Prove cyclic inequality

Question

Please help me to prove this.

Assume$\forall n,a_n>0$, then $${a_1a_2\over a_3}+{a_2a_3\over a_4}+\dots+{a_{n-1}a_n\over a_1}+{a_na_1\over a_2}\geq a_1+\dots+a_n.$$

I can prove for $n=3$, but it seems impossible to extend it to the general case. Please help to prove. Thanks.

For $n=3$, it is easy to show $${ab\over c}+{bc\over a}+{ca\over{{{{b}}}}}\geq a+b+c\iff{1\over c^2}+{1\over a^2}+{1\over {{{{b}}}}^2}\geq{1\over{ {{cb}}}}+{1\over{ac}}+{1\over{a {{b}}}}$$ which holds following$$a^2+b^2+c^2\geq ab+bc+ca.$$ But it seems hard to extend it.

Answer 1

Hint: Set $a_i = e^{x_i}$ so that we need to show: $$\sum_{cyc} e^{x_1+x_2-x_3} \ge \sum_{cyc} e^{x_1}$$ and use Karamata's inequality.

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To show the majorization, apart from showing the double stochastic matrix, here is a more elementary though cumbersome way: Let $x_{i_1}+x_{i_1+1}-x_{i_1+2} \ge x_{i_2}+x_{i_2+1}-x_{i_2+2} \ge \dots \ge x_{i_n}+x_{i_n+1}-x_{i_n+2}$ and $x_{j_1} \ge x_{j_2} \ge \dots \ge x_{j_n}$ for some indices $i_k, j_k \in \{1, 2, ... n\}$.

Then, obviously $ x_{i_1}+x_{i_1+1}-x_{i_1+2} \ge x_{j_1}+x_{j_1+1}-x_{j_1+2} \ge x_{j_1}$.

Similarly, $(x_{i_1}+x_{i_1+1}-x_{i_1+2})+(x_{i_2}+x_{i_2+1}-x_{i_2+2}) \ge (x_{j_1}+x_{j_1+1}-x_{j_1+2})+(x_{j_2}+x_{j_2+1}-x_{j_2+2}) \ge x_{j_1}+x_{j_2}$ and so on.

Answer 2

It's wrong!

Try $n=4$ and $$(a_1,a_2,a_3,a_4)=(1,20,2,6)$$