An inequality for twice differentiable convex functions and applications for the Shannon and Rényi's entropies (Q2709725)

The following Jensen's type inequality is proved. Let \(f\) be twice differentiable on \([a,b]\) and \(m\leq f''(x)\leq M\) for all \(x\in [a,b]\). If \(x_{i}\in [a,b]\) \((i=1,\dots,n)\) and \(p=( p_{i}) _{i=1,\dots,n}\) is a probability distribution, then NEWLINE\[NEWLINE\frac{1}{2} \sum_{i,j=1}^{n}p_{i}p_{j}(x_{i}-x_{j})(f'(x_{i})-f'(x_{j}))-\frac{M}{4}\sum _{i,j=1}^{n}p_{i}p_{j}(x_{i}-x_{j})^{2}NEWLINE\]NEWLINE NEWLINE\[NEWLINE\leq \sum_{i=1}^{n}p_{i}f(x_{i}) -f\left( \sum_{i=1}^{n}p_{i}x_{i}\right)NEWLINE\]NEWLINE NEWLINE\[NEWLINE\leq \frac{1}{2} \sum_{i,j=1}^{n}p_{i}p_{j}(x_{i}-x_{j})(f'(x_{i})-f'(x_{j}))-\frac{m}{4}\sum_{i,j=1}^{n}p_{i}p_{j}(x_{i}-x_{j})^{2}.NEWLINE\]NEWLINE Some applications for means and entropies are also given.