Abstract:

Using the Nabla integral on time scales, this paper establishes the monotonicity rules for the ratios of parametric Nabla integrals

and the ratios of the parametric Nabla integrals with variable limits

In the part of monotonicity rules for the ratios of parametric Nabla integrals, some different special cases are considered in detail, including the ratio of two polynomials on time scales and the ratio of two Nabla Laplace transforms. Using these monotonicity rules, the monotonicity of the functions $ s\mapsto\frac{\sum\limits_{i=1}^n \mathcal{J}{u_i}(s)}{n \mathcal{J}{\bar{u}}(s)} $, $ s\mapsto\frac{\sum\limits_{i=1}^n \mathcal{J}{v}(u_is)}{n \mathcal{J}{v}(\bar{u}s)} $, $ s\mapsto\frac{\sum\limits_{i=1}^n K_{u_i}(s)}{n K_{\bar{u}}(s)} $, $ s\mapsto\frac{\sum\limits_{i=1}^n \mathcal{Y}{u_i}(s)}{n \mathcal{Y}{\bar{u}}(s)} $ and $ s\mapsto\frac{\sum\limits_{i=1}^n \mathcal{Y}{v}(u_is)}{n \mathcal{Y}{v}(\bar{u}s)} $ is proved, where $ \bar{u}=\sum\limits_{i=1}^n u_i/n $, $ I_u(\cdot), K_u(\cdot) $ are the modified Bessel functions of the first and second kind, respectively, $ \mathcal{J}_u(s):= \big( \frac{s}{2} \big)^{-u} I_{u}(s) $ and $ \mathcal{Y}_u(s):=K_u(s)-K_0(s) $.

Key words: Monotonicity rules, Time scales, Nabla integral, Modified Bessel functions, Laplace transforms