Sufficient conditions for strong starlikeness

Abstract

Let p be an analytic function defined on the open unit disc D with p(0) = 1 and 0 < α ≤ 1. The conditions on complex valued functions C, D and E are obtained for p to be subordinate to (Sufficient conditions for confluent (Kummer) hypergeometric function and generalized and normalized Bessel function of the first kind of complex order to be subordinate to ((1 + z)/(1 − z)) α are obtained as applications. The conditions on α and β are derived for p to be subordinate to ((1 + z)/(1 − z)) α when 1 + βzp (z)/p n (z) with n = 1, 2 is subordinate to 1 + 4z/3 + 2z 2 /3 =: ϕ CAR (z). Similar problems were investigated for Re p(z) > 0 when the functions p(z) + βzp (z)/p n (z) with n = 0, 2 is subordinate to ϕ CAR (z). The condition on β is determined for p to be subordinate to ((1 + z)/(1 − z)) α when p(z) + βzp (z)/p n (z) with n = 0, 1, 2 is subordinate to ((1 + z)/(1 − z)) α .