Subordination Involving Regular Coulomb Wave Functions

Abstract

The functions 1+z, ez, 1+Az, A∈(0,1] map the unit disc D to a domain which is symmetric about the x-axis. The Regular Coulomb wave function (RCWF) FL,η is a function involving two parameters L and η, and FL,η is symmetric about these. In this article, we derive conditions on the parameter L and η for which the normalized form fL of FL,η are subordinated by 1+z. We also consider the subordination by ez and 1+Az, A∈(0,1]. A few more subordination properties involving RCWF are discussed, which leads to the star-likeness of normalized Regular Coulomb wave functions.

Geometric Nature of Special Functions on Domain Enclosed by Nephroid and Leminscate Curve

“…Recently, research into the theory of geometric functions related to the nephroid and leminscate domains has gained prominence [1][2][3][4][5][6]. Here, the leminscate domain refers to the image of D = {z : |z| < 1} by the function P l (z) = √ 1 + z, while the image of D by the function φ N e (z) = 1 + z − z 3 /3 is known as nephroid domain.…”

“…A few studies [1][2][3][4]12] in the literature have examined Problem 1, taking account of special a(z), b(z), and d(z). Nonetheless, in this investigation we first present and evaluate a broad interpretation of the findings in Section 2, and then we present examples in Section 3.…”