Problem
ALG-B2-M05-P022 General shifted product
#22
★★★★★ Level 5 of 5
Let \(x_1,\ldots,x_n\ge0\) and \(x_1+\cdots+x_n=S\). Prove \[\prod_{i=1}^n(1+x_i)\le\left(1+\frac{S}{n}\right)^n.\]
Hint. Take logarithms and apply Jensen to \(\ln(1+x)\).
The function \(\ln(1+x)\) is concave for \(x\ge0\). Hence \[\frac1n\sum_{i=1}^n\ln(1+x_i)\le\ln\left(1+\frac{S}{n}\right).\] Multiplying by \(n\), we get \(\ln\prod(1+x_i)\le n\ln(1+S/n)\). Exponentiating gives the result.
The difficulty is choosing \(\ln(1+x)\), not \(\ln x\), and returning carefully to the product.