UPPER BOUNDS FOR RADIUS PROBLEMS INVOLVING RATIOS OF ANALYTIC FUNCTIONS

In recent years, the problem of finding the sharp radii bounds for certain properties in geometric function theory has attracted several researchers. However, there are several instances where only lower bounds for the radius problems have been established. In this paper, we have worked in the similar direction to compute the upper bounds in these cases which coincides with the conjectured values. Moreover, explicit functions are provided which yield that these bounds are attainable.

By making use of the concept of subordination, Ma and Minda in 1992, integrated several subclasses of functions which map the unit disk D onto a starlike domain and defined the class S^*(φ) (for a specific φ) consisting of functions f ∈ A with zf'(z)/f(z) ≺ φ(z) for all z ∈ D, where the function φ is univalent, with positive real part that maps D onto a domain symmetric with respect to real axis and starlike with respect to φ(0) = 1 and φ'(0) > 0. In 2021, Lecko, Ravichandran and Sebastian introduced the class G involving ratio of analytic functions and computed various S^*(φ)-radii for the class G. But for the five choices of φ associated with parabolic domain, exponential function, cardioid domain, lune shaped domain and rational function, the calculated radii by Lecko et al. were not sharp. Consequently, this research paper is an attempt to tackle these type of radii problems.