Applications of subordination theory to starlike functions

Let p be an analytic function defined on the open unit disc D with p(0)=1. The conditions on α and β are derived for p(z) to be subordinate to 1+4z/3+2z²/3=:φC(z) when (1−α)p(z)+αp²(z)+βzp′(z)/p(z) is subordinate to ez. Similar problems were investigated for p(z) to lie in a region bounded by lemniscate of Bernoulli |w2−1|=1 when the functions (1−α)p(z)+αp²(z)+βzp′(z) , (1−α)p(z)+αp²(z)+βzp′(z)/p(z) or p(z)+βzp′(z)/p²(z) is subordinate to φC(z). Related results for p to be in the parabolic region bounded by the REw=|w−1| are investigated.