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A stability result concerning the Shannon entropy. (English) Zbl 0991.39021
Let $$\Gamma_n=\{ (p_1,\dots,p_n)\in{\mathbb R}^n:p_1,\dots,p_n\geq 0$$, $$\sum_{i=1}^np_i=1\}$$. The sequence of functions $$I_n:\Gamma_n\to{\mathbb R}$$ ($$n=2,3,\dots$$) given by $$I_n(p_1,\dots,p_n)=-\sum_{i=1}^np_i\log_2p_i$$ is called a Shannon entropy. By Faddeev’s theorem it is characterized by the following conditions:
F1) for every integer $$n\geq 2$$, the function $$I_n$$ is symmetric in all its arguments;
F2) for every $$n\geq 2$$, $$(p_1,\dots,p_n)\in\Gamma_n$$ and $$t\in[0,1]$$ $I_{n+1}\bigl(p_1(1-t),p_1t,p_2,\dots,p_n\bigr)=I_n(p_1,\dots,p_n)+p_1I_2(1-t,t);$
F3) the function $$f(x):=I_2(1-x,x)$$ ($$0\leq x\leq 1$$) is continuous in $$[0,1]$$ and $$f({1\over 2})=1$$.
\smallskip The following problem is studied: Let $$\{I_n\}$$ be a sequence of functions satisfying F1), F3) and (instead of F2)) $\Bigl|I_{n+1}\bigl(p_1(1-t),p_1t,p_2,\dots,p_n\bigr)-I_n(p_1,\dots,p_n)-p_1I_2(1-t,t) \Bigr|\leq\sigma_n$ for every $$n\geq 2$$, $$(p_1,\dots,p_n)\in\Gamma_n$$, $$t\in[0,1]$$ and suitable $$\sigma_n=\sigma_n(t;p_1,\dots,p_n)$$. Is it then true that $\Bigl|I_n(p_1,\dots,p_n)+\sum_{i=1}^np_i\log_2p_i \Bigr|\leq K$ for every $$n\geq 2$$, $$(p_1,\dots,p_n)\in\Gamma_n$$ and $$K$$ dependent on $$\{\sigma_n\}$$?
\smallskip It is proved that for a particular form of functions $$\sigma_n$$ the answer is positive.

##### MSC:
 39B82 Stability, separation, extension, and related topics for functional equations 94A17 Measures of information, entropy 39B72 Systems of functional equations and inequalities
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