Abstract
First we present and discuss an important proof of Hardy's inequality via Jensen's inequality which Hardy and his collaborators did not discover during the 10 years of research until Hardy finally proved his famous inequality in 1925. If Hardy had discovered this proof, it obviously would have changed this prehistory, and in this article the authors argue that this discovery would probably also have changed the dramatic development of Hardy type inequalities in an essential way. In particular, in this article some results concerning power-weight cases in the finite interval case are proved and discussed in this historical perspective. Moreover, a new Hardy type inequality for piecewise constant p = p(x) is proved with this technique, limiting cases are pointed out and put into this frame. Mathematics Subject Classification: 26D15.
Highlights
IntroductionHardy’s inequality in its original continuous form reads: If f is non-negative and p-integrable on (0,∞),
Hardy’s inequality in its original continuous form reads: If f is non-negative and p-integrable on (0,∞), ∞⎛ x ⎞p⎝ 1 f (y)dy⎠ dx ≤ x p p−1 p∞ f p(x)dx, p > 1. (1:1)The dramatic more than 10 years period of research until Hardy proved (1.1)in 1925, [1], was recently described by Kufner et al.[3]
We shall point out some limit cases of inequalities (1.1) and (4.1)
Summary
Hardy’s inequality in its original continuous form reads: If f is non-negative and p-integrable on (0,∞),. Hardy type inequalities are equivalent, all constants are sharp and the inequalities hold for p < 0. Our proof of Theorem 2.4 shows that all constants in this inequality must be sharp These facts have not been pointed out before in this generality, but for sure it should have been done early if Hardy had discovered the proof above. In this case the inequality is in a sense sharp (see Theorems 3.1 and 3.6).
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