Differential subordination for functions associated with the lemniscate of bernoulli

Abstract

Conditions on β are determined so that 1 + βzp′(z) subordinated to √1 +z implies p is subordinated to √ 1 + z. Analogous results are also obtained involving the expressions 1+βzp′ (z)/p(z) and 1+βzp′(z)/p2(z). These results are applied to obtain sufficient conditions for normalized analytic functions f to satisfy the condition |(zf′(z)/f(z))2 -1| < 1.