## 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.)
Full Text:

### References:

  Miller SS, Differential Subordinations: Theory and Applications, Monographs and Textbooks in Pure and Applied Mathematics 225 (2000)  DOI: 10.1016/0022-247X(82)90155-X · Zbl 0487.30007  Goodman AW, Trans. Amer. Math. Soc. 68 pp 204– (1950)  Srivastava HM, Current Topics in Analytic Function Theory (1992)  DOI: 10.1090/S0002-9939-1965-0178131-2  DOI: 10.1090/S0002-9947-1969-0232920-2  DOI: 10.1006/jmaa.1993.1204 · Zbl 0774.30008  DOI: 10.1016/j.camwa.2006.08.022 · Zbl 1132.30311  Uralegadi BA, J. Math. Res. Expo. 15 pp 14– (1995)  DOI: 10.1016/0022-247X(72)90081-9 · Zbl 0246.30031  Li JL, Soochow J. Math. 25 pp 91– (1999)  Liu JL, Chinese Quar. J. Math. 15 pp 27– (2000)  Özkan Ö, J. Inequal. Appl. pp 1– (2006)  Patel J, Indian J. Pure. Appl. Math. 35 pp 1167– (2004)  DOI: 10.1155/IJMMS/2006/94572 · Zbl 1137.30006  Whittaker ET, A Course on Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; With an Account of the Principal Transcendental Functions,, 4. ed. (1927) · JFM 53.0180.04  DOI: 10.1307/mmj/1029002507 · Zbl 0439.30015  DOI: 10.1090/S0002-9939-1975-0374403-3  DOI: 10.1016/0022-0396(85)90082-8 · Zbl 0507.34009  DOI: 10.1112/jlms/s2-21.2.287 · Zbl 0431.30007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.