Problem
ALG-B2-M09-P013 The fraction x/(1+x)
#13
★★★★☆ Level 4 of 5
Let \(a,b,c\ge0\), \(a+b+c=3\). Prove \[\frac{a}{1+a}+\frac{b}{1+b}+\frac{c}{1+c}\le\frac32.\]
Hint. The function \(x/(1+x)\) is concave for \(x\ge0\).
By Jensen for the concave function \(f(x)=x/(1+x)\), \(\frac13\sum f(a)\le f(1)=1/2\). Hence the sum is at most \(3/2\). Equality holds at \(a=b=c=1\).
Typical extremum of a fractional sum under a fixed sum.