Sufficient Conditions for Strong Starlikeness

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 ((1 + z)/(1 − z))^α when C(z)z² p (z) + D(z)zp (z) + E(z)p(z) = 0. 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+2z2/3 =: ϕCAR(z). Similar problems were investigated for Re p(z) 0 when the function p(z) + βzp (z)/p^n(z) with n = 0, 2 is subordinate to 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.