For a convex function ff and points 𝐱(1),…,𝐱(t)\mathbf{x}^{(1)},…,\mathbf{x}^{(t)},

f(1t⋅𝐱(1)+...+1t⋅𝐱(t))≤1t⋅f(𝐱(1))+...+1t⋅f(𝐱(t))f\left(\frac{1}{t}\cdot\mathbf{x}^{(1)}+...+\frac{1}{t}\cdot\mathbf{x}^{(t)}\right) \leq \frac{1}{t}\cdot f(\mathbf{x}^{(1)}) + ... + \frac{1}{t}\cdot f(\mathbf{x}^{(t)})

More generally, deriving from variance and expectation,
𝔼[f(X)]≥f(𝔼[X])\mathbb{E}[f(X)] \geq f(\mathbb{E}[X])

For concave function,

References:

  1. https://www.chrismusco.com/amlds2023/lectures/lec8_annotated.pdf
  2. https://www.probabilitycourse.com/chapter6/6_2_5_jensen's_inequality.php
  3. https://mathworld.wolfram.com/JensensInequality.html