## 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 $$\mathcal I^\lambda_{p,\delta}$$.

### MSC:

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

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