Functional equation approach to inequalities. VI
In connection with the backward Cauchy inequality \[ (a_ 1b_ 1- \sum^{n}_{j=2}a_ jb_ j)^ 2\geq (a^ 2_ 1- \sum^{n}_{j=2}a^ 2_ j)(\quad b^ 2_ 1-\sum^{n}_{j=2}b^ 2_ j) \] (with equality holding if and only if \(b_ j=ca_ j\), where \(a_ j,b_ j\geq 0\), \(1\leq j\leq n\), c is a constant), this paper establishes the Hölder-Lorentz inequality, the Minkowski-Lorentz inequality and the Beckenbach-Lorentz inequality in a unique manner by means of the functional equation approach of dynamic programming. In fact, they are inverse inequalities of those established in Part I of this series [J. Math. Anal. Appl. 71, 423-430 (1979; Zbl 0425.26009)]. [For Parts II-IV cf.: J. Math. Anal. Appl. 78, 522-530 (1980; Zbl 0461.26010); ibid. 80, 31-35 (1981; Zbl 0472.26009); ibid. 86, 96-98 (1982; Zbl 0486.26009).]
functional equation approach of dynamic programming
scientific article; zbMATH DE number 3959826
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