Abstract
AbstractIn this paper we derive some new inequalities involving the Hardy operator, using some estimates of the Jensen functional, continuous form generalization of the Bellman inequality and a Banach space variant of it. Some results are generalized to the case of Banach lattices on .
Highlights
Where Φ = xφ(x) + Φ(0) with φ convex increasing on [0, d), 0 < d ≤ ∞, we can derive in a similar way the following estimate for the Hardy operator: ( Hf p+1(x))
Milman writes in [12] the subtle point is that Hardy’s operator H = H(f ) is not invertible on Lp(0, ∞)-spaces, and it is not possible to find a reverse Hardy inequality of the form
Remember that the Banach lattice E is q-concave if there exists a positive constant M such that, for every finite set x1, x2, ... , xn of elements in E, we have
Summary
As applications we get some well-known such reverse Hardy-type inequalities but including the case with finite intervals. By the Jensen inequality mentioned above, Jφ(f ) ≥ 0 and, since this inequality holds in the reversed direction for concave functions, that Jφ(f ) ≤ 0 in this case. If instead φ is concave, these inequalities hold in the reversed directions.
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