The resulted theory taken from the concept of convex functions, may be applied to topics occuring in economics and real analysis. Convexity has a relation back to Archimedes.It is natural notion. By using convexity Archimedes estimated the value of Pai.
In developed Era, there occurs a fast and active development in the concept of convex functions.There exist many reasons, we see, behind it: first of all so many areas in modern analysis involve directly or indirectly the applications of convex function,Also secondly convex functions have close relation to the theory of inequalities. Many important inequalities are resultant of the application of convex functions.
On majorization and convexity I described some results that the measure of diversity about components of n-dimensional vector (n-tuple) arose as a measure and it related to the convexity closely. It has been discussed in the volumes of Hardy, Polya and Littlewood in 1934 and 1952; I restricted my attention to the results of majorization that involved the convex functions directly and it was in detailed treated by Olkin and Marshall 1979. In order to avoid the repetition of effort I referred to their books for the proofs.
The main component of majorization as per definition is diversity of two n-tuples components. In the following case I state a quite alike definition of functions that are integrable. Let’s suppose , be the functions having real value described on an interval as [a, b] such that , both of them occur for each w [f, g].
A 2nd source for majorization is demonstrated in the work done by Schur 1923 in the inequality which are determinant of Hadamard’s. As an introduction for proof of this dissimilarity, Schur showed that the elements in diagonal ,..., of matrix i.e Hermitian, that are positive definite, they are majorized with eigenvalues ,..., , i.e., ( ,..., ) ≺ ( ,..., ).(i)
Afterwards, Horn (1954i) solved that (i) in fact symbolizes those numbers ,..., and ,..., respectively which can come together as the element of diagonal and the eigenvalues in the matrix i.e Hermitian. Via recognizing whole function of that fulfils the definition v ≺ w, => (v) ≤ (w) each time v, w ∈ .
Schur pointed out each of the probable inequalities in place of semi-definite non-negative of matrix i.e Hermitian; which equate a function in diagonal element having identical function for the eigenvalues but inequality of Hadamard, determinant is any example. Similarly many further problems of categorization have solutions that contain majorization. In each of the problems minor extension or results of Schur leads to different cases of inequality.
It is interesting to foresee the probability of exploring the order of partial type, of majorization for cope through the difficult in its place of real valued vectors. Whichever description of majorization demands elements by arrangement of vectors doesn’t serve well in the multifaceted settings; it is so because it demands complex number selection in ordering.
In developed Era, there occurs a fast and active development in the concept of convex functions.There exist many reasons, we see, behind it: first of all so many areas in modern analysis involve directly or indirectly the applications of convex function,Also secondly convex functions have close relation to the theory of inequalities. Many important inequalities are resultant of the application of convex functions.
On majorization and convexity I described some results that the measure of diversity about components of n-dimensional vector (n-tuple) arose as a measure and it related to the convexity closely. It has been discussed in the volumes of Hardy, Polya and Littlewood in 1934 and 1952; I restricted my attention to the results of majorization that involved the convex functions directly and it was in detailed treated by Olkin and Marshall 1979. In order to avoid the repetition of effort I referred to their books for the proofs.
The main component of majorization as per definition is diversity of two n-tuples components. In the following case I state a quite alike definition of functions that are integrable. Let’s suppose , be the functions having real value described on an interval as [a, b] such that , both of them occur for each w [f, g].
A 2nd source for majorization is demonstrated in the work done by Schur 1923 in the inequality which are determinant of Hadamard’s. As an introduction for proof of this dissimilarity, Schur showed that the elements in diagonal ,..., of matrix i.e Hermitian, that are positive definite, they are majorized with eigenvalues ,..., , i.e., ( ,..., ) ≺ ( ,..., ).(i)
Afterwards, Horn (1954i) solved that (i) in fact symbolizes those numbers ,..., and ,..., respectively which can come together as the element of diagonal and the eigenvalues in the matrix i.e Hermitian. Via recognizing whole function of that fulfils the definition v ≺ w, => (v) ≤ (w) each time v, w ∈ .
Schur pointed out each of the probable inequalities in place of semi-definite non-negative of matrix i.e Hermitian; which equate a function in diagonal element having identical function for the eigenvalues but inequality of Hadamard, determinant is any example. Similarly many further problems of categorization have solutions that contain majorization. In each of the problems minor extension or results of Schur leads to different cases of inequality.
It is interesting to foresee the probability of exploring the order of partial type, of majorization for cope through the difficult in its place of real valued vectors. Whichever description of majorization demands elements by arrangement of vectors doesn’t serve well in the multifaceted settings; it is so because it demands complex number selection in ordering.
