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

Abstract

In this work, the geometric nature of solutions to two second-order differential equations, zy′′(z)+a(z)y′(z)+b(z)y(z)=0 and z2y′′(z)+a(z)y′(z)+b(z)y(z)=d(z), is studied. Here, a(z), b(z), and d(z) are analytic functions defined on the unit disc. Using differential subordination, we established that the normalized solution F(z) (with F(0) = 1) of above differential equations maps the unit disc to the domain bounded by the leminscate curve 1+z. We construct several examples by the judicious choice of a(z), b(z), and d(z). The examples include Bessel functions, Struve functions, the Bessel–Sturve kernel, confluent hypergeometric functions, and many other special functions. We also established a connection with the nephroid domain. Directly using subordination, we construct functions that are subordinated by a nephroid function. Two open problems are also suggested in the conclusion.