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An inequality with applications in potential theory. (English) Zbl 0531.26010

Let \(x_ 0<x_ 1<...<x_ n\) be a collection of points on the real line for which the consecutive differences \(\tau_ i=x_ i-x_{i-1}\), \(i=1,...,n\), are either nonincreasing or nondecreasing. Let \(\alpha\) be a number satisfying \(0<\alpha<1\). The principal result of the paper is to show that for any set \(m_ 0,...,m_ n\) of nonnegative numbers we have \[ \int^{x_ n}_{x_ 0}(\sum^{n}_{i=1}\frac{m_ i}{(x-x_ i)^ 2})^{\alpha /1+\alpha}dx\leq \frac{20}{1-\alpha}(\sum m_ i)^{\alpha /1+\alpha}(\sum \tau_ i)^{\alpha /1+\alpha}. \] It is subsequently shown how this inequality changes if the monotonicity hypothesis on the \(\tau_ i\) is relaxed, although it is shown that monotonicity cannot be dropped completely. This inequality, which originally arose in connection with some problems in potential theory, also has an interpretation in terms of information theory which is sketched out in the final section of the paper.

MSC:

26D15 Inequalities for sums, series and integrals
31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
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References:

[1] Boris Korenblum, An extension of the Nevanlinna theory, Acta Math. 135 (1975), no. 3-4, 187 – 219. · Zbl 0323.30030 · doi:10.1007/BF02392019
[2] -, Description of Riesz measures for some classes of subharmonic functions (preliminary report), Abstracts Amer. Math. Soc. 2 (1981), 433.
[3] Boris Korenblum, Some problems in potential theory and the notion of harmonic entropy, Bull. Amer. Math. Soc. (N.S.) 8 (1983), no. 3, 459 – 462. · Zbl 0519.31001
[4] A. Hinkkanen and R. C. Vaughan, An analytic inequality, manuscript communicated by W. K. Hayman. · Zbl 0577.26014
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