Convexity Theorem for Hyperbolic Functions in Complex Analysis

This paper studies the convexity of hyperbolic complex functions, where hyperbolic numbers are commutative rings that contain zero divisors and are composed of two real numbers. Based on the zero-divisor factorization theorem of hyperbolic numbers and other properties of functions of hyperbolic numbers, this article establishes necessary and sufficient conditions for the convexity of -differentiable functions with hyperbolic complex variables. This study, which generalizes the convexity characterization theorems from real analysis to the hyperbolic complex plane, will further establish a theoretical research foundation for the function theory of hyperbolic complex analysis and meanwhile provide impetus for the application of hyperbolic functions in physics.