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Zbl 0541.26008
Vasić, Petar M.; Pečarić, Josip E.
On the Hölder and some related inequalities.
(English)
[J] Math., Rev. Anal. Numér. Théor. Approximation, Math. 25(48), 95-103 (1983).

This paper contains a very useful summary of various inequalities due to Hölder, Minkowski, Aczél, Popoviciu and Bellman: the statements are usually followed by short proofs, indications of proofs or references where proofs can be found. In each case the results are extended and refined, again the proofs are all extremely neat. An example of such a result that refines an inequality due to Aczél is: let $a\sb k\ge 0$, 1$\le k\le n$, $-\infty<r<\infty$ and put $$f(x)=(a\sb 1\sp{r+x}b\sb 1\sp{r-x}-\sum\sp{n}\sb{k=2}a\sb k\sp{r+x}b\sb x\sp{r-x})(a\sb 1\sp{r- x}b\sb 1\sp{r+x}-\sum\sp{n}\sb{k=2}a\sb k\sp{r-x}b\sb k\sp{r+x}).$$ Then if $a\sb k=\lambda b\sb k$, 1$\le k\le n$, f is constant and if $\vert y\vert>\vert x\vert$ and if both terms in f(y) are positive then $f(x)\ge f(y).$
[P.S.Bullen]
MSC 2000:
*26D15 Inequalities for sums, series and integrals of real functions

Keywords: inequalities due to Hölder, Minkowski, Aczél, Popoviciu; Bellman

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