Convolution Properties of Some Harmonic Mappings in the Right Half-Plane

Abstract

It is well known that harmonic convolution of two normalized right halfplane mappings is convex in the direction of the real axis, provided the convolution function is locally univalent and sense-preserving in E = {z : |z| < 1}. Further, it is also known that the condition of local univalence and sense-preserving in E on the convolution function can be dropped when one of the convoluting functions is the standard right half-plane mapping with dilatation −z and other is the right halfplane mapping with dilatation e iθ z n , n = 1, 2, θ ∈ R. This result does not hold for n = 3, 4, 5, . . . . In this paper, we generalize this result by taking the dilatation of one of the right half-plane mappings as e iθ z n ( n ∈ N, θ ∈ R) and that of the other as (a − z)/(1 − az), a ∈ (−1, 1). We shall prove that our result holds true for all n ∈ N, provided the real constant a is restricted in the interval [(n − 2)/(n + 2), 1). The range of the real constant a is shown to be sharp.

Convolutions of harmonic right half-plane mappings

“…Some related works have been done on these topics, one can refer to [1,3,6,13,15,16]. In [1], the authors proved the following result.…”

“…However, corresponding questions for the class of univalent harmonic mappings seem to be difficult to handle as can be seen from the recent investigations of the authors [13][14][15][16][17]. In [1], Kumar et al constructed the harmonic functions f a D h a C g a 2 K H in the right half-plane, which satisfy the conditions h a C g a D z=.1 z/ with ! a .z/ D .a z/=.1 az/ .…”

“…The main difference of our work from [1] is that we construct a sequence of functions for finding all zeros of polynomials which are in U, and we will show that the dilatation of f a f n satisfies je ! 1 .z/j D j.g a g/ 0 =.h a h/ 0 j < 1 by using mathematical induction, which greatly simplifies the calculation compared with the proof of Theorem 2.2 in [1]. We also show that L c f a is univalent and convex in the horizontal direction for 0 < c Ä 2.1 C a/=.1 a/, and derive the following theorem.…”

Convolution of two harmonic mappings in the right-half plane

“…A harmonic mapping f is locally univalent and sense-preserving in D if and only if J f = |h | 2 − | | 2 > 0 in D; or equivalently if h 0 in D and the dilatation w = /h has the property that |w| < 1 in D [10]. Let S H be the subclass of H consisting of univalent and sense-preserving functions.…”

On Harmonic Convolutions Involving a Vertical Strip Mapping

“…Kumar et al [4], defined harmonic right half-plane mappings F a = H a + G a , where H a + G a = z 1 − z with dilatations ω a (z) = a − z 1 − az , a ∈ (−1, 1).…”