Generalizations of Hadamard products of functions with negative coefficients (Q1916840)

\({\mathcal I}(n)\) denotes the family of regular functions in the unit circle \(| z|<1\) such that \[ f(z)=z-\sum^\infty_{k=n+1}a_kz^k\quad (a_k\geq 0). \] If \(f(z)\in{\mathcal I}(n)\) satisfies the following condition \[ \text{Re} \left\{ {zf'(z)\over f(z)} \right\}>\alpha \quad (0\leq\alpha<1), \] then \(f(z)\) is called to be in the family \({\mathcal I}^*(n,\alpha)\). If \(f(z)\in{\mathcal I}(n)\) satisfies the following condition \[ \text{Re}\left\{1+{zf'(z)\over f''(z)} \right\}> \alpha\quad(0\leq\alpha<1). \] then \(f(z)\) is called to be in the family \({\mathcal C}(n,\alpha)\). \({\mathcal I}^*(n,\alpha)\) or \({\mathcal C}(n,\alpha)\) are considered with respect to generalized Hadamard product by means of \textit{S. K. Chatterjea}'s results [J. Pure Math. 1, 23-26 (1981; Zbl 0514.30010)].