On Harmonic Convolutions Involving a Vertical Strip Mapping

Abstract

Abstract. Let f β = h β + g β and Fa = Ha + Ga be harmonic mappings obtained by shearing of analytic mappingsand Ha + Ga = z/(1 − z), respectively. Kumar et al. [7] conjectured that if ω(z) = e iθ z n (θ ∈ R, n ∈ N) and ωa(z) = (a − z)/(1 − az), a ∈ (−1, 1) are dilatations of f β and Fa, respectively, then Fa * f β ∈ S 0 H and is convex in the direction of the real axis, provided a ∈ [(n − 2)/(n + 2), 1). They claimed to have verified the result for n = 1, 2, 3 and 4 only. In the present paper, we settle the above conjecture, in the affirmative, for β = π/2 and for all n ∈ N.

Convolutions of harmonic right half-plane mappings

“…Many research papers in recent years have studied the convolution or Hadamard product of planar harmonic mappings, see [2][3][4][5][6][7][8][9][10][11][12]. 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].…”

Some new harmonic mappings convex in one direction and their convolution

“…It is interesting to explore the properties of convolutions of harmonic functions as they are quite different from those of analytic functions. Many fruitful convolution results have been established in the recent past (see [3,5,6,8,13,[16][17][18]). We mention below few of them.…”

“…In 2012, Dorff et al [6] established the following result. In 2015, Kumar et al [8] considered more general class of right half plane mappings F a = H a + G a , given by…”

On logarithmic coefficients of certain starlike functions related to the vertical strip

“…The function B δ (z) due to Dorff [2] and studied in [3], [4], [12] and [13]. The function B δ (z) is convex univalent in ∆ and has the form…”