Power sum inequality
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.
real-analysis inequality
asked 18 hours ago
FatsWallers
46318
add a comment |
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.
real-analysis inequality
asked 18 hours ago
FatsWallers
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
17 hours ago
add a comment |
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.
real-analysis inequality
asked 18 hours ago
FatsWallers
46318
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.
real-analysis inequality
real-analysis inequality
asked 18 hours ago
FatsWallers
46318
asked 18 hours ago
FatsWallers
46318
asked 18 hours ago
FatsWallers
46318
asked 18 hours ago
FatsWallers
46318
asked 18 hours ago
FatsWallers
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
17 hours ago
add a comment |
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
17 hours ago
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
17 hours ago
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
17 hours ago
add a comment |
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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
17 hours ago