Fractional integral operators and distortion theorems for a class of multivalent functions with negative coefficients (Q679195)

Let \(f\) be an analytic function \(f\) in the unit disc \(\Delta= \{|z|<1\}\) defined by the power series \(f(z)= z^p- \sum^\infty_{k=n} |a_k |z^{k+p}\), \(n=1,2, \dots\). Let \({\mathcal S}_\delta [p,\alpha,n]\), \(\delta\in (-\infty, \infty)\), \(p=1,2, \dots, 0\leq\alpha <1\), denote the class of all \(f\) satisfying a coefficient condition that \(\sum^\infty_{k=n} ((k+p)/p)^\delta (k+p-p \alpha) |a_k|\leq p(1- \alpha)\). In the present paper, the authors prove the distortion theorems involving fractional derivative and fractional integral. A covering theorem for functions \(f\) in \({\mathcal S}_\delta [p,\alpha,n]\) follows as a consequence of the bounds on \(|f(z)|\) and \(|f'(z) |\). These results unify and generalize the earlier results of several authors.