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 $frac{1}{e^2}leqalphaleq 1$ 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.
real-analysis optimization
edited Dec 28 '18 at 14:10
uniquesolution
8,6311823
asked Dec 28 '18 at 13:49
FatsWallersFatsWallers
46318
add a comment |
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 $frac{1}{e^2}leqalphaleq 1$ 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.
real-analysis optimization
edited Dec 28 '18 at 14:10
uniquesolution
8,6311823
asked Dec 28 '18 at 13:49
FatsWallersFatsWallers
46318
add a comment |
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 $frac{1}{e^2}leqalphaleq 1$ 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.
real-analysis optimization
edited Dec 28 '18 at 14:10
uniquesolution
8,6311823
asked Dec 28 '18 at 13:49
FatsWallersFatsWallers
46318
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 $frac{1}{e^2}leqalphaleq 1$ 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.
real-analysis optimization
real-analysis optimization
edited Dec 28 '18 at 14:10
uniquesolution
8,6311823
asked Dec 28 '18 at 13:49
FatsWallersFatsWallers
46318
edited Dec 28 '18 at 14:10
uniquesolution
8,6311823
asked Dec 28 '18 at 13:49
FatsWallersFatsWallers
46318
edited Dec 28 '18 at 14:10
uniquesolution
8,6311823
edited Dec 28 '18 at 14:10
uniquesolution
8,6311823
edited Dec 28 '18 at 14:10
uniquesolution
8,6311823
8,6311823
asked Dec 28 '18 at 13:49
FatsWallersFatsWallers
46318
asked Dec 28 '18 at 13:49
FatsWallersFatsWallers
46318
asked Dec 28 '18 at 13:49
FatsWallersFatsWallers
46318
46318
add a comment |
add a comment |
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