[/quote]
,
be reals such that 
and 
Prove that
be a cyclic quadrilateral. The point 
is the reflection of 
the intersection of 
and 
,
is the reflection of 
.
be the reflection of 
.
concyclic.
. Prove that: 
be a triangle, and let 
be a point on side 
at 
and ray 
at 
. The circumcircle of triangle 
intersects the circumcircle 
of triangle 
distinct from point 
and 
intersect 
and 
respectively.
such that 
for all real number 
and 
and 
on the plane such that 
is an equilateral . Let 
and 
at 
and 
at 
,
on 
satisfies 
and 
.
be the point where the perpendicular from 
to line 
meets the angle bisector of 
.
holds.
be a positive integer such that 
.
.
be lines in the plane such that no two of them are parallel and no three of them are concurrent. Let 
.
on 
and 
on 
such that 
Interior points.
)
.
,
be the orthocenter of it and 
be any point on the side 
.
are on the segments 
,
and 
are cyclic. The segments 
and 
intersect at 
is a point on 
such that 
is tangent to the circumcircle of triangle 
at 
and 
intersect at 
.
and 
and 
then prove 
be the set of positive real numbers.
such that there exists a function 
with the property:![\[
f(y^2f(x) + y + c) = xf(x+y^2)
\]](http://latex.artofproblemsolving.com/e/3/0/e30ee6d9f69cc43422a5c2b9813192a632d359cb.png)
for all 
.
be non-negative real numbers such that 
Prove that
Y by jatin, az360, hctb00, dan23, dantx5, cobbler, dizzy, Einstein314, champion999, 62861, Kgxtixigct, richrow12, Adventure10, Mango247, and 18 other users
Since I was asked about it I'm posting it.
We begin with a definition:
1. Let
, 
, ..., 
, 
, 
, ..., 
be arbitrary real numbers satisfying 
and 
. Now, if we have all the following conditions fulfilled:
;
;
;
...
generally
for any natural k with 
;
and
(mind the equality sign in the last condition; it is not a
sign!), then we say that the number array 
majorizes the number array 
. We write this in the form 
.
Now, if instead of
;
;
;
...
generally for any natural k with ;
and,
we have the conditions
;
;
;
...
generally
for any natural k with ;
and
fulfilled (i. e., the same conditions with all's replaced by 
's), then we say that the number array minorizes the number array . We write this in the form 
. Of course, this is equivalent to 
.
2. Thus we have defined the notions "majorizes" and "minorizes" only for non-increasing arrays (i. e. for arrays satisfying and ). Now, assume that , , ..., , , , ..., are just arbitrary real numbers, without any conditions. Then, let 
be the non-increasing permutation of the array , i. e. the permutation of the array that satisfies 
. Similarly, let 
be the non-increasing permutation of the array , i. e. the permutation of the array that satisfies 
. The number arrays and are both non-increasing, and hence we have defined majorization for such arrays.
Then, we say that the number array majorizes the number array if and only if the number array majorizes the number array . In this case, we write .
Similarly, we say that the number array minorizes the number array if and only if the number array minorizes the number array . In this case, we write . Again this is equivalent to .
3. So we have defined the terms "majorize" and "minorize" for any two number arrays. It should be noted that majorization is a partial order on the set of number arrays, not a total order - i. e., not for every pair of two number arrays and one can say that either the first one majorizes the second one, or the second one majorizes the first one. It often happens that none of the arrays majorizes or minorizes the other one. But sometimes when you have some special arrays, you can prove that one of them majorizes the other one.
4. Now, the Karamata inequality, also called the Majorization Inequality or the Hardy-Littlewood inequality, states that if, , ..., , , , ..., are 
reals from an interval 
such that the number array majorizes the number array , and 
is any convex function, then
.
If is a concave function instead, then we instead have
.
If the number array minorizes the number array instead of majorizing it, then both inequalities are reversed.
5. The Jensen inequality for real numbers is a special case of the Karamata inequality. In fact, if
is the arithmetic mean of the numbers , , ..., , then it is easy to show that the number array majorizes the number array 
. Hence, the Karamata inequality yields:
If f(x) is any convex function, then
,
i. e.
.
If f(x) is a concave function instead, then we instead have
,
i. e.
.
Of course, this is exactly the Jensen inequality for
reals.
6. The definition of majorizing and minorizing number arrays given above is somewhat unsatisfying from an intuitive point of view, since it does not help one to imagine how an array majorizing another array looks like. Unfortunately, this is partly inherent to the notion of majorization, which indeed is quite unintuitive. For a - rather facile - visualization of the notion, you can imagine that a number array majorizes a number array if the two arrays have the same sum of numbers, but the numbers of the first array are set wider apart than those of the second array, while those of the second array lie closer together. From this intuitive viewpoint, it is clear why the number array majorizes the number array , where is the arithmetic mean of the numbers , , ..., : In fact, the two number arrays have the same sum of elements, but the elements of the second number array lie nearer to each other (they are all equal). Alas, this viewpoint does not help one to really understand what majorization is about.
Well, I know there is more to say. For instance, the Karamata inequality has a kind of converse, but I am not sure how it is formulated, so I leave this to the other MathLinkers more used to inequalities.
Darij
We begin with a definition:
1. Let
, 
, ..., 
, 
, 
, ..., 
be arbitrary real numbers satisfying 
and 
. Now, if we have all the following conditions fulfilled:
;
;
;...
generally
for any natural k with 
;and
(mind the equality sign in the last condition; it is not a
sign!), then we say that the number array 
majorizes the number array 
. We write this in the form 
.Now, if instead of
;;;...
generally
for any natural k with ;and
,we have the conditions
;
;
;...
generally
for any natural k with ;and
fulfilled (i. e., the same conditions with all
's replaced by 
's), then we say that the number array minorizes the number array . We write this in the form 
. Of course, this is equivalent to 
.2. Thus we have defined the notions "majorizes" and "minorizes" only for non-increasing arrays (i. e. for arrays satisfying
and ). Now, assume that , , ..., , , , ..., are just arbitrary real numbers, without any conditions. Then, let 
be the non-increasing permutation of the array , i. e. the permutation of the array that satisfies 
. Similarly, let 
be the non-increasing permutation of the array , i. e. the permutation of the array that satisfies 
. The number arrays and are both non-increasing, and hence we have defined majorization for such arrays.Then, we say that the number array
majorizes the number array if and only if the number array majorizes the number array . In this case, we write .Similarly, we say that the number array
minorizes the number array if and only if the number array minorizes the number array . In this case, we write . Again this is equivalent to .3. So we have defined the terms "majorize" and "minorize" for any two number arrays. It should be noted that majorization is a partial order on the set of number arrays, not a total order - i. e., not for every pair of two number arrays
and one can say that either the first one majorizes the second one, or the second one majorizes the first one. It often happens that none of the arrays majorizes or minorizes the other one. But sometimes when you have some special arrays, you can prove that one of them majorizes the other one.4. Now, the Karamata inequality, also called the Majorization Inequality or the Hardy-Littlewood inequality, states that if
, , ..., , , , ..., are 
reals from an interval 
such that the number array majorizes the number array , and 
is any convex function, then
.If
is a concave function instead, then we instead have
.If the number array
minorizes the number array instead of majorizing it, then both inequalities are reversed.5. The Jensen inequality for real numbers is a special case of the Karamata inequality. In fact, if
is the arithmetic mean of the numbers , , ..., , then it is easy to show that the number array majorizes the number array 
. Hence, the Karamata inequality yields:If f(x) is any convex function, then
,i. e.
.If f(x) is a concave function instead, then we instead have
,i. e.
.Of course, this is exactly the Jensen inequality for
reals.6. The definition of majorizing and minorizing number arrays given above is somewhat unsatisfying from an intuitive point of view, since it does not help one to imagine how an array majorizing another array looks like. Unfortunately, this is partly inherent to the notion of majorization, which indeed is quite unintuitive. For a - rather facile - visualization of the notion, you can imagine that a number array
majorizes a number array if the two arrays have the same sum of numbers, but the numbers of the first array are set wider apart than those of the second array, while those of the second array lie closer together. From this intuitive viewpoint, it is clear why the number array majorizes the number array , where is the arithmetic mean of the numbers , , ..., : In fact, the two number arrays have the same sum of elements, but the elements of the second number array lie nearer to each other (they are all equal). Alas, this viewpoint does not help one to really understand what majorization is about.Well, I know there is more to say. For instance, the Karamata inequality has a kind of converse, but I am not sure how it is formulated, so I leave this to the other MathLinkers more used to inequalities.
Darij
This post has been edited 5 times. Last edited by darij grinberg, Mar 29, 2009, 12:10 AM
a 
matrix having properties:
,
for 
is given in 
and 
( where 
denotes the transpose of 
)
?
.
and 
such that 
with the properties 1) and 2) such that 
.
![\[a_1f(x_1)+a_2f(x_2)+\dots+a_nf(x_n)\geq f(a_1x_1+\dots+a_nx_n)\]](http://latex.artofproblemsolving.com/5/7/5/5754cb3b57716e75963d3135c67dd2c546c493b0.png)
,
is convex function, ![$a_i\in[0,1]$](http://latex.artofproblemsolving.com/c/f/5/cf5c874062885702b41ac30bb2066012d49bb8c5.png)
and 
.
is a particular case only.
![$f:[a,b]\to \mathbb{R}$](http://latex.artofproblemsolving.com/b/7/3/b7344c80f2c235aaadafedfbca0e2734001b156d.png)
.![\[k\cdot f(x)+(1-k)\cdot f(y)\geq f(kx+(1-k)y)\quad \forall x,y\in [a,b],\ \forall k\in[0,1].\]](http://latex.artofproblemsolving.com/a/0/a/a0a7b7d852cb8965215cb2261f310aef9010310f.png)
for all ![$x\in[a,b]$](http://latex.artofproblemsolving.com/4/9/4/49432e6cff349a39878727bec9bfcb0efc9fbb7c.png)
.
.
for every convex function 
.
)
(conclusion: the majorization conditions).
holds for every function f of the form f(x) = |x - u| (with u constant), then 
.
is the arithmetic mean of the numbers 
,
,
,
majorizes the number array 
.
,
.
,
.
T
and 
,
would need to be convex, to convert it back to the additive form.
and 
be two functions defined on interval 
satisfying:
for any 
;
.![\[\int_{a}^{b}f\left(\varphi(x)\right)\geq \int_{a}^{b}f\left(\psi(x)\right).\]](http://latex.artofproblemsolving.com/e/0/e/e0e96f79a8227a0de04ccaaf236619b96cdb4c65.png)
I hope someone can prove it or give a counterexample.
be a smooth nonnegative function which is convex on ![$[a,b]$](http://latex.artofproblemsolving.com/8/e/c/8ecbd1ba3da8f2adef66a63f2ab32c47e63fa734.png)
and let ![$[x,y]\subset[a,b]$](http://latex.artofproblemsolving.com/c/e/7/ce70161202a64abd224f277a3bebb8d24f6673f5.png)
such that 
.
?
