Properties of certain classes of analytic functions with negative coefficients (Q1808745)

Let \(T\) denote the class of functions \[ f(z)= a_1z- \sum^\infty_{k=2} a_kz^k,\quad a_1>0,\quad a_k\geq 0\tag{1} \] which are analytic in the unit disc \(U= \{z:|z|< 1\}\) and let \(T_0\), \(T_1\) denote the subclasses of \(T\) satisfying \(f(z_0)= z_0\) and \(f'(z_0)= 1\), respectively. Let \(D^nf(z)= z(z^{n- 1}f(z))^{(n)}/n!\) denote the \(n\)th-order Ruscheweyh derivative and let \(K^{**}_n\) be the subclass of \(T\) such that \[ \text{Re}\Biggl\{{D^{n+ 1}f(z)\over D^nf(z)}\Biggr\}> {1\over 2}. \] In this paper the coefficient estimates, distortion theorems, closure theorems and radius of convexity of order \(\gamma\) for the classes \(K^{**}_{n,i}(z_0)= K^{**}_n\cap T_i\), \(i= 0,1\) are determined. Some other classes and problems are also investigated.