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.
Feng et al. (Sat,) studied this question.