Let $I$ and $J$ be two intervals, and let $f, g: I \rightarrow \mathbb{ R}$. If for any points $a$ and $b$ in $I$ and any positive numbers $p$ and $q$ such that $p + q = 1$, we have \begin{ align} \nonumber p f(a) + q f(b) + g(pa + qb) \in J, \end{ align} then for any points $x_{ 1}, \ldots, x_{ n}$ in $I$ and any positive numbers $\lambda_{ 1}, \ldots, \lambda_{ n}$ such that $\sum_{ i=1}^{ n}\lambda_{ i} = 1$, we have \begin{ align} \nonumber \sum_{ i=1}^{ n}\lambda_{ i} f(x_{ i}) + g\left( \sum_{ i=1}^{ n}\lambda_{ i}x_{ i} \right) \in J. \end{ align} If we take $g = -f$ and $J = [0, +\infty)$, then the Jensen's inequality. The conclusion is only a short glimpse of the grand picture behind Jensen's inequality shows in this paper.
Categories: math.GM