This is a paper on bounds for the normalized Jensen functional. Jensen’s inequality for convex functions is one of the best known and extensively used inequalities in various fields. It is a source of many classical inequalities including the generalized triangle inequality, the arithmetic mean-geometric mean-harmonic mean inequality, the positivity of relative entropy, Shannon’s inequality, Levinson’s inequality and other results. Let
\[ \mathcal P_n : = \left\{ (p_1, p_2, \dots, p_n) \mid p_j \geq 0 \;(j=1, \dots, n),\;\sum_{j=1}^n p_j = 1\right\}. \]
The author considers the normalized Jensen functional
\[ \mathcal J_n(f, X, P): = \sum_{j=1}^n p_j f(x_j) - f\left(\sum_{j=1}^n p_j x_j\right) \geq 0 , \]
where \(f: C \rightarrow \mathbb R\) is a convex function on the convex set \(C\), \(X= (x_1, \dots, x_n) \in C^n\) and \(P=(p_1, \dots, p_n) \in \mathcal P_n\), and proves that if \(P, Q \in \mathcal P_n\), \(q_j >0 \;(j=1, \dots, n)\), then
\[ \max_{1\leq j\leq n} \{{p_j}/{q_j}\} \mathcal J_n(f, X, Q)\geq \mathcal J_n(f, X, P) \geq \min_{1\leq j\leq n}\{{p_j}/{q_j}\} \mathcal J_n(f, X, Q) \geq 0 . \]
As applications, a generalized arithmetic mean-geometric mean-harmonic mean inequality, the positivity of Kullback-Leibler divergence and a refinement of Shannon’s inequality are obtained.