Solution to a Function Equation and Divergence Measures

As early as in 1952, Chernoff 1 used the α-divergence to evaluate classification errors. Since then, the study of various divergence measures has been attracting many researchers. So far, we have known that the Csiszár f-divergence is a unique class of divergences having information monotonicity, from which the dual α geometrical structure with the Fisher metric is derived, and the Bregman divergence is another class of divergences that gives a dually flat geometrical structure different from the α-structure in general. Actually, a divergence measure between two probability distributions or positive measures have been proved a useful tool for solving optimization problems in optimization, signal processing, machine learning, and statistical inference. For more information on the theory of divergence measures, please see, for example, 2–5 and references therein. Motivated by these studies, we investigate in this paper the solution to the following function equation