Certain classes of analytic functions with negative coefficients (Q1086376)

Let T be the class of functions f having the power series \[ f(z)=z- \sum^{\infty}_{m=2}a_ mz^ m,\quad a_ m\geq 0. \] Let \(S_ n(A,B)\) denote the class of functions \(f\in T\) such that \[ D^{n+1}f(z)/D^ nf(z)=(1+AW(z))/(1+BW(z)),\quad -1\leq A<B\leq 1, \] \(n\in N\cup \{0\}\), w(z) is a regular function with \(| w(z)| <1\), \(z\in E\) (the unit disc) and \[ D^{\alpha}f(z)=f(z)*z/(1-z)^{\alpha} \] where * denotes the usual Hadamard product. Also, let \(K_ n(A,B)\) denote the class of functions \(f\in T\) such that \(zf'(z)\in S_ n(A,B)\). In this paper, the authors find coefficient inequalities, extreme points, support points and radius of convexity for these classes. Typical are the following two results: (1) Let \(f\in T\). Then \(f\in S_ n(A,B)\) if and only if \[ \sum^{\infty}_{m=2}[(n+m-1)!/(n+1)!(m-1)!]C_ ma_ m\leq B-A, \] where \(C_ m=(B+1)(n+m)-(A+1)(n+1)\). (2) If \(f(z)\in S_ n(A,B)\), then f is convex for \(| z| <r_ 2\) where \(r_ 2=\inf_{m}[\frac{(n+m-1)!C_ m}{m(n+1)!m!(B-A)}]^{(1-m)}\).