Delta Complete Monotonicity and Completely Monotonic Degree on Time Scales

The theory of time scale was proposed to unite continuous and discrete statements. In this paper, we introduce the concepts of complete monotonicity, logarithmic complete monotonicity, and absolute monotonicity of multivariate functions on time scales utilizing the delta derivative. Following that, we investigate the properties of delta complete monotonicity on time scales and present some judgment rules. Using judgment rules, three functions are shown to be delta complete monotonicity on time scales. And then, in order to quantitatively measure two delta completely monotonic functions on time scales, we present the concept of completely monotonic degree of univariate and multivariate functions on time scales and explore some properties of them.

Automated summary

The theory of time scale is proposed to combine continuous and discrete statements. This paper introduces the concepts of complete monotonicity, logarithmic complete monotonicity, and absolute monotonicity of multivariate functions on time scales using the delta derivative. The properties of delta complete monotonicity on time scales are investigated and some judgment rules are presented. The concept of completely monotonic degree of univariate and multivariate functions on time scales is presented to quantitatively measure two delta completely monotonic functions on time scales, and some properties of them are explored.