Delta L’Hospital-, Laplace- and Variable Limit-Type Monotonicity Rules on Time Scales

Monotonicity rules for the quotient of two functions play foundational roles in the fields of mathematics and applied mathematics. In this paper, we establish monotonicity rules for the L’Hospital type, the quotient of two Laplace transforms, and the quotient of two variable limit integrals with delta derivatives and delta integrals on time scales. Some generalized forms for these monotonicity rules are also considered in detail. As an application, we utilize monotonicity rules to demonstrate that a function is strictly decreasing.