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A characterization of entropies of degree α

Authors:	L Losonczi

Institution:	(1) Department of Mathematics, L. Kossuth University, Pf. 12, 4010 Debrecen, Hungary

Abstract:	The functional equation $\mathop \sum \limits_{i = 1}^k \mathop \sum \limits_{j = 1}^l f(p_i q_j ) = \mathop \sum \limits_{i = 1}^k f(p_i ) + \mathop \sum \limits_{j = 1}^l f(q_j ) + \lambda \mathop \sum \limits_{i = 1}^k f(p_i )\mathop \sum \limits_{j = 1}^l f(q_j )p_i \geqslant 0,q_j \geqslant 0,\mathop \sum \limits_{i = 1}^k p_i = \mathop \sum \limits_{j = 1}^l q_j = 1$ has been studied by several authors under various assumptions onf and onk, l. Here we solve this equation iff is measurable andk 3,l 2 are fixed integers. Using the solution we characterize the entropies of degree for all real . Our results generalize the results ofBehara/Nath 1973],Kannappan 1974] andMittal 1976].

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