Some $q$-analysis variants of Hardy type inequalities of the form $${\int }_0^b{\left(x^{\alpha -1}{\int }_0^x{t}^{-\alpha }f\left(t\right){\mathrm{d}}_{q}t\right)}^p{\mathrm{d}}_{q}x\le C{\int }_0^b{f}^p\left(t\right){\mathrm{d}}_{q}t$$ with sharp constant $C$ are proved and discussed. A similar result with the Riemann-Liouville operator involved is also proved. Finally, it is pointed out that by using these techniques we can also obtain some new discrete Hardy and Copson type inequalities in the classical case.

This paper is divided in two parts. In the first part we study a convergent $q$-analog of the divergent Euler series, with $q\in (0,1)$, and we show how the Borel sum of a generic Gevrey formal solution to a differential equation can be uniformly approximated on a convenient sector by a meromorphic solution of a corresponding $q$-difference equation. In the second part, we work under the assumption $q\in (1,+\infty )$. In this case, at least four different $q$-Borel sums of a divergent power series solution of an irregular singular...

We present a version of Opial's inequality for time scales and point out some of its applications to so-called dynamic equations. Such dynamic equations contain both differential equations and difference equations as special cases. Various extensions of our inequality are offered as well.