Problem
ALG-B2-M09-P014 Shifted reciprocal
#14
★★★★☆ Level 4 of 5
Let \(a,b,c\ge0\), \(a+b+c=3\). Prove \[\frac1{1+a}+\frac1{1+b}+\frac1{1+c}\ge\frac32.\]
Hint. The function \(1/(1+x)\) is convex.
By Jensen for the convex function \(f(x)=1/(1+x)\), \(\frac13\sum f(a)\ge f(1)=1/2\). Therefore the sum is at least \(3/2\).
Companion to the previous problem: the direction changes because of convexity.