A characterization of entropies of degree α

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 α for all real α. Our results generalize the results ofBehara/Nath [1973],Kannappan [1974] andMittal [1976].