linguistic:translation, structural:reformatting
A sharpened and generalized form

Let X be a one-dimensional random variable with mean μ {\displaystyle \mu } and variance σ 2 ≥ 0 {\displaystyle \sigma ^{2}\geq 0}. Let φ (x) {\displaystyle \varphi (x)} be a twice differentiable function, and define the function

Then

In particular, when φ (x) {\displaystyle \varphi (x)} is convex, then φ ″ (x) ≥ 0 {\displaystyle \varphi ''(x)\geq 0}, and the standard form of Jensen's inequality immediately follows for the case where φ (x) {\displaystyle \varphi (x)} is additionally assumed to be twice differentiable.