# $a^{ab}+b^{bc}+c^{cd}+d^{da}\geq a^{2a^2b^2}+b^{2b^2c^2}+c^{2c^2d^2}+d^{2d^2a^2}$ with some conditions

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 ?