A new approach to signal denoising in the analytic univalent function space

Abstract

UDC 517.54

We introduce a new Gegenbauer-polynomial-based subclass of analytic bi-univalent functions and establish sharp coefficients and functional bounds, which reveal a previously unexplored relationship between the geometric function theory and signal denoising. Unlike classical Fourier and wavelet approaches, the proposed framework enforces intrinsic analytic coherence across the coefficients yielding hierarchical stability on individual, pairwise, and higher-order levels. The analytic bounds supported by the graphical evidence demonstrate how geometric constraints naturally suppress noise-induced distortions and provide a mathematically rigorous and structurally novel approach to denoising based on the univalent function theory. This is achieved by developing a robust new class of analytic $F$-starlike functions with the help of normalized bivariate Gegenbauer polynomials, the sigmiod function, and starlike-Bazilevic functions denoted by $BS_q^\mu$ via the generalized discrete probability distribution, convolution, and subordination principles. The consequences of the  obtained results are demonstrated in the corollaries and graphically. We present three new coefficient inequalities that extend and refine several standard topics in the geometric function theory, including the Bieberbach inequality, the Fekete–Szego problem, and the Zalcman or Hankel-type quadratic relation. Beyond their theoretical significance for complex analysis, we use a linear combination of functions to gain an insight into their application to denoising in signal processing. It is concluded that the method and the results are novel in their reinterpretation for signal processing, where the coefficient bounds support denoising by enforcing thresholding, structural consistency, and energy stability.