Some properties of an integral transform

Let AA be the class of normalized analytic functions in the unit disk UU, and S,S∗(α)S,S∗(α) and K(α)K(α) denote the subclasses of AA consisting of univalent functions, starlike functions of order αα and convex functions of order αα in UU, respectively. Y.C. Kim gave the following conjecture: Let 0≤α<1,β>10≤α<1,β>1. If f∈Sf∈S, or S∗(α)S∗(α), or K(α)K(α), then ϕ(3,3+β;z)∗f(z)ϕ(3,3+β;z)∗f(z) belongs to the same class, where ϕ(a,c;z)ϕ(a,c;z) is the incomplete beta function. In this work, we prove that Kim’s conjecture is true for f∈S∗(α)f∈S∗(α) or f∈K(α)f∈K(α) and improve some other results.