Cyclic to symmetric conversion
by MathPassionForever, Sep 4, 2019, 7:54 AM
A comparatively much smaller post (as of now), as I don't have much information on this as of now.
We can clearly see that the SD theorems are strong. But then, some inequalities are cyclic, sigh. So the basic objective of this post is to serve as a collection of inequalities which can make cyclic inequalities symmetric (and these results need to be strong themselves,for our inequality shouldn't become false after their use
)
Here is the first one:
Proof: This inequality is obviously cyclic. So we may work with just 2 cases:
and 
In either case, we have:
Luckily, I have a question to post
Question: For nonnegative reals
, prove that
And in case someone doesn't find it that sharp, we have the famous and super strong Vasc inequality here!
![\[ (a^2 + b^2 + c^2)^2 \geq 3(a^3b + b^3c + c^3a) \ \forall a,b,c \in \mathbb{R} \]](//latex.artofproblemsolving.com/7/4/6/74654a43f2ccd3f8db453256b7b230baa274c910.png)
I shall take leave here. Bye, see you soon!
We can clearly see that the SD theorems are strong. But then, some inequalities are cyclic, sigh. So the basic objective of this post is to serve as a collection of inequalities which can make cyclic inequalities symmetric (and these results need to be strong themselves,for our inequality shouldn't become false after their use
)Here is the first one:
Proof: This inequality is obviously cyclic. So we may work with just 2 cases:
and 
In either case, we have:
Luckily, I have a question to post
Question: For nonnegative reals
, prove that
![\[ (a^2 + b^2 + c^2)^2 \geq (\sqrt{2}-1)(a+b+c)(\sqrt{2}\left(a^3+b^3+c^3)+\dfrac{4}{27}(a+b+c)^3\right)\]](http://latex.artofproblemsolving.com/e/d/1/ed1caa4cef3ce1702be9fe4c4f33e485135ec567.png)
Well well, looks ugly. But what is the SD for? The first check is 
and the second one is 
.
,And in case someone doesn't find it that sharp, we have the famous and super strong Vasc inequality here!
![\[ (a^2 + b^2 + c^2)^2 \geq 3(a^3b + b^3c + c^3a) \ \forall a,b,c \in \mathbb{R} \]](http://latex.artofproblemsolving.com/7/4/6/74654a43f2ccd3f8db453256b7b230baa274c910.png)
I shall take leave here. Bye, see you soon!This post has been edited 5 times. Last edited by MathPassionForever, Sep 5, 2019, 8:32 PM
September 2019