×

Generalizations of Dobrushin’s inequalities and applications. (English) Zbl 0873.60006

Summary: Let \(f:\mathbb{R}^n\to \mathbb{R}\) be a seminorm and let \((e_i)_{1\leq i\leq n}\) be the canonical base of \(\mathbb{R}^n\). Denote \(M={1\over 2}\max_{r,s}f(e_r-e_s)\), \(K=\max_rf(e_r)\). We prove the inequality \[ f(x)\leq M\Biggl( \sum_{i=1}^n |x_i|\Biggr)+ (K-M)\Biggl|\sum_{i=1}^n x_i\Biggr|,\qquad x=(x_1,x_2,\dots,x_n)\in \mathbb{R}^n. \] We use the above inequality to prove some generalizations of Dobrushin’s inequalities and a generalization of an inequality due to J. E. Cohen, Y. Iwasa, Gh. Răutu, M. B. Ruskai, E. Seneta and Gh. Zbăganu [Linear Algebra Appl. 179, 211-235 (1993; Zbl 0764.60068)]. Hilbert space generalizations of the above inequalities are proved using Levi’s reduction theorem. As special cases of our results we obtain several inequalities given by D. D. Adamović [Mat. Vesn., N. Ser. 1(16), 39-43 (1964; Zbl 0132.08901)], D. Ž. Đoković [Period. Math. Phys. Astron., II. Ser. 18, 169-175 (1963; Zbl 0127.32503)], and H. Hornich [Math. Z. 48, 268-274 (1942; Zbl 0027.13203)].

MSC:

60E15 Inequalities; stochastic orderings
PDFBibTeX XMLCite
Full Text: DOI