Problem
ALG-B3-M09-P008 Jensen with a Bound
#8
★★★☆☆ Level 3 of 5
Let \(f\) satisfy \(f\left(\frac{x+y}{2}\right)=\frac{f(x)+f(y)}{2}\) and be bounded above on \([0,1]\). Prove that \(f(x)=ax+b\).
Reduce to a Jensen-linear function with \(g(0)=0\).
Let \(g(x)=f(x)-f(0)\). Then \(g(0)=0\), \(g\) satisfies the same Jensen equation and is bounded above on an interval. The standard result gives continuity and linearity \(g(x)=ax\). Hence \(f(x)=ax+b\).
Acceptable theoretical fact for this module.