Power sum inequality












0














Today I have this to propose :




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




My try :



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



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



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 , $3leq aleq 3.3$ and $0leq bleq 0.5$ and $bgeq 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 .






real-analysis inequality







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  • Are you working on this inequality by any chance? If so, please visit my chatroom as I have made progress with it as well :)
    – TheSimpliFire
    Jan 4 at 14:13


















0














Today I have this to propose :




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




My try :



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



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



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 , $3leq aleq 3.3$ and $0leq bleq 0.5$ and $bgeq 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 .






real-analysis inequality







share|cite|improve this question
























  • Are you working on this inequality by any chance? If so, please visit my chatroom as I have made progress with it as well :)
    – TheSimpliFire
    Jan 4 at 14:13
















0












0








0


1





Today I have this to propose :




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




My try :



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



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



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 , $3leq aleq 3.3$ and $0leq bleq 0.5$ and $bgeq 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 .






real-analysis inequality







share|cite|improve this question















Today I have this to propose :




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




My try :



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



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



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 , $3leq aleq 3.3$ and $0leq bleq 0.5$ and $bgeq 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 .






real-analysis inequality




real-analysis inequality






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




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edited Jan 5 at 14:50







FatsWallers

















asked Jan 4 at 13:48









FatsWallersFatsWallers

46318




46318












  • Are you working on this inequality by any chance? If so, please visit my chatroom as I have made progress with it as well :)
    – TheSimpliFire
    Jan 4 at 14:13




















  • Are you working on this inequality by any chance? If so, please visit my chatroom as I have made progress with it as well :)
    – TheSimpliFire
    Jan 4 at 14:13


















Are you working on this inequality by any chance? If so, please visit my chatroom as I have made progress with it as well :)
– TheSimpliFire
Jan 4 at 14:13






Are you working on this inequality by any chance? If so, please visit my chatroom as I have made progress with it as well :)
– TheSimpliFire
Jan 4 at 14:13












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