Problem
ALG-B3-M09-P015 Monotone Jensen
#15
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Let \(f\) be nondecreasing and satisfy \(f\left(\frac{x+y}{2}\right)=\frac{f(x)+f(y)}{2}\). Prove that \(f(x)=ax+b\).
Monotonicity gives regularity for Jensen's equation.
Let \(g(x)=f(x)-f(0)\). Then \(g(0)=0\), \(g\) satisfies Jensen's equation and is nondecreasing. By the standard fact, a Jensen function with monotonicity is linear: \(g(x)=ax\). Hence \(f(x)=ax+b\).
Jensen with monotonicity instead of boundedness.