Abstract
We formulate and prove a converse for a generalization of the classical Minkowski's inequality. The case when is also considered. Applying the same technique, we obtain an analog converse theorem for integral Minkowski's type inequality.
Highlights
If p > 1, ai ≥ 0, and bi ≥ 0 i 1, . . . , n are real numbers, by the classical Minkowski’s inequality n 1/p n 1/p ai bi p ≤ api bip i1This inequality was published by Minkowski 1, pages 115–117 hundred years ago in his famous book “Geometrie der Zahlen.”It is known see 2 that for 0 < p < 1 the above inequality is satisfied with “≥” instead of “≤”.Many extensions and generalizations of Minkowski’s inequality can be found in 2, 3
We want to point out the following inequality:
If m ≤ n, inequality 1.4 is sharp in the cases when 1 ≤ q < p and 0 < q ≤ 1 ≤ p
Summary
If p > 1, ai ≥ 0, and bi ≥ 0 i 1, . . . , n are real numbers, by the classical Minkowski’s inequality n 1/p n 1/p ai bi p ≤ api bip i1It is known see 2 that for 0 < p < 1 the above inequality is satisfied with “≥” instead of “≤”.Many extensions and generalizations of Minkowski’s inequality can be found in 2, 3. We obtain an analog converse theorem for integral Minkowski’s type inequality. If m ≤ n, inequality 1.4 is sharp in the cases when 1 ≤ q < p and 0 < q ≤ 1 ≤ p.
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