On certain subclasses of analytic functions with negative coefficients (Q1775331)

Let \(S^{*}(n,\lambda,A,B)\) denote the class of functions \[ f(z)=a_{1}z-\sum_{m=2}^{\infty}a_{m}z^{m} \quad (a_{1}>0,a_{m}\geq 0) \] analytic and univalent in the disc \(| z| <1\) for which \[ (1-\lambda)\dfrac{D^{n}f(z)}{z}+\lambda\frac{D^{n+1}f(z)}{z}\prec a_{1}\frac{1+Az}{1+Bz}, \] where \(D^{n}\) denotes Ruscheweyh derivative, \(\lambda\geq0\), \(-1\leq A<B\leq 1\), and \(n\in N\cup{0}\). The authors obtained basic properties which include the coefficient estimates, distortion theorem, closure theorems and the radius of convexity for the classes \(S_{i}^{*}(n,\lambda,A,B)\) \((i=1,2)\).