps. In 2017, using harmonic mappings properties, Constantin and Martin [4] obtained a complete solution of classifying all two dimensionall fluid flows. A complex-valued function f = u + iv defined on a simply connected domain ⊆ C is harmonic, if both u and v are real-valued harmonic functions in . Let H denote the class of complex-valued harmonic functions f in the unit disk D := {z ∈ C : |z| < 1}, normalized by f (0) = 0 = f z (0) − 1. Each function f in H can be expressed as f = h + g, where h and g are analytic functions in D with the following series representations h(z) = z +
∞
an z n and g(z) =
n=2
∞
bn z n .
n=1
We call h and g the analytic and co-analytic parts of f respectively. By Lewy’s theorem [11], f ∈ H is locally univalent and sense-preserving if, and only if, J f (z) > 0 in D, where J f (z) = |h (z)|2 − |g (z)|2 denotes the Jacobian of f . We note that J f (z) > 0 in D is equivalent to the existence of an analytic function ω, called the dilatation of f , given by ω(z) = g (z)/h (z), with |ω(z)| < 1 for all z ∈ D, and where h (z) = 0 in D. Let SH be the subclass of H consisting of univalent i.e., one-to-one and sense0 = { f ∈ S : f (0) = 0}. Hence for preserving harmonic mappings on D, and SH z H 0 any function f = h + g in SH , its analytic and co-analytic parts can be represented by h(z) = z +
∞
an z n and g(z) =
n=2
∞
bn z n
n=2
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