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On the Inequality

Published online by Cambridge University Press:  20 November 2018

Pal Fischer
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In this article, we are concerned with the following inequality

(1)

where 0<pi<1, 0<q<1, (i=l, 2,…,n), n is a fixed positive integer, n≥2 and f(p)≠0 for <p<l.

This inequality was first considered by A. Renyi, who gave the general differentiate solution of (1) for n≥3, [1]. With the help of this inequality one can characterize Renyi’s entropy [2].

We shall state later the Renyi’s result, which will be a special case of the Theorem 3.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 17 , Issue 2 , 01 June 1974 , pp. 193 - 199
Copyright
Copyright © Canadian Mathematical Society 1974

References

1. Renyi, A., Unpublished.Google Scholar
2. Renyi, A., On the Foundations of Information Theory, Rev. Inst. Internat. Stat. 33 (1965), 1-14.Google Scholar
3. Aczel, J. and Pfanzagl, G., Remarks on the Measurement of Subjective Probability and Information, Metrika II (1966), 91-105.Google Scholar
4. Aczel, J. and Daroczy, Z., Measures of Information and their Characterizations, Academic Press, New York, in preparation.Google Scholar
5. Fischer, P., On the inequality Σpif(pi) ≥ Σ pif(qi), Metrika, Vol. 18 1972, 199-208.Google Scholar
6. Fischer, P., On the inequality Σg(pi)f(pi)≥ Σg(pi)f(qi), in print, Aequation. Math.Google Scholar
7. M, J.. Kemperman, B., A general functional equation Transactions of the American Math. Society, 1957, pp. 28-56, 1957.Google Scholar