Univalence and convexity in one direction of the convolution of harmonic mappings

作者：Nagpal Sumit; Ravichandran V^*

来源：Complex Variables and Elliptic Equations, 2014, 59(9): 1328-1341.

DOI：10.1080/17476933.2013.836188

• univalent harmonic mappings
• sense-preserving
• convex in one direction
• convolution

Let H denote the class of all complex-valued harmonic functions f in the open unit disc normalized by f (0) = 0 = f(z) (0) - 1 = f((z) over bar)(0), and let A be the subclass of H consisting of normalized analytic functions. For phi is an element of A, let W-H(-)(phi) := {f = h + (g) over bar is an element of H : h - g = phi} and W-H(+)(phi) := {f = h + (g) over bar is an element of H : h + g = phi} be subfamilies of H. In this paper, we shall determine the conditions under which the harmonic convolution f(1) * f(2) is univalent and convex in one direction if f(1) is an element of W-H(-)(z) and f(2) is an element of W-H(-)(phi). A similar analysis is carried out if f(1) is an element of W-H(-)(z) and f(2) is an element of W-H(+)(phi). Examples of univalent harmonic mappings constructed by way of convolution are also presented.

• 出版日期2014