Some applications of fractional calculus operators to a new class of analytic functions with negative coefficients (Q2731552)

Let \(A(n)\), \(n\in\mathbb{N}\), denote the class of functions \(f\) of the form \(f(z)=z- \sum^\infty_{k=n+1} \alpha_kz^k\), \(\alpha_k\geq 0\) for \(k=n+1, \dots\) which are holomorphic in the unit disc \(U\). The author considers the class \(P(n,\lambda, \alpha,r)\), \(0\leq\lambda\), \(\alpha\leq 1\), \(r=1,2,\dots\), of functions \(f\in A(n)\) which satisfies the condition NEWLINE\[NEWLINE\text{Re} \left\{z{ (\lambda rz^{r-1}+1 -\lambda)f'(z)+ \lambda z^rf''(z)\over \lambda z^rf'(z)+(1-\lambda)f(z)} \right\}> \alpha\quad \text{for all }z\in U.NEWLINE\]NEWLINE He proves various distortion theorems for the fractional calculus of the functions \(f\) in the class \(P(n,\lambda, \alpha,r)\).