Some monotonicity rules for quotient of integrals on time scales

TL;DR

This paper develops new monotonicity rules for quotients of various integrals on time scales, including Delta, Nabla, and Diamond-Alpha integrals, with applications to products and power series.

Contribution

It introduces novel monotonicity rules for quotient integrals on time scales, extending existing analysis tools to more complex integral forms and parameter settings.

Findings

Established monotonicity rules for quotient of Delta, Nabla, and Diamond-Alpha integrals.

Extended rules to products of multiple integrals with parameters.

Connected power series as special cases of integral with parameters.

Abstract

As an efficient mathematical tool, monotonicity rules play an extremely crucial role in the real analysis field. In this paper, we explore some monotonicity rules for quotient of Delta, Nabla and Diamond-Alpha integrals with variable upper limits and parameters on time scales, respectively. Moreover, we consider the monotonicity rules for quotient of the product of multiple Delta integrals with parameters on time scales. Power series is also concerned for being a special case of integral with parameters on time scales.