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I have this to propose it's related to this Prove that $a^{ab}+b^{bc}+c^{cd}+d^{da} \geq \pi$ :

Let $a,b,c,d$ be real positive numbers such that $abcd=\alpha$ with $\frac{1}{e^2}\leq\alpha\leq 1$ then we have : $$\sum_{cyc}a^{ab}\geq (\alpha^{0.25})^{\alpha^{0.5}}$$

I have finally found a condition such that the inequality works .

But some questions remain : Why the minimum changes if $\alpha$ is close to zero ?

How to pass to the original condition $a+b+c+d=4$ to the condition $abcd=\alpha$ ?

Can we use convexity to solve this problem ?

Many thanks if you can solve one of these questions.

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