Differential Subordination for Functions Associated With the Lemniscate of Bernoulli

Conditions on β are determined so that 1 + β zp'(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.