Geometric analysis of integral operators related to Touchard polynomials and generalized Bessel functions

Manas Kumar Giri; Narjes Alabkary; Raghavendar Kondooru; Saiful R. Mondal; Manas Kumar Giri; Narjes Alabkary; Raghavendar Kondooru; Saiful R. Mondal

doi:10.3934/math.20251265

AIMS Mathematics

2025, Volume 10, Issue 12: 28753-28784. doi: 10.3934/math.20251265

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Research article

Geometric analysis of integral operators related to Touchard polynomials and generalized Bessel functions

1.
Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology Vellore, Vellore 632014, India
2.
Department of Mathematics and Statistics, College of Science, King Faisal University, Al-Ahsa 31982, Saudi Arabia

Received: 25 August 2025 Revised: 25 October 2025 Accepted: 25 November 2025 Published: 05 December 2025
MSC : 30C45, 30C50, 33C10

This article investigated the inclusion properties of Touchard polynomials $ Y_m(p, z) $ and generalized Bessel functions of the first kind $ \mathcal{G}_a(z) $, along with their associated convolution and integral operators, within the analytic function classes $ \mathcal{R}_{s, d}^{w}(\delta) $ and $ \mathcal{M}_{b, s}^{w}(\delta) $. Using coefficient bounds of certain function classes, we derived sufficient conditions for the convolution operators $ \mathcal{Y}_m^p(\psi, z) $ and $ \mathcal{E}_{l_{a}, c}(\psi, z) $, and the integral operators $ \mathfrak{Y}_m(p, z) $ and $ \mathfrak{G}_{a}(z) $ to belong to various subclasses of starlike and convex functions. Furthermore, inclusion results among these subclasses were obtained, thereby extending and unifying several existing results in geometric function theory.
- special functions,
- convex functions,
- starlike functions,
- spirallike functions,
- integral operators
Citation: Manas Kumar Giri, Narjes Alabkary, Raghavendar Kondooru, Saiful R. Mondal. Geometric analysis of integral operators related to Touchard polynomials and generalized Bessel functions[J]. AIMS Mathematics, 2025, 10(12): 28753-28784. doi: 10.3934/math.20251265

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Abstract

This article investigated the inclusion properties of Touchard polynomials $ Y_m(p, z) $ and generalized Bessel functions of the first kind $ \mathcal{G}_a(z) $, along with their associated convolution and integral operators, within the analytic function classes $ \mathcal{R}_{s, d}^{w}(\delta) $ and $ \mathcal{M}_{b, s}^{w}(\delta) $. Using coefficient bounds of certain function classes, we derived sufficient conditions for the convolution operators $ \mathcal{Y}_m^p(\psi, z) $ and $ \mathcal{E}_{l_{a}, c}(\psi, z) $, and the integral operators $ \mathfrak{Y}_m(p, z) $ and $ \mathfrak{G}_{a}(z) $ to belong to various subclasses of starlike and convex functions. Furthermore, inclusion results among these subclasses were obtained, thereby extending and unifying several existing results in geometric function theory.

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