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I'm interested by the following problem :

Let $a,b,c,d>0$ such that $a\geq b \geq c \geq d $ then we have : $$|a+d-\sqrt{ad}-\Big(a^ad^d\Big)^{\frac{1}{a+d}}|\geq|b+c-\sqrt{bc}-\Big(b^bc^c\Big)^{\frac{1}{b+c}}|$$

In fact it's a new refinement of my last inequality New bound for Am-Gm of 2 variables

It's very very sharp and I try to use the niculescu inequality :

Let $f(x)$ be a twice differentiable function and convex on $I$ then for $a\geq b \geq c \geq d $ and $a,b,c,d\in I$the Niculescu inequality states that : $$f(a)+f(d)-f\Big(\frac{a+d}{2}\Big)\geq f(b)+f(c)-f\Big(\frac{b+c}{2}\Big)$$

But I think it's too weak and I'm really lost . I recognize that I'm just an amateur and create problem is easy but solve it is hard .

If you have a hint it would be nice .

Thanks again and again !

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