Vol. 70, No. 2, 1977

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On some new generalizations of Shannon’s inequality

Pál Fischer

Vol. 70 (1977), No. 2, 351–360
Abstract

Let An = {P Rn : P = (p1,p2,,pn), where i=1npi = 1 and pi > 0 for i = 1,2,,n} and let Bn = {P An : p1 p2 pn}. We show that the inequality

n∑          ∑n
pif(pi) ≧   pif(qi)
i=1         i=1
(1)

for all P,Q Bn and some integer n 3, implies that f(p) = clog p + d, where c is an arbitrary nonnegative number and d is an arbitrary real number. We show, furthermore, that if we restrict the domain of the inequality (1) to those P,Q Bn for which P Q (Hardy-Littlewood-Pólya order), then any function that is convex and increasing satisfies (1).

Mathematical Subject Classification 2000
Primary: 94A15
Milestones
Received: 20 April 1977
Published: 1 June 1977
Authors
Pál Fischer