Starlike, convex and close-to convex functions of complex order (Q1855983)

The main purpose of this paper is to introduce two classes \(P_\beta (\lambda,b)\) and \(R_\beta (\lambda,b)\) \((b,a\) complex number with \(\text{Re} (b)>0)\) of functions which are analytic in the unit disc. Coefficient estimates for functions in these classes and radii of close-to convexity, starlikeness and convexity are obtained. Further applications in the fractional calculus are given for functions in \(P_\beta(\lambda,b)\) and \(R_\beta( \lambda, b)\). \(P_0(\lambda,b)\) and \(R_0(\lambda,b)\) are classes of starlike and convex functions of complex order \(b\). \(R_0(0,b)\) is the class of close-to convex functions of complex order \(b\), \(P_0(\lambda, 1-\alpha)\) and \(R_0(\lambda,1-\alpha)\) were studied by Altintaş (1990). \(R_0(\alpha,1)\) was investigated by Altinaş and Ertekin (1991,1992). The classes \(P_0(0,b)\) and \(P_0(1,b)\) were studied by Owa (1978,1989). Owa gave the following conjecture: 1. if \(f(z) \in P_0(0,b)\) then \(\sum^\infty_{n=2} (n+|b|-1) a_n\leq|b|\); 2. if \(f(z)\in P_0(1,b)\) then \(\sum^\infty_{n=2} n(n+|b|-1)a_n \leq |b|\); 3. if \(f(z)\in R_0(0,b)\) then \(\sum^\infty_{n=2} na_n\leq |b|\). \(P(\lambda,b)\) and \(R(\lambda,b)\) were studied by Altintaş and Özkan (1999). Therefore, the classes \(P_\beta(\lambda,b)\) and \(R_\beta (\lambda,b)\) are a generalization of Altintaş and Özkan's works.