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On an inequality from Information Theory
Horst, Alzer
2014-12-23
Abstract
We prove that the inequalities
$$
\sum_{j=1}^n \frac{q_j (q_j-p_j)^2}{q_j^2 +m_j^{\alpha} M_j^{1-\alpha}}
\leq
\sum_{j=1}^n p_j \log \frac{p_j}{q_j}
\leq
\sum_{j=1}^n \frac{q_j (q_j-p_j)^2}{q_j^2 +m_j^{\beta} M_j^{1-\beta}}
\quad{(\alpha, \beta \in \mathbb{R})},
$$
where
$$
m_j=\min(p_j^2, q_j^2)
\quad\mbox{and}
\quad{M_j=\max(p_j^2, q_j^2)}
\quad(j=1,...,n),
$$
hold for all positive real numbers
$p_j, q_j$ $(j=1,...,n; n\geq 2)$ with
$\sum_{j=1}^n p_j=\sum_{j=1}^n q_j$ if and
only if $\alpha\leq 1/3$ and $\beta\geq 2/3$.
This refines a result of Halliwell and Mercer, who showed that the inequalities
are valid with $\alpha=0$ and $\beta=1$.
Series
Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics
46 (2014)
Languages
en
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