On extremal problems related to integral transforms of a class of analytic functions

For γ⩾0γ⩾0 and β<1β<1 given, let Pγ(β)Pγ(β) denote the class of all analytic functions f   in the unit disk with the normalization f(0)=f′(0)−1=0f(0)=f′(0)−1=0 and satisfying the conditionRe{eiϕ(f′(z)+γzf″(z)−β)}>0,z∈D, for some ϕ∈Rϕ∈R. We shall investigate the integral transformVλ(f)(z)=∫01λ(t)f(tz)tdt, where λ   is a nonnegative real valued function normalized by ∫01λ(t)dt=1. From our main results we get conditions on the number β and the function λ   such that Vλ(f)Vλ(f) is starlike of order μ   (0⩽μ⩽1/20⩽μ⩽1/2) when f∈Pγ(β)f∈Pγ(β). As applications we study various choices of λ(t)λ(t), related to classical integral transforms.