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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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