On two subclasses of analytic functions related to Carlson-Shaffer operator defined by subordination

Srinivasan, Ramalingam Sathish; Sivasubramanian, Srikandan; Ezhilarasi, Raman; Sudharsan, Thirumalai Vinjimur

doi:10.22075/ijnaa.2026.38882.5533

On two subclasses of analytic functions related to Carlson-Shaffer operator defined by subordination

Articles in Press

Document Type : Research Paper

Authors

Ramalingam Sathish Srinivasan ¹

Srikandan Sivasubramanian ²

Raman Ezhilarasi ³

Thirumalai Vinjimur Sudharsan ³

¹ University of Madras and Department of Mathematics, S.I.V.E.T. College, Gowrivakkam, Chennai-600073, Tamil Nadu, India

² Department of Mathematics, University College of Engineering Tindivanam, Anna University, Tindivanam-604001, Tamil Nadu, India

³ Department of Mathematics, S.I.V.E.T. College, Gowrivakkam, Chennai-600073, Tamil Nadu, India

10.22075/ijnaa.2026.38882.5533

Abstract

Let $\mathscr{A}$ denote the class of analytic functions defined in the open unit disk, and let $\mathcal{P}$ be the family of analytic and univalent functions with positive real part that map the unit disk onto a convex domain. In this paper, we introduce the new subclasses $\mathcal{K}_{p,q}^{\alpha,\beta}(\psi,\phi)$ and $\mathcal{S}_{p,q}^{\alpha,\beta}(\psi,\phi)$ associated with the Carlson--Shaffer operator. These classes are defined by means of suitable subordination conditions involving analytic functions related to convex and starlike mappings. We investigate several geometric properties of functions belonging to these subclasses and establish coefficient estimates along with other related results. The results obtained in this work generalize and unify various earlier studies in the theory of analytic and univalent functions.

Keywords

Analytic function

close to convex

close to starlike

subordination

Carlson-Shaffer operator

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