Certain subclasses of multivalent functions defined by new multiplier transformations

TL;DR

This paper introduces new multiplier transformations for multivalent functions, defines two novel subclasses, and explores their properties, including inclusion, neighborhoods, and applications in fractional calculus and convolution.

Contribution

The paper defines new multiplier transformations and introduces two new subclasses of multivalent functions, analyzing their properties and relationships with existing classes.

Findings

Established inclusion relationships and neighborhoods for the subclasses.

Derived properties related to fractional calculus and quasi-convolution.

Connected new subclasses with earlier known results in the literature.

Abstract

In the present paper the new multiplier transformations $\mathrm{{\mathcal{J}% }}_{p}^{\delta }(\lambda ,\mu ,l)$ $(\delta ,l\geq 0,\;\lambda \geq \mu \geq 0;\;p\in \mathrm{% }%\mathbb{N} )}$ of multivalent functions is defined. Making use of the operator $\mathrm{% {\mathcal{J}}}_{p}^{\delta }(\lambda ,\mu ,l),$ two new subclasses $\mathcal{% P}_{\lambda ,\mu ,l}^{\delta }(A,B;\sigma ,p)$ and $\widetilde{\mathcal{P}}% _{\lambda ,\mu ,l}^{\delta }(A,B;\sigma ,p)$\textbf{\ }of multivalent analytic functions are introduced and investigated in the open unit disk. Some interesting relations and characteristics such as inclusion relationships, neighborhoods, partial sums, some applications of fractional calculus and quasi-convolution properties of functions belonging to each of these subclasses $\mathcal{P}_{\lambda ,\mu ,l}^{\delta }(A,B;\sigma ,p)$ and $\widetilde{\mathcal{P}}_{\lambda ,\mu ,l}^{\delta }(A,B;\sigma ,p)$ are investigated. Relevant connections of the definitions and results presented in this paper with those obtained in several earlier works on the subject are also pointed out.