Comparison of two vectors sometimes leads to interesting inequalities, which is an important tool in comparing vectors. In general it is said that vectors are not comparable. They are either equal or unequal, yes we have been comparing the vectors in terms of their norms, but the readers will learn a new type of comparison of vectors in term their co-ordinates, and how it will be applied on functions. Majorization theory provides a method to compare the vectors with same number of coordinates. I hope that the result in this work will lead readers to discover further applications and extensions. 

For this Rajendra Bhatia has given details in the “Matrix Analysis" (Bhatia, 1997). If we talk about history of majorization, indeed many of the key ideas related to majorization were discussed in the volume entitled “Inequalities” by Hardy, Littlewood and Polya (Hardy. E. & G. Polya. 1934). The appearance of Marshall and Olkin (W. Marshal. & Olkin. 1979) book on inequalities with special emphasize on majorization generated a surge of interest in majorazation and Schur Convexity. probability of covering the circle by random arcs, by Shepp and Huffer (Huffer & A Shepp, 1987) effect of unequal catchability on estimates of the number of classes in a population by Nayak and Christman (Nayak & Christman, 1992) the mean wai-ting time for a pattern probability by Ross (Ross, 1999). Balinski and Young provided a good survey of the methods usually considered in (L. Balinski and young, 2010). 

Martin Mittelbach and Eduard Jorswieck applied majorization theory to compare different tap correlation scenarios for perfect, partial, and no CSI at the transmitter (Mittelbech, 2008)  Eduard Jorswieck and Martin Mittelbach used majorization for functions to show that the average rate with perfectly informed receiver is largest for uncorrelated scattering if the transmitter is uninformed (lorseck & Mittelbech, 2009).

Basic Notations and Preliminaries

Majorization is an important order notion that arises in several areas of mathematics. The following notations will be used in the subsequent discussion

Let   , then a^↑ and a^↓ denotes the vectors which obtained by rearranging the coordinates of a in increasing order and decreasing order respectively, 

Where, 

R^n Shows set of n tuples.

For any two real numbers a and b, maximum of a and b, and minimum of a and b are denoted as (a∨b) and (a∧b) respectively.

Definition 2.1

For any real number a∈R the function a^+, replaces the negative real number to 0, that is, a^+=a∨0.

Definition 2.2 

For any real number a∈R, |a|=a∨(-a).

Now we will extend these to 

R^n.

Let   a,b∈R^n,a∧b=(a_1∧b_1,a_2∧b_2,...,a_n∧b_n).

Now for a∈R^n, the function a^+ replaces the negative co-ordinates of a by 0. that is a^+=(a_1∨0,a_2∨0,…,a_n∨0).

Also for any a∈R^n,

|a|=(a_1∨(-a_1),a_2∨(-a_2),...,a_n∨(-a_n)). 

Definition 2.3 

Majorization

Let a,b∈R^n, we call a is majorized by b in symbol a≺b, if

In case a,b are in decreasing order.

The order of the entries of the vectors does not affect the majorization that is if we arrange the vectors in increasing order, then

Let e denotes the vector (1,1,1,....,1) and for any subset I of {1,2,3,...,n}, let e_j denotes the vector whose j^th = component is 1 if j∈I and 0 if j∉I. Given a vector a∈R^n,

Then 

Where ⟨⋅,⋅⟩ denotes the inner product in R^n and tr stands for trace.

,

Also a≺b if and only for each subset I of {1,2,3,...,n}, there exist a subset J with |I|=|J|, such that ⟨a,e_I⟩=⟨b,e_J⟩ and tr a=tr b

Definition 2.4 

Let a,b∈R^n, we call a is weakly submajorized by b in symbol a≺_w b, if

Definition 2.5 

Let a,b∈R^n, we call a is weakly supermajorized by b in symbol a≺^w b, if

Definition 2.6 

A real valued function f from R^n to R^m be isotone if

Definition 2.7

A real valued function f from R^n in to R^m be strongly isotone if

Definition 2.8 

Function f from R^n in to R^m be strictly isotone if

More ever if m=1, we have f:R^n→R, then isotone maps are precisely schur convex maps.

Lemma 2.9 

For a,b∈R^n the following statements are equivalent.

a≺b.

a  is obtained from b by a finite number of T-transformations.

a is in the convex hull of all vectors obtained by permuting the co-ordinates of b.

a=Ab for some doubly stochastic matrix A.

Lemma 2.10

Let a,b be two vectors with non-negative co-ordinates, then a≺_w b if and only if a=Qb for some doubly substochastic matrix Q

In some other terms the following import-ant results about the properties of majorization, weakly submajorization and weakly super majorization and relations between them was established in (Bhatia, 1997). We provide the proof for convenience.

This theorem will show the relation of majorization, weakly submajorization and weakly super majorization 

Theorem 3.1 

For a,b∈R^n

Proof 1   

Let a≺b, then by definition we have

Also if a≺b, then by definition we have

Conversely, 

Let a≺_w b and a≺^w b.

To prove a≺b, when a≺^w b, this implies

And we have the relation

Similarly for b we have  

Using (2) and (3) in equation (1) we have

Since we have a≺_w b, that is

Similarly by a≺_w b we can show that

From equations (4) and (5) we have

This result together with the given conditions imply a≺b.

(2) Let  a≺^w b,  to show 

αa≺^w αb.

a≺^w b, implies

Now for α>0 we have

Also let a≺_w b, which implies

Now for α>0 we have

Note that using this, we see that -a≺^w-b, 

(4)     If a≺b, which implies

To prove αa≺αb, we multiply both sides of the last two equations by α≥0,

Now again consider a≺b, which implies

Multiplying both sides by α<0,  

We get,

Theorem 3.2 

The relation of majorization, weakly supermajorization and weakly submajorization are reflexive and transitive.

Proof 1

Let a∈R^n, if we rearrange a in decreasing order, then we have

So ≺ is reflexive, from this we can conclude that ≺_w and ≺^w are also reflexive.

Now let a^↓=(a_1,a_2,...,a_n), b^↓=(b_1,b_2,...,b_n) and c^↓=(c_1,c_2,...,c_n) in decreasing order with a≺b and b≺c, we want to show that 

a≺c.

Since a≺b, this implies

Also, since b≺c this implies

From equations (6) and (7) we have

This shows a≺c, which implies ≺ is transitive. Hence ≺^w and ≺_w are also transitive.

As we have already seen that the relations of majorization, weakly submajorization and weakly supermajorization are all reflexive and transitive. Next, we see that none of these are a partial order relation.

For e.g.: 

Let 

a=(1,2,3,4,5) and 

b=(3,2,1,5,4), 

Clearly 

a≠b.

One can easily check that 

a≺b And 

b≺a.

Same example will work for weakly submajorization and weakly supermajorization also. In fact, for two vectors a and b, if a is obtained by permuting the co-ordinates of b, one can easily see that a≺b and vice versa but they are not equal vectors, also none of these are symmetric, clearly ≺^w and ≺_w are not symmetric too, the symmetric relation is only possible, if a=pb for some permutation matrix p, otherwise ≺ is anti-symmetric, reflexive and transitive on the set {a∈R^n:a_1≥a_2≥...≥a_n}.

Theorem 3.3

Some Equivalent Properties

Let a,b∈R^n, then

• a≺_w b if and only if for all t∈R

• a≺^w b if and only if for all t∈R

• a≺b if and only if for all t∈R

Proof 1 

Let a≺_w b, that is

Now if t>a_i^↓ for each 

  i, 

Then 

(a_i-t)^+=0, 

For each i.

Hence

Now let t∈R be such, that a_(k+1)^↓≤t≤a_k^↓, for some 1≤k≤n, then

Conversely, let

To prove for some 1≤k≤n, 

Then

But

From the relations (8), (9) and (10) we get

(2)  Let a≺^w b, this implies

To prove

If  

a^e b, this implies that -b≺_w-a.

By part (1), -b≺_w-a if and only if for every real number  t,

Which is same as saying

(3)   Let a≺b, which implies

Also a≺b if and only if a≺_w b and a≺^w b

by part (1) and (2) this holds if and only if

That is if and only if

Majorization in Convex and Monotonic Functions

In this section we give importance to the functions from Rⁿ to R^m, which preserve ordering majorization. Let f:R→R be function, we will denote the map induced by f on Rⁿ also by f, that is

f(a)=(f(a₁ )…f(a_n ))  for a∈R_n

Similarly the function f:Rⁿ→R^m is convex if

f(ta+(1-t)b)≤tf(a)+(1-t)f(b)" for " 0≤t≤1.

To show the function |a-r|=f_r (a) is convex, let a₁,a₂∈Rⁿ and 0≤t≤1,

A useful characterization of majorization is the following:

Theorem 3.4 

Let a,b∈R^n, then the following two conditions are equivalent:

1. a≺b.
2. tr f(a)≤trf(b) for all convex functions f from R to R.

Proof (1)    Let a≺b, then a=Ab, for some doubly stochastic matrix A, so 

Where Hence for every convex function f,

Hence

That is for all convex functions f from R to R,

tr f(a)≤trf(b)

(2)    Let tr f(a)≤trf(b)for all convex functions f from R to R, that is

Now let be convex function, then we have

Which implies

And

Let us firstly consider

And let then

But

From (11), (13) and (14) we get

Now consider (12), that is

Let , then we have

For the following inequality

We must have

Now from (15) and (16) we get a≺b.

Next we consider majorization on monotonic functions.

Theorem 3.5 

For a,b∈Rⁿ, the following two conditions are equivalent:

1. a≺_w b.
2. tr f(a)≤trf(b) for all monotonically increasing convex functions f from R to R.

Proof: 

Let 

Consider a function f_r (a)=(a-r), firstly we will show that this function is convex.

To show f_r (a) is monotonically increasing function, that is if a_1≤a_2," then   " f_r (a₁)≤f_r (a₂):

Let a∈Rⁿ, be in decreasing order. Consider r=a_k, then

Conversely, let

To prove  

But

From (17), (18) and (19) we get

Furthermore, a real-valued function f on Rⁿ satisfying is Schur-convex, which expresses preservation of order rather then convexity.

Let   f:Rⁿ→R^m, the domain of f is either all of R^n, or some convex set invariant under coordinate permutations of its elements. Such a map is monotonically increasing if

Monotonically decreasing if -f is monotonically increasing,

Also it is convex if

 And concave if -f is convex.

Theorem 3.6 

Let f:Rⁿ→R^m be a convex map, suppose it, for any P∈S_n, then there exist P∈S_m such that                         

Where S_n denotes the grope of n×n permutation matrix. In addition if f is monotonically increasing, then f is strongly isotone.

Proof 1 

Let a≺b in Rⁿ, by the lemma (2.9) there exists permutation matrices and real positive numbers with , such that

So by (20) and convexity of f we have

Then

Now suppose f is monotonically increasing, let d≺_w b then, by (2.10) there exists a, such that d≤a≺b, hence 

Recall the remark from the literature (Bhatia, 1997)as follows.

1. If f:R→R is convex, then the induced map f:Rⁿ→Rⁿ is isotone.
2. If f:R→R is convex and monotonically increasing, then the induced map f:Rⁿ→Rⁿ is strongly isotone.

The above results imply that:

   Here stands for the collection of vectors

Proof 1   

Let a≺b in Rⁿ, and let f(a)=|a| we want to show that f(a)=|a| is convex in R.

So by first result the function f(a)=|a| is isotone in Rⁿ.

(2)    Let a≺b in Rⁿ, and let f(a)=a² we want to show that f(a)=a² is convex in R,

Then, by first result the function f(a)=a² is isotone in Rⁿ.

(3)    Let a≺_w b in , and let   f(a)=a^p, we want to show that f(a)=a^p is convex and monotonically increasing in R₊.

To show the function f(a)=a^p is convex in R₊ for p>1,

f″(a)≥0," since " p>1," so it is convex".

Let     a≤b⇒a^p≤b^p, since a,b∈R₊  and   P>1, so it is monotonically increasing in R₊.

This implies that, for p>1⇒f(a)=a^p is strongly isotone in   

(4)    Let f(a)=a⁺, and a≺_w b in Rⁿ

Since a⁺ is obtained from a by replacing the negative coordinates of a, by 0 so if there is negative coordinates in a, then

Similarly for b

Returning to Theorem (3.6) we note that for m=1, the condition (20), that is

Says f is permutation invariant, that is

In this case Theorem (3.6) says that if a function f:Rⁿ→R is convex and permutation invariant, then it is isotone.

Also every isotone function f:Rⁿ→R has to be permutation invariant, because pa and a majorized each other, that is pa≺a, hence being isotone of f implies f(pa)=f(a) in this case.

Theorem 3.7

Let f:Rⁿ→R be a convex function and let.

Then  g is isotone, if in addition f is monotonically increasing, then g is strongly isotone.

Proof 1 

Let P∈S_n be permutation matrix,

Now pp′ is again a permutation matrix so we can write . Since Q is

Which implies g is convex.

So we can conclude that, g is isotone. 

Let 

So, g is monotonically increasing, hence g is strongly isotone.

There are many conditions on vectors which implies Majorization. Some of these conditions presents weak-ly submajorization and some shows weakly super-majorization which are forms of majorization, we defined all of them in this paper. The properties of Majorization, Weakly supermajorization and weakly submajorization and the relations between them are explored in this work with examples.  By considering the concept of our title Majorization and its applications on some Functions, we summarized the application of Majorization on some functions like monotonic functions, convex functions and so on with some properties. Such results are explained by theorems and examples.

We would like to thank Assistant Professor Mustafa Danesh for reviewing our work for grammatical and scientific errors. We also thank the editors and referees for their insightful comments and suggestions. We are all pleased to publish our work online after a lengthy process and frequent reviews of the literature.

All authors have contributed to this research, and pub-lishing it has no potential conflicts of interest.