On an inequality from Information Theory

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 $\beta=1$.