A. I. Saied
Mathematical Institute, Slovak Academy of Sciences, Gresakova 6, 040 01, Kosice, Slovakia.
Abstract: The main objective of this work is to give several dynamic inequalities of Hardytype on time scales which represent the unified approach of the classical integral and discrete inequalities. These results include a new reformulation of Hardy-type inequality which is valid for p = 1 and Hardy-type inequalities with a negative parameter p < 0.
Furthermore, we present an inequality involving the Hardy-Steklov operator which is a generalization of Hardy operator, where the lower and upper limits of the integration are increasing functions. In addition, we show the characterizations of weighted functions that are suitable for the validity of the weighted Hardy-type inequality with two different spaces ℓp, ℓq for 1 < p ≤ q < ∞ and two different weights involving a nonnegative general kernel.
Finally, the last aim is to get a global measure of the dispersion of the function f around its integral mean (1/x) R b a f(x)Δx which gives the standard deviation when p = 2.
Keywords: Hardy type inequalities, Time scale, H¨older’s inequality, Jensen’s inequality,
Hardy-Steklov operator, The global measure of the dispersion.
Mathematics Subject Classification: 26D10, 26D15, 34N05, 47B38, 39A12