Problem
ALG-B2-M05-P021 Exponential with fixed average
#21
★★★★★ Level 5 of 5
Let \(x_1,\ldots,x_n\) be real numbers. Prove \[\frac{e^{x_1}+\cdots+e^{x_n}}{n}\ge e^{(x_1+\cdots+x_n)/n}.\]
Hint. This is direct Jensen for the convex function \(e^x\).
The function \(e^x\) is convex on the whole real line. Therefore Jensen says that the average of the function values is at least the function value at the average argument: \[\frac1n\sum e^{x_i}\ge e^{(\sum x_i)/n}.\]
A generalizing problem, useful as a bridge to logarithmic and exponential substitutions.