DIFFERENTIAL SUBORDINATION FOR FUNCTIONS ASSOCIATED WITH THE LEMNISCATE OF BERNOULLI

Abstract

Conditions on $\beta$ are determined so that $1 + \beta zp'(z)$ subordinated to $\sqrt{1+z}$ implies $p$ is subordinated to $\sqrt{1+z}$. Analogous results are also obtained involving the expressions $1 + \beta{zp'(z)}/{p(z)}$ and $1 + \beta{zp'(z)}/{p^2(z)}$. These results are applied to obtain sufficient conditions for normalized analytic functions $f$ to satisfy the condition $|\left({zf'(z)}/{f(z)}\right)^2-1| \lt 1$.