Analytic normalization and geometric behavior of generalized Bessel functions

Syed Ali Haider Shah; Hafsa; Hajer Zaway; Salma Trabelsi; Syed Ali Haider Shah; Hafsa; Hajer Zaway; Salma Trabelsi

doi:10.3934/math.20251327

AIMS Mathematics

2025, Volume 10, Issue 12: 30206-30228. doi: 10.3934/math.20251327

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Analytic normalization and geometric behavior of generalized Bessel functions

1.
Department of Mathematics, University of Sargodha, Sargodha, Pakistan
2.
Department of Mathematics and Statistics, College of Science, King Faisal University, Al Ahsa, Saudi Arabia

Received: 05 August 2025 Revised: 01 December 2025 Accepted: 17 December 2025 Published: 24 December 2025
MSC : 30C45, 33B15, 33C10

This work introduced normalized representations of generalized Bessel functions, denoted by $ _{k}H_{\beta, g}(\wp) $ and $ _{k}M_{\beta, g}(\wp; s) $ in terms of $ k $ and $ (s, k) $, where $ s > k > 0 $ and $ s < 1 $. The motivation stemed from the increasing relevance of such functions in diverse fields such as mathematical physics, wave dynamics and fractional calculus, where classical Bessel functions may be limited. This study also gave detailed geometric investigation of these generalized Bessel functions. We derived sufficient conditions for these functions to be starlike and convex of order $ \zeta $ in the open unit disk $ \mathbb{U} $. The results of our analysis revealed enhanced structural flexibility over classical counterparts, making them well-suited for advanced applications in complex modeling theory. Several illustrative examples and plots were provided to validate and visualize the behavior of the proposed functions, highlighting their potential in mathematical modeling and theoretical studies. These findings enhanced the geometric function theory framework for special functions and provided a solid foundation for future analytical and applied investigations.
- starlikeness,
- convexity,
- Gamma function,
- generalized Bessel function
Citation: Syed Ali Haider Shah, Hafsa, Hajer Zaway, Salma Trabelsi. Analytic normalization and geometric behavior of generalized Bessel functions[J]. AIMS Mathematics, 2025, 10(12): 30206-30228. doi: 10.3934/math.20251327

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Abstract

This work introduced normalized representations of generalized Bessel functions, denoted by $ _{k}H_{\beta, g}(\wp) $ and $ _{k}M_{\beta, g}(\wp; s) $ in terms of $ k $ and $ (s, k) $, where $ s > k > 0 $ and $ s < 1 $. The motivation stemed from the increasing relevance of such functions in diverse fields such as mathematical physics, wave dynamics and fractional calculus, where classical Bessel functions may be limited. This study also gave detailed geometric investigation of these generalized Bessel functions. We derived sufficient conditions for these functions to be starlike and convex of order $ \zeta $ in the open unit disk $ \mathbb{U} $. The results of our analysis revealed enhanced structural flexibility over classical counterparts, making them well-suited for advanced applications in complex modeling theory. Several illustrative examples and plots were provided to validate and visualize the behavior of the proposed functions, highlighting their potential in mathematical modeling and theoretical studies. These findings enhanced the geometric function theory framework for special functions and provided a solid foundation for future analytical and applied investigations.

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