Monotonicity and bounded variation for harmonic mappings with applications to special functions

Abstract

We investigate sufficient conditions for the close-to-convexity and univalence of planar harmonic mappings. First, we establish a unified monotonicity criterion for harmonic mappings that incorporates the ordinary and odd harmonic cases. We then develop a criterion based on the total variation of successive coefficient differences. Unlike conventional monotonicity conditions, this approach allows the coefficient sequences to exhibit non-monotone behavior. Several previously established criteria for harmonic close-to-convexity are recovered as particular consequences. Applications are developed for harmonic mappings associated with Gaussian hypergeometric functions and Legendre polynomials, with explicit restrictions on their parameters.