# Question concerning a minimum

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.