On certain classes of analytic functions defined by Noor integral operator
Let \(E\) be the open unit disc of the complex plane. Let \(R^*_k (\alpha)\) denote the class of all analytic functions on \(E\) with \(f(0)=0\), \(f' (0)=1\) such that \(zf'(z)/f(z)\) has a representation \[ {zf'(z)\over f(z)}=\left( {k\over 4}+{1\over 2}\right) p_1(z)-\left( {k\over 4}-{1\over 2}\right) p_2(z) \] for some \(k\geq 2\), \(0\leq\alpha <1\) and for some functions \(p_1(z)\), \(p_2(z)\) with \(p_1(0)=1= p_2(0)\), \(\text{Re} (p_i(z))> \alpha\), \((i=1,2)\) \((z\in E)\). Let * denote the Hadamard convolution and \(f_n(z)={z\over (1-z)^{n+1}}(n\) a positive integer). Let \(f_n^{(-1)}(z)\) be defined by \(f_n^{(-1)} (z)*f_n(z) ={z\over 1-z}\) and put \((I_nf) (z)=f_n^{-1}(z)*f(z)\). We shall now denote by \(R^*_k(n, \alpha)\), (where \(n\) is a positive integer and \(0\leq \alpha <1)\) the class of all analytic functions \(f\) on \(E\) with \(f(0)=0\), \(f'(0)=1\) for which \(I_n(f)\in R^*_k (\alpha)\). The authors obtain several results like coefficient estimates, effect of certain integral operators etc. for \(R^*_k (\alpha)\).
scientific article; zbMATH DE number 1956858
scientific article