Abstract
A time-scale version of the Hardy inequality is presented, which unifies and extends well-known Hardy inequalities in the continuous and in the discrete setting. An application in the oscillation theory of half-linear dynamic equations is given.
Highlights
Introduction and preliminariesOne gets more than two hundred papers when searching by the keywords “Hardy” and “inequality” in the review journals Zentralblatt fur Mathematik or Mathematical Reviews
Almost half of these publications appeared after 1990. These papers deal with various generalizations, extensions and improvements of the wellknown Hardy inequality (HI) presented in monograph [8], namely, for example, HI in several variables, weighted HI, inequalities of Hardy’s type involving certain transforms and forms, HI involving higher order derivatives, HI on certain manifolds, in various spaces, and many others
Let us mention at least a few papers [5, 11, 15], among many others dealing with various types of HI’s, and nice monographs [12, 13, 14]
Summary
One gets more than two hundred papers when searching by the keywords “Hardy” and “inequality” in the review journals Zentralblatt fur Mathematik or Mathematical Reviews. Many related topics can be found when one looks for inequalities involving functions and their integrals and derivatives. 496 Hardy inequality on time scales a generalized Euler dynamic equation. Those results turn out to be new even in the special linear case. B ∈ T and a delta differentiable function f , the Cauchy integral is defined by b a f. Note that similar observations as in the lemma can be done without difficulties when the integrals are taken over finite intervals, and when the integrand is replaced by a nonincreasing function. By virtue of the definition of the delta Riemann integrability, there exists a time scale TD containing a and satisfying (1.6), such that. Unless either f , g are proportional, or at least one of the functions is identically zero
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