Sufficient Conditions for Starlike Functions Associated with the Lemniscate of Bernoulli

TL;DR

This paper establishes sufficient conditions involving complex functions and derivatives that guarantee certain starlike properties related to the lemniscate of Bernoulli, expanding the understanding of geometric function theory.

Contribution

It introduces new conditions on parameters ensuring functions are subordinate to specific starlike functions associated with the lemniscate of Bernoulli.

Findings

Derived conditions on eta for subordination to +Az)/(1+Bz)

Established criteria for p(z) related to +Az)/(1+Bz) and +z

Explored additional problems in similar subordination contexts

Abstract

Let -1\leq B<A\leq 1. Condition on \beta, is determined so that 1+\beta zp'(z)/p^k(z)\prec(1+Az)/(1+Bz)\;(-1<k\leq3) implies p(z)\prec \sqrt{1+z}. Similarly, condition on \beta is determined so that 1+\beta zp'(z)/p^n(z) or p(z)+\beta zp'(z)/p^n(z)\prec\sqrt{1+z}\;(n=0, 1, 2) implies p(z)\prec(1+Az)/(1+Bz) or \sqrt{1+z}. In addition to that condition on \beta is derived so that p(z)\prec(1+Az)/(1+Bz) when p(z)+\beta zp'(z)/p(z)\prec\sqrt{1+z}. Few more problems of the similar flavor are also considered.