1 Applications of Jensen’s inequality
In this chapter, \(h\) denotes the function \(h(x) := x \log \frac{1}{x}\) for \(x \in [0,1]\).
\(h\) is strictly concave on \([0,\infty )\).
Check that \(h'\) is strictly monotone decreasing.
If \(S\) is a finite set, and \(\sum _{s \in S} w_s = 1\) for some non-negative \(w_s\), and \(p_s \in [0,1]\) for all \(s \in S\), then
Apply Jensen and Lemma 1.1.
If equality holds in the above lemma, then \(p_s = \sum _{s \in S} w_s h(p_s)\) whenever \(w_s \neq 0\).
Need some converse form of Jensen, not sure if it is already in Mathlib. May also wish to state it as an if and only if.
If \(S\) is a finite set, and \(a_s,b_s\) are non-negative for \(s\in S\), then
with the convention \(0\log \frac{0}{b}=0\) for any \(b\ge 0\) and \(0\log \frac{a}{0}=\infty \) for any \(a{\gt}0\).
Let \(B:=\sum _{s\in S} b_s\). Apply Jensen and Lemma 1.1 to show that \(\sum _{s\in S} \frac{b_s}{B} h(\frac{a_s}{b_s})\ge h(\frac{\sum _{s\in S} a_s}{B})\).
If equality holds in Lemma 1.4, then \(a_s=r\cdot b_s\) for every \(s\in S\), for some constant \(r\in \mathbb {R}\).
By the fact that \(h\) is strictly concave and the equality condition of Jensen.