New developments of dynamic inequalities on time scales

Abstract

UDC 517.98

We establish new results for $\diamond_\alpha$-inequalities on time scales and formulate some dynamic Hilbert-type inequalities on the $\diamond_\alpha$-calculus of time scales for functions  $\diamond_\alpha$-differentiable with respect to one and two variables. We obtain discrete and continuous inequalities as exceptional cases of our results ($\mathbb{T}=\mathbb{Z},$ $\mathbb{T}=\mathbb{R},$ and $\mathbb{T}=k\mathbb{Z},$ where $k>0$). In addition, we can derive some other inequalities on different time scales, such as $\mathbb{T}=q^{\mathbb{Z}},$ where $q>1.$ These inequalities are proved by using H\"older's inequality and the mean inequality.