APPLICATIONS OF CERTAIN FUNCTIONS ASSOCIATED WITH LEMNISCATE BERNOULLI

For J(alpha, f(z)) = (1 - alpha) zf'(z)/f(z) + alpha[1 + zf''(z)/f'(z)] (alpha >= 0), denote SL(alpha) and SLc as classes of alpha-convex and convex function which respectively satisy condition vertical bar[J(alpha, f(z))](2) - 1 vertical bar < 1 and vertical bar[1 + zf''(z)/f'(z)](2) - 1 vertical bar < 1. Using established results, namely 1 + beta zp'(z) < 1+ Dz/1+ Ez , 1 + beta zp'(z)/p(z) < 1+ Dz/1+ Ez and 1 + 1 + beta zp'(z)/p(2)(z) < 1+ Dz/1+ Ez imply p (z) < root 1+z where p(z) is an analytic function defined on the open unit disk D with p(0) = 1. This article obtains conditions so that analytic functions f belong to the classes SL(alpha) and SLc.