Invariance of some subclass of multivalent functions under a differintegral operator. (English) Zbl 1195.30028

Summary: Let \(\mathcal A_p\) denote the class of functions analytic in the open unit disc \(\mathcal U=\{z\in\mathbb C : | z|<1\}\) and given by the series
\[ f(z)=z^p+\sum_{n=p+1}^\infty a_n z^n. \]
For \(f\in\mathcal A_p\), let
\[ \mathcal I^\lambda_{p,\delta} f(z)=z^p +\sum_{n=p+1}^\infty\bigg(\frac{p+\delta}{n+\delta}\bigg)^\lambda a_n z^n\qquad (\lambda\in \mathbb R,\; \delta>-p,\; z\in \mathcal U). \]
A function \(f\in\mathcal A\) is in the class \(\mathcal S^\lambda_\delta(\alpha;A,B)\) (\(0\leq \alpha<p,\; -1\leq B<A\leq a\)) if
\[ \frac{1}{p-\alpha}\bigg(\frac{z(\mathcal I^\lambda_{p,\delta}f(z))'}{\mathcal I^\lambda_{p,\delta}f(z))}-\alpha\bigg)\prec\frac{1+Az}{1+Bz}\qquad(z\in \mathcal U) \]
is satisfied, where \(\prec\) stands for subordination. The main result of this article is the invariance of the class \(\mathcal S^\lambda_\delta(\alpha;A,B)\) under the operator \(\mathcal I^1_{p,\mu}\) when \(\mu\) varies. Other results include Strohäcker-Marx-like inequalities involving the operator \(\mathcal I^\lambda_{p,\delta}\).


30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
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