Entropy functions and functional equations (Q2897916)

Let a real number \(q\) or real numbers \(\alpha\) and \(\beta\) be given. The author finds all solutions \(f: (0,1] \to \mathbb{R}\) of the functional equations NEWLINE\[NEWLINE f(xy) + f((1 - x)y) - f(y) =(f(x) + f(1 - x))y^{q} \tag{1} NEWLINE\]NEWLINE and NEWLINE\[NEWLINE f(xy) = \frac{x^{\alpha} + x^{\beta}}{2}f(y) + \frac{y^{\alpha} + y^{\beta}}{2}f(x) NEWLINE\]NEWLINE connected with the Shannon entropy and also with the Tsallis entropy.NEWLINENEWLINEThe results are applied to find all bounded above or below on a subset of \((0,1)\) of positive Lebesgue measure (or Lebesgue measurable on \((0,1)\)) solutions of equations \((1)\) and NEWLINE\[NEWLINE f(xy) = x^{\alpha}f(y) + y^{\alpha}f(x). NEWLINE\]