On convexity and close-to-convexity of an integral transform on subclasses of univalent functions (Q1897926)

Let \(S^0\), \(S^*\), and \(K\) denote the classes of convex, starlike, and close-to-convex functions in the unit disk. The author considers the integral transform \[ F(z)= \int^z_0 \prod^p_{i= 1} (f_i'(t))^{\alpha_i} \prod^q_{j= 1} \Biggl({g_j(t)\over t}\Biggr)^{\beta_j} dt, \] provided (1) \(f_1,\dots, f_k, g_1,\dots, g_l\in S^0\); \(f_{k+ 1},\dots,f_m, g_{l+ 1},\dots, g_n\in S^*\); \(f_{m+ 1},\dots, f_p\), \(g_{n+ 1},\dots, g_q\in K\); where \(0\leq k\leq m\leq p\); \(0\leq l\leq n\leq q\); and \(\alpha_i\), \(\beta_j\) are complex numbers. For a class \(\mathfrak M\) of functions, let \(\Lambda({\mathfrak M})\subset C^{p+ q}\) denote the set of all parameters \(\lambda= (\alpha_1,\dots, \alpha_p,\) \(\beta_1,\dots, \beta_q)\) such that whenever conditions (1) are satisfied, then \(F\in {\mathfrak M}\). The author gives a precise description of the sets \(\Lambda(S^0)\) and \(\Lambda(K)\) and provides an outline of the proof. Previously known results follow as special cases of the author's theorem.