Chebyshev inequality in function spaces
type: Item
instance of: scholarly article
title: Chebyshev inequality in function spaces
zbMATH Open document ID: 0748.26011
author: H. P. Heinig
author: Lech Maligranda
published in: Real Analysis Exchange
publication date: 1992-09-26T00:00:00Z
review text: This paper gives new variants, generalizations and abstractions of the well-known Chebyshev inequality for monotonic functions. For example, the following result was proved by reviewer's method: Let \(K\) be a positive continuous function on \(I^ 2\;(I=[0,a],a>0)\) and suppose \(f:I^ 2\to[0,\infty)\) is a continuous positive set function. a) If for all \(t\in I\) \(\int^ a_ t \int^ t_ 0 K(u,v)dv du = \int^ t_ 0 \int^ a_ tK(u,v)dv du\) then \[ \int^ a_ 0 \int^ a_ 0K(s,t)f(s,t)ds dt\leq\int^ a_ 0 \left(\int^ a_ 0K(s,t)ds\right)f(t,t)dt. \] b) If for all \(s,t\in I\) \(\int^ a_ 0K(u,t)du=Bw(t)\) and \(\int^ a_ 0K(s,v)dv=Bw(s)\), then \[ \int^ a_ 0 \int^ a_ 0K(s,t)f(s;t)ds dt\leq B\int^ a_ 0w(s)f(s,s)ds. \]
Mathematics Subject Classification ID: 26D15
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 54504
zbMATH Keywords: Chebyshev inequality
zbMATH Keywords: Banach function spaces
zbMATH Keywords: symmetric spaces
zbMATH Keywords: Ky Fan inequality
reviewed by: J. Pečarić
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 54504