Integral transforms of functions to be in certain class defined by the combination of starlike and convex functions

Let Pγ(β)Pγ(β), β<1β<1, denote the class of all normalized analytic functions ff in the unit disc D={z∈C:|z|<1}D={z∈C:|z|<1} such that Re (eiϕ((1−γ)f(z)z+γf′(z)−β))>0,z∈D for some ϕ∈Rϕ∈R. Let M(μ,α)M(μ,α), 0≤μ<10≤μ<1, denote the Pascu class of αα-convex functions of order μμ and given by the analytic condition Re αz(zf′(z))′+(1−α)zf′(z)αzf′(z)+(1−α)f(z)>μ which unifies S∗(μ)S∗(μ) and C(μ)C(μ), respectively, the classes of analytic functions that map DD onto the starlike and convex domain. In this work, we consider integral transforms of the form Vλ(f)(z)=∫01λ(t)f(tz)tdt. The aim of this paper is to find conditions on λ(t)λ(t) so that the above transformation carry Pγ(β)Pγ(β) into M(μ,α)M(μ,α). As applications, for specific values of λ(t)λ(t), it is found that several known integral operators carry Pγ(β)Pγ(β) into M(μ,α)M(μ,α).