It is well known that the traditional Jensen inequality is proved by lower bounding the given convex function, $f(x)$, by the tangential affine function that passes through the point $(E\{X\},f(E\{X\}))$, where $E\{X\}$ is the expectation of the random variable $X$. While this tangential affine function yields the tightest lower bound among all lower bounds induced by affine functions that are tangential to $f$, it turns out that when the function $f$ is just part of a more complicated expression whose expectation is to be bounded, the tightest lower bound might belong to a tangential affine function that passes through a point different than $(E\{X\},f(E\{X\}))$. In this paper, we take advantage of this observation, by optimizing the point of tangency with regard to the specific given expression, in a variety of cases, and thereby derive several families of inequalities, henceforth referred to as ``Jensen-like'' inequalities, which are new to the best knowledge of the author. The degree of tightness and the potential usefulness of these inequalities is demonstrated in several application examples related to information theory.

翻译：众所周知，传统的Jensen不等式是通过用经过点$(E\{X\},f(E\{X\}))$（其中$E\{X\}$为随机变量$X$的期望）的切向仿射函数对给定凸函数$f(x)$进行下界估计而得证的。尽管该切向仿射函数在所有与$f$相切的仿射函数所诱导的下界中能提供最紧的下界，但当函数$f$仅是需要对其期望进行界定的复杂表达式的一部分时，最紧的下界可能属于经过不同于$(E\{X\},f(E\{X\}))$的点的切向仿射函数。本文利用这一观察，通过针对不同情况下的具体表达式优化切点位置，推导出若干不等式族，即作者所知的文献中未曾出现过的“Jensen型”不等式。通过多个与信息论相关的应用实例，展示了这些不等式的紧致程度及其潜在实用性。