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Review Article | Open Access | Aust. J. Eng. Innov. Technol., 2023; 5(1), 1-14 | doi: 10.34104/ajeit.023.01014

Majorization and its Applications on Some Functions

Suhaila Barekzai* Mail Img ,
BiBi Amina Salimi Mail Img ,
Mustafa Danesh Mail Img

Abstract

This paper reviews the special case of an order which is called Majorization ordering. It generalizes vector Majorization and some applications that have come after the publication of Marshall and Olkin Inequalities. It presents the basic properties of Majorization and two important kinds of Majorization which are Weakly Supermajorization and Weakly Submajorization and some relations between them. Furthermore, this paper also contains maps from R^n to R^m which preserve various orders that most of these orders are elementary and useful characterizations of Majorization, as Majorization together with the strongly related concept of Schur-convexity gives an important characterization of convex functions that expresses preservation of order rather than convexity. Also in this study, examples are used to explore the characteristics of majorization, weakly supermajorization, and weakly submajorization as well as the relationships between them. We described the application of majorization on various functions, such as monotonic functions, convex functions, and so on, with some properties by taking into account the concept of our title majorization and its applications on some Functions. Theorems and examples are used to explain such outcomes. 

INTRODUCTION

Comparison of two vectors sometimes leads to interes-ting inequalities, which is an important tool in compar-ing 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 com-parison 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 co-ordinates. 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 majo-rization theory to compare different tap correlation scenarios for perfect, partial, and no CSI at the trans-mitter (Mittelbech, 2008)  Eduard Jorswieck and Mart-in 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   a∈R^n, then a^↑ and a^↓ denotes the vectors which obtained by rearranging the co-ordinates 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

a≺b □(⇒┬ ) ∑_(i=1)^n▒a_i =∑_(i=1)^n▒b_i   and ∑_(i=1)^k▒a_i^↓ ≤∑_(i=1)^k▒b_i^↓   for  1≤k≤n

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

∑_(i=1)^n▒a_i^↑ =∑_(i=1)^n▒b_i^↑   and ∑_(i=1)^k▒a_i^↑ ≥∑_(i=1)^k▒b_i^↑   ∀  1≤k≤n

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.

∑_(i=1)^k▒a_i^↓ =("max" )┬(|I|=k) ⟨a,e_I⟩,

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

∑_(=1)^k▒a_i^↓ ≤∑_(i=1)^k▒b_i^↓  " for " 1≤k≤n.

Definition 2.5 

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

∑_(i=1)^k▒a_i^↑ ≥∑_(i=1)^k▒b_i^↑  " for " 1≤k≤n.

Definition 2.6 

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

a≺b □(⇒┴ ) f( a) ≺_w f(b) 

Definition 2.7

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

a≺_w b □(⇒┴ ) f( a) ≺_w f(b) 

Definition 2.8 

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

a≺_ b □(⇒┴ ) f( a) ≺_ f(b) 

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

RESULTS

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

a≺b⇔a≺_w b,a≺^w b.

a≺_w b⇒αa≺_w αb" and " a≺^w b⇒αa≺^w αb,   for every  α>0.

a≺_w b⇔-a≺^w-b.

a≺b⇒αa≺αb, For every α∈R.

Proof 1   

Let a≺b, then by definition we have

∑_(i=1)^n▒a_i =∑_(i=1)^n▒b_i   and ∑_(i=1)^k▒a_i^↓ ≤∑_(i=1)^k▒b_i^↓   for  1≤k≤n

Also if a≺b, then by definition we have

∑_(i=1)^n▒a_i^↑ =∑_(i=1)^n▒b_i^↑   and ∑_(i=1)^k▒a_i^↑ ≥∑_(i=1)^k▒b_i^↑   ∀  1≤k≤⇒a≺┴w b.

□(⇒┬ ) a≺^w b

Conversely, 

Let a≺_w b and a≺^w b.

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

∑_(i=1)^k▒a_i^↑ ≥∑_(i=1)^k▒a_i^↑                                                 (1)

And we have the relation

   ∑_(i=1)^k▒a_i^↑ =∑_(i=1)^n▒a_i -∑_(i=1)^(n-k)▒a_i^↓                             (2)

Similarly for b we have  

∑_(i=1)^k▒b_i^↑ =∑_(i=1)^n▒b_i -∑_(i=1)^(n-k)▒b_i^↓                                (3)

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

∑_(i=1)^n▒a_i -∑_(i=1)^(n-k)▒a_i^↓ ≥∑_(i=1)^n▒b_i -∑_(i=1)^(n-k)▒b_i^↓ ,

Since we have a≺_w b, that is

 ∑_(i=1)^(n-k)▒a_i^↑ ≥∑_(i=1)^(n-k)▒a_i^↑  □(⇒┬ ) ∑_(i=1)^n▒a_i ≥∑_(i=1)^n▒b_i .           (4)

Similarly by a≺_w b we can show that

∑_(i=1)^n▒a_i ≤∑_(i=1)^n▒b_i .                                               (5)

From equations (4) and (5) we have

∑_(i=1)^n▒a_i =∑_(i=1)^n▒b_i ,

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

(2) Let  a≺^w b,  to show 

αa≺^w αb.

a≺^w b, implies

∑_(i=1)^k▒a_i^↑ ≥∑_(i=1)^k▒b_i^↑ .

Now for α>0 we have

Also let a≺_w b, which implies

∑_(i=1)^k▒a_i^↓ ≤∑_(i=1)^k▒b_i^↓ ,

Now for α>0 we have

(3)  a≺_w b⇔-a≺^w-b.

Note that (-a)_i^↑=-a_i^↓ using this, we see that -a≺^w-b, 

□(⇔┬ ) ∑_(i=1)^k▒〖-a〗_i^↓ ≥∑_(i=1)^k▒〖-b_i^↓ 〗

□(⇔┬ ) ∑_(i=1)^k▒a_i^↓ ≤∑_(i=1)^k▒b_i^↓  □(⇔┬ )  a≺_w b

(4)     If a≺b, which implies

■(&∑_(i=1)^k▒a_i^↓ ≤∑_(i=1)^k▒b_i^↓  " for " 1≤k≤n" and " ∑_(i=1)^n▒a_i =∑_(i=1)^n▒b_i .)

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

■(α&∑_(i=1)^k▒a_i^↓ ≤α∑_(i=1)^k▒b_i^↓  " and " α∑_(i=1)^n▒a_i =α∑_(i=1)^n▒b_i @&∑_(i=1)^k▒α a_i^↓≤∑_(i=1)^k▒α b_i^↓," for " 1≤k≤n@&∑_(i=1)^n▒α a_i=∑_(i=1)^n▒α b_i⇒αa≺αb.)

Now again consider a≺b, which implies

■(&∑_(i=1)^k▒a_i^↓ ≤∑_(i=1)^k▒b_i^↓  " for " 1≤k≤n" and " ∑_(i=1)^n▒a_i =∑_(i=1)^n▒b_i .)

Multiplying both sides by α<0,  

We get,

■(&α∑_(i=1)^k▒a_i^↓ ≥α∑_(i=1)^k▒b_i^↓  " for " 1≤k≤n⇒∑_(i=1)^k▒α a_i^↑≥∑_(i=1)^k▒α b_i^↑@" and " &α∑_(i=1)^n▒a_i =α∑_(i=1)^n▒b_i ⇒∑_(i=1)^n▒α a_i=∑_(i=1)^n▒α b_i⇒αa≺αb.)

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

∑_(i=1)^n▒α a_i=∑_(i=1)^n▒α b_(i and)  ∑_(i=1)^k▒a_i^↓ ≤∑_(i=1)^k▒b_(i    for     1≤k≤n ⇒a≺α)^↓  

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

∑_(i=1)^n▒( a_i-t)^+≤∑_(i=1)^n▒( b_i-t)^+.

a≺^w b if and only if for all t∈R

∑_(i=1)^n▒( t-a_i )^+≤∑_(i=1)^n▒( t-b_i )^+.

a≺b if and only if for all t∈R

∑_(i=1)^n▒| a_i-t|≤∑_(i=1)^n▒| b_i-t|.

Proof 1 

Let a≺_w b, that is

∑_(i=1)^k▒a_i^↓ ≤∑_(i=1)^k▒b_i^↓ .

Now if t>a_i^↓ for each 

  i, 

Then 

(a_i-t)^+=0, 

For each i.

Hence

∑_(i=1)^n▒( a_i-t)^+≤∑_(i=1)^n▒( b_i-t)^+.

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

■(& @=&∑_(i=1)^k▒( b_i^↓-t)≤∑_(i=1)^k▒( b_i^↓-t)^+≤∑_(i=1)^n▒( b_i^↓-t)^+⇒∑_(i=1)^n▒( a_i^↓-t)^+≤∑_(i=1)^n▒( b_i^↓-t)^+.)

Conversely, let

  ∑_(i=1)^n▒( a_i^↓-t)^+≤∑_(i=1)^n▒( b_i^↓-t)^+                                 (8)

To prove a≺_w b, if t=b_k^↓ for some 1≤k≤n, 

Then

    ∑_(i=1)^n▒( b_i-t)^+=∑_(i=1)^K▒( b_i^↓-t)=∑_(i=1)^K▒b^↓ -kt         (9)

But

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

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

(2)  Let a≺^w b, this implies

∑_(i=1)^k▒a_i^↑ ≥∑_(i=1)^k▒b_i^↑ ,

To prove

∑_(i=1)^n▒( t-a_i )^+≤∑_(i=1)^n▒( t-b_i )^+.

If  

a≺^w b, this implies that -b≺_w-a.

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

∑_(i=1)^n▒(-a_i+t)^+≤∑_(i=1)^n▒(-b_i+t)^+,  

Which is same as saying

∑_(i=1)^n▒( t-a_i )^+≤∑_(i=1)^n▒( t-b_i )^+.

(3)   Let a≺b, which implies

∑_(i=1)^n▒a_i^↓ =∑_(i=1)^n▒b_i^↓  " and " ∑_(i=1)^k▒a_i^↓ ≤∑_(i=1)^k▒b_i^↓  " for " 1≤k≤n.

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

∑_(i=1)^n▒( a_i-t)^+≤∑_(i=1)^n▒( b_i-t)^+ " and " ∑_(i=1)^n▒( t-a_i )^+≤∑_(i=1)^n▒( t-b_i )^+,

That is if and only if

∑_(i=1)^n▒| a_i-t|≤∑_(i=1)^n▒| b_i-t|.

Majorization in Convex and Monotonic Functions

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

f(a)=(f(a_1 )…f(a_n ))  for a∈R^n

Similarly the function f:R^n→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 a_1,a_2∈R^n and 0≤t≤1,

f_r (ta_1+(1-t)a_2)=|ta_1+(1-t)a_2-r|

≤|t||a_1-r|+|(1-t)||a_2-r|

=t|a_1-r|+(1-t)|a_2-r|

=tf(a_1)+(1-t)f(a_2)," for " 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:

a≺b.

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 

a_i=∑_(j=1)^n▒a_ij  b_j 

Where a_ij≥0 and ∑_(j=1)^n▒a_ij =1∀i Hence for every convex function f,

f(a_i)=f(∑_(j=1)^n▒a_ij  b_j )≤∑_(j=1)^n▒a_ij  f(b_j).

Hence

∑_(i=1)^n▒f(a_i)≤∑_(i,j)^n▒a_ij  f(b_j)=∑_(j=1)^n▒f(b_j)⇒∑_(i=1)^n▒f(a_i)≤∑_(j=1)^n▒f(b_j)

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

∑_(i=1)^n▒f(a_i)≤∑_(j=1)^n▒f(b_j).

Now let f_r (a)=|a-r| and f_r (b)=|b-r| be convex function, then we have

Which implies

∑_(i=1)^n▒| a-r|≤∑_(i=1)^n▒| b-r|     

And

Let us firstly consider

Let us firstly consider

∑_(i=1)^n▒( a_i-r)^+≤∑_(i=1)^n▒( b_i-r)^+

And let b_k^↓=r, then

∑_(i=1)^n▒( b_i-r)≤∑_(i=1)^k▒b_i^↓ -kr                      (13)    

But

∑_(i=1)^k▒a_i^↓ -kr≤∑_(i=1)^n▒( a_i-r)^+                     (14)    

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

Now consider (12), that is

∑_(i=1)^n▒( r-a_i )^+≤∑_(i=1)^n▒( r-b_i )^+

Let r=b_k^↓, then we have

∑_(i=1)^n▒( r-b_i^↓ )^+≥∑_(i=1)^k▒( r-b_i^↓)=kr-∑_(i=1)^k▒b_i^↓ ," but " kr-∑_(i=1)^k▒a_i^↑ ≥∑_(i=1)^n▒( r-a_i^↑ )^+.

For the following inequality

∑_(i=1)^n▒( r-a_i^↑ )^+≤∑_(i=1)^n▒( r-b_i^↑ )^+,

We must have

kr-∑_(i=1)^k▒a_i^↑ ≤kr-∑_(i=1)^k▒b_i^↑ ⇒a≺┴w b.      (16)    

Now from (15) and (16) we get a≺b.

Next we consider majorization on monotonic functions.

Theorem 3.5 

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

a≺_w b.

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

Proof: 

Let 

a≺_w b, for a,b∈R^n, that is ∑_(i=1)^k▒a_i^↓ ≤∑_(i=1)^k▒b_i^↓ .

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

■(&f_r (ta_1+(1-t)a_2)@&≤(t(a_1-r)+(1-t)(a_2-r))^+@&=tf(a_1)+(1-t)f(a_2)" for " 0≤t≤1)

⇒f_r (a)   is convex.

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

■(&f_r (a_1)=(a_1-r)≤(a_1-r)^+=a_1-r@&f_r (a_2)=(a_2-r)≤(a_2-r)^+=a_2-r@&⇒f_r (a_1)≤f_r (a_2).)

Let a∈R^n, be in decreasing order. Consider r=a_k, then 〖(a〗_i^↓-r)^+=0∀i≥k

 trf_r (a)=∑_(i=1)^n▒(a_i-r) ≤∑_(i=1)^k▒( a_i^↓-r)^+

=■(&∑_(i=1)^k▒a_i^↓ -kr≤∑_(i=1)^k▒b_i^↓ -kr," for " r=b_k )

≤∑_(i=1)^n▒( b_i^↓-r)

⇒〖tr f〗_r (a)≤trf_r (b).   

Conversely, let

〖tr f〗_r (a)≤trf_r (b) For a,b∈R^n                  (17)

To prove    a≺_w b, let a_(k+1)≤r≤a_k

=∑_(i=1)^k▒a_i^↓ -kr,                                                    (18)    

But

■(&∑_(i=1)^k▒b_i^↓ -kr≤∑_(i=1)^k▒( b_i^↓-r)^+," for " r=b_k )

≤∑_(i=1)^n▒( b_i^↓-r)^+=∑_(i=1)^n▒( b_i^↓-r) ,                  (19)    

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

∑_(i=1)^k▒a_i^↓ ≤∑_(i=1)^k▒b_i^↓   □(⇒┬ ) a≺_w b

Furthermore, a real valued function f on R^n satisfying a≺b □(⇒┬ ) f(a)≤f(b) is Schur-convex, which expresses preservation of order rather then convexity.

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

Monotonically decreasing if -f is monotonically increasing,

Also it is convex if

f(ta+(1-t)b)≤tf(b)+(1-t)f(b)  0≤t≤1

 And concave if -f is convex.

Theorem 3.6 

Let f:R^n→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 R^n, by the lemma (2.9) there exists P_1,P_2,...,P_N ∈S_n permutation matrices and real positive numbers λ_1,λ_2,...,λ_N with ∑_(j=1)^n▒λ_j =1, such that

a=∑_(j=1)^n▒λ_j  P_j b.

So by (20) and convexity of f we have

■(f(a)=&f(∑_(j=1)^n▒λ_j  p_j b)≤∑_(j=1)^n▒λ_j  f(p_j b)@=&∑_(j=1)^n▒λ_j  p_j f(b)=c" say " ,)

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.

If f:R→R is convex, then the induced map f:R^n→R^n is isotone.

If f:R→R is convex and monotonically increasing, then the induced map

f:R^n→R^n is strongly isotone.

The above results imply that:

R_+^n   Here stands for the collection of vectors a≥0, |a|=(|a_1 |,|a_2 |,...,|a_n |) and a^+=a∨0.

Proof 1   

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

■(&f(a)=|a|," for some " 0≤t≤1@&f(ta+(1-t)b)≤|t|a|+(1-t)|b|@&=tf(a)+(1-t)f(b)," for " 0≤t≤1)

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

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

■(&f(ta+(1-t)b)=t^2 a^2+(1-t)^2 b^2+2ta(1-t)b@≤&ta^2+(1-t)b^2@=&tf(a)+(1-t)f(b)" for " 0≤t≤1)

Then, by first result the function f(a)=a^2 is isotone in R^n.

(3)    Let a≺_w b in R_+^n, 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   R_+^n, so a^p ≺_w b^p.

(4)    Let f(a)=a^+, and a≺_w b in R^n □(⇒┬ ) ∑_(i=1)^k▒a_i^↓ =∑_(i=1)^k▒b_i^↓ 

Since a^+ is obtained from a by replacing the negative co-ordinates of a, by 0 so if there is negative co-ordinates in a, then

∑_(i=1)^k▒a_i^↓ ≤∑_(i=1)^k▒( a_i^↓ )^+.

Similarly for b

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

[f(pa)=pf(a)∀a∈R^n],

Says f is permutation invariant, that is

f(pa)=f(a)∀a∈R^n " and " p∈S_n.

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

Also every isotone function f:R^n→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:R^n→R be a convex function and let.

g(a)=("max" )┬(P∈S_n ) f(Pa),

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,

g(Pa)=("max" )┬(p∈S_n ) f(PPa).

Now ????????′ is again a permutation matrix so we can write ????????′=????, so ????????′ (????)=????(????). Since ???? is

g(Pa)=("max" )┬(Q∈S_n ) f(Qa),

Which implies  g is convex.

So we can conclude that, g is isotone. 

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

CONCLUSION

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 appli-cations on some Functions, we summarized the appli-cation of Majorization on some functions like mono-tonic functions, convex functions and so on with some properties. Such results are explained by theorems and examples.

ACKNOWLEGDEMENT

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.

CONFLICTS OF INTEREST

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

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Article Info:

Academic Editor

Dr. Wiyanti Fransisca Simanullang, Assistant Professor, Department of Chemical Engineering, Universitas Katolik Widya Mandala Surabaya, East Java, Indonesia.

Received

December 12, 2022

Accepted

January 19, 2023

Published

January 31, 2023

Article DOI: 10.34104/ajeit.023.01014

Corresponding author

Suhaila Barekzai*

Department of Mathematics, Kabul Education University, Kabul City, Afghanistan

Cite this article

 Barekzai S, Salimi BA, and Danesh M. (2023). Majorization and its applications on some functions. Aust. J. Eng. Innov. Technol., 5(1), 1-14. https://doi.org/10.34104/ajeit.023.01014 

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