I'm interested by the following problem :

Let $a,b,c,d$ be real positive numbers suc that $a\geq1$,$b\leq 1$ , $c\leq 1$ , $d\leq 1$ with conditions that I precise below then we have : $$a^{ab}+b^{bc}+c^{cd}+d^{da}\geq a^{2a^2b^2}+b^{2b^2c^2}+c^{2c^2d^2}+d^{2d^2a^2}$$ The conditions are : $$ab\ln(a)\geq -bc\ln(b)\geq -cd\ln(c)\geq -da\ln(d)$$ $$a^2b^2\ln(a^2)\geq -b^2c^2\ln(b^2)\geq -c^2d^2\ln(c^2)\geq -d^2a^2\ln(d^2)$$ $0.5\geq ab$ and $a\leq c$ and $2bc^2d\geq ab $ and finally $4c^2d^2\geq 1$

My proof is too long to explain but I use the following theorem (Koenig's theorem of Majorization) :

Let $a_i>0$ be $n$ real numbers and $b_i>0$ be $n$ real numbers then we have : $$\prod_{i=1}^{k}a_i\leq \prod_{i=1}^{k}b_i,1\leq k \leq n \implies \sum_{i=1}^{k}a_i\leq \sum_{i=1}^{k}b_i,1\leq k \leq n $$

I apply this to my majorization with sum to get a majorization with product and I get this conditions (the two first conditions are arbitrary) .

My question : Have you an alternative proof ?

Thanks in advance for your time .


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