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This is a general method to have power sum inequality .

We work with $x_i> 1$ $n$ real numbers .

We want to show this kind of inequality :

Let $x_i> 1$ be $n$ real positive numbers and $a_i>0$ be $n$ real numbers and $b_i>0$ be $n$ real numbers then we have : $$\sum_{i=1}^{n}x_i^{a_i}\geq \sum_{i=1}^{n} x_i^{b_i}$$ With the conditions : $$a_1\ln(x_1)\geq a_{2}\ln(x_{2})\geq \cdots \geq a_n\ln(x_n)$$ $$b_1\ln(x_1)\geq b_{2}\ln(x_{2})\geq \cdots \geq b_n\ln(x_n)$$ And : $$\prod_{i=1}^{k} a_k\geq \prod_{i=1}^{k}b_k,1\leq k\leq n$$

In fact log-majorization implies majorization with sum . And finally use Karamata's inequality .

Example :

Let $a,b,c,d>1$ be real numbers then we have $$\sum_{cyc}a^{\frac{1}{ab}}\geq \sum_{cyc}a^{\frac{2}{(ab)^2}}$$ With the condition : $$\frac{\ln(a)}{ab}\geq \frac{\ln(b)}{bc}\geq \frac{\ln(c)}{cd}\geq \frac{\ln(d)}{da}$$ $$\frac{2\ln(a)}{(ab)^2}\geq \frac{2\ln(b)}{(bc)^2}\geq \frac{2\ln(c)}{(cd)^2}\geq \frac{2\ln(d)}{(da)^2}$$ And : $$ab\geq 2$$ $$ab^2c\geq 4$$ $$ab^2c^2d\geq 8$$ $$a^2b^2c^2d^2\geq 16$$

My question have you others examples of this kind ? And there exists a proof without Karamata's inequality ?

Thanks in advance for your time .

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