Probabilistic and Neutrosophic Coefficient Conditions in Geometric Function Theory with Applications to Sound Denoising Associated With Mersenne Polynomials

Ibrahim Tunji Awolere; Abimbola Adeniyi Ayeni; Abiodun Tinuoye Oladipo

Authors

Ibrahim Tunji Awolere Department of Mathematical Sciences, Olusegun Agagu University of Science and Technology, Okitipupa, Ondo State, Nigeria;
Abimbola Adeniyi Ayeni Department of Mathematics, Federal College of Education (Special), Oyo, Nigeria;
Abiodun Tinuoye Oladipo Department of Pure and Applied Mathematics, Ladoke Akintola University of Technology, Ogbomoso, Oyo State, Nigeria;

Keywords:

Analytic functions, Generalized distribution series, Neutrosophic logic, Poisson distribution, Co efficient estimates, Sigmoid function.

Abstract

The investigation of generalized discrete probability distributions within advanced analytical frame
works is essential for developing rigorous models in probability theory and complex analysis. The Neutrosophic
Poisson Distribution (NPD), characterized by its capacity to address uncertainty and indeterminacy, poses sig
nificant challenges that traditional probabilistic approaches often fail to resolve, particularly within the domain
of univalent function theory.
This paper investigates the analytical properties of generalized discrete probability series, Neutrosophic
Poisson distributions, and sigmoid functions. We introduce a novel Advanced Differential-Sigmoid Operator
based on Mersenne polynomials to enhance analytical precision and structural adaptability. Utilizing this
operator, we derive several functional inequalities, growth estimates, and coefficient bounds for associated classes
of analytic univalent functions. Furthermore, we explore convex combinations and the geometric behavior of
the sigmoid function within this framework.
By integrating neutrosophic logic, special functions, and operator theory, this research provides a unified
framework that advances the treatment of uncertainty and supports practical applications in statistical model
ing, signal processing, and neural networks.

DOI 10.5281/zenodo.20348875