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Today I have this to propose :

Let $a,b,c,d$ be real positive numbers such that $a+b+c+d=4$ and $a\geq 3>1\geq b\ge c \geq d$ and $0\leq\varepsilon\leq d$ then we have : $$a^{ab}+b^{bc}+c^{cd}+d^{da} \geq (a+2\varepsilon)^{(a+2\varepsilon)(b-\varepsilon)}+(b-\varepsilon)^{(b-\varepsilon)c}+c^{c(d-\varepsilon)}+(d-\varepsilon)^{(d-\varepsilon)(a+2\varepsilon)} $$

My try :

I want to prove that we have $f(\varepsilon)$ convex with :

$$f(\varepsilon)=(a+2\varepsilon)^{(a+2\varepsilon)(b-\varepsilon)}+(b-\varepsilon)^{(b-\varepsilon)c}+c^{c(d-\varepsilon)}+(d-\varepsilon)^{(d-\varepsilon)(a+2\varepsilon)} $$

And work with this but I can't go further .

Any hints would be appreciable

Thanks.

Edit : it's a conjecture but with the condition of the minimum I mean , $3\leq a\leq 3.3$ and $0\leq b\leq 0.5$ and $b\geq c \geq d $ we have the following refinement : $$a^{ab}+b^{bc}+c^{cd}+d^{da}> \sum_{cyc}e^{\frac{-a^2}{2.55^2}}> \pi$$

@Simplifire Thanks for the invitation I will speak with you the next week if you want :) . And yes we are working on the same things . Have a good day .

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  • $\begingroup$ Are you working on this inequality by any chance? If so, please visit my chatroom as I have made progress with it as well :) $\endgroup$ – TheSimpliFire Jan 4 at 14:13
  • $\begingroup$ I have completed every case except for $a>1,b,c,d<1$ and $a>1,b<1,c>1,d<1$. Drop in a message (and ping me @TheSimpliFire) so that I know you're there. Thanks! $\endgroup$ – TheSimpliFire Jan 12 at 17:14

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