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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!

Martin R
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Tatarusan Marius
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    "Nice looking"????? What does that mean or have to do with anything? or the double exclamation points? – David G. Stork Jan 07 '20 at 19:57
  • The exclamation points are there because you should keep in mind they're greather or equal to 1. Nice looking because it's nice/simple looking but pretty hard I think. – Tatarusan Marius Jan 07 '20 at 20:00
  • "Nice looking because it's nice/simple looking..." Do you realize how vacuous that statement is? And that adding double exclamation points (on a trivial matter) is ungrammatical? – David G. Stork Jan 07 '20 at 20:04
  • I don't know I just wanted an idea on the inequality I didn't focus on the title. – Tatarusan Marius Jan 07 '20 at 20:12
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    @DavidG.Stork $a_n \in \mathbb{R}^+$ is not the same as $a_n \geq 1$ (re: your edit) -- in fact this does not hold if we only assume $a_n \in \mathbb{R}^+$ – user125932 Jan 07 '20 at 20:16

2 Answers2

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

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

WimC
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