Showing $\sum_{k=1}^n \frac{1}{1+a_k} \ge\frac{n}{1+\sqrt[n]{\prod_{k=1}^{n} a_k}} .$

Question

Suppose $$a_1 , a_2 , ... , a_n$$ are given real numbers that are greater or equal than $1$.

Prove that $$\sum_{k=1}^n \frac{1}{1+a_k} ≥ \frac{n}{1+\sqrt[n]{\prod_{k=1}^{n} a_k}} $$ I tried using Bergstrom but the it results the opposite inequality(less or equal). And I tried writing them as $$a_1 = x_1 + 1$$ so I can work on positive real numbers but it gets really complicated.

Please help. Thanks!

Answer 1

Write $a_k = e^{b_k}$ for some $b_k \geq 0$. Then use Jensen’s inequality on the convex function $$x \to \frac{1}{1+e^x}.$$

Answer 2

Try to prove that $f(x)=\frac{1}{1+e^x}$ is a convex function for $x\geq0$ and use Jensen.