Schur Convexity for a Class of Symmetric Functions
نویسندگان
چکیده
In this paper we derive some general conditions in order to prove the Schurconvexity of a class of symmetric functions. The log-convexity conditions which appear in this paper will contradicts one of the results of K. Guan from [2]. Also, we prove that a special class of rational maps are Schur-convex functions in Rn +. As an application, Ky-Fan’s inequality is generalized. 2010 Mathematics Subject Classification. Primary 34K20, 34K25; Secondary 26C15, 26D15.
منابع مشابه
Schur–convexity, Schur Geometric and Schur Harmonic Convexities of Dual Form of a Class Symmetric Functions
By the properties of Schur-convex function, Schur geometrically convex function and Schur harmonically convex function, Schur-convexity, Schur geometric and Schur harmonic convexities of the dual form for a class of symmetric functions are simply proved. As an application, several inequalities are obtained, some of which extend the known ones. Mathematics subject classification (2010): 26D15, 0...
متن کاملSchur-Convexity for a Class of Symmetric Functions and Its Applications
For x x1, x2, . . . , xn ∈ R , the symmetric function φn x, r is defined by φn x, r φn x1, x2, . . . , xn; r ∏ 1≤i1<i2 ···<ir≤n ∑r j 1 xij / 1 xij 1/r , where r 1, 2, . . . , n and i1, i2, . . . , in are positive integers. In this article, the Schur convexity, Schur multiplicative convexity and Schur harmonic convexity of φn x, r are discussed. As applications, some inequalities are established...
متن کاملSchur Convexity with Respect to a Class of Symmetric Functions and Their Applications
For x = (x1, x2, · · · , xn) ∈ R+, the symmetric function φn(x, r) is defined by φn(x, r) = φn(x1, x2, · · · , xn; r) = ∏ 1≤i1<i2···<ir≤n r ∑ j=1 1 + xij xij , where r = 1, 2, · · · , n, and i1, i2, · · · , in are positive integers. In this article, the Schur convexity, Schur multiplicative convexity and Schur harmonic convexity of φn(x, r) are discussed. As applications, some inequalitie...
متن کاملOn Schur Convexity of Some Symmetric Functions
For x x1, x2, . . . , xn ∈ 0, 1 n and r ∈ {1, 2, . . . , n}, the symmetric function Fn x, r is defined as Fn x, r Fn x1, x2, . . . , xn; r ∑ 1≤i1<i2 ···<ir≤n ∏r j 1 1 xij / 1−xij , where i1, i2, . . . , in are positive integers. In this paper, the Schur convexity, Schur multiplicative convexity, and Schur harmonic convexity of Fn x, r are discussed. As consequences, several inequalities are est...
متن کاملSchur-convexity, Schur-geometric and Schur-harmonic convexity for a composite function of complete symmetric function
In this paper, using the properties of Schur-convex function, Schur-geometrically convex function and Schur-harmonically convex function, we provide much simpler proofs of the Schur-convexity, Schur-geometric convexity and Schur-harmonic convexity for a composite function of the complete symmetric function.
متن کامل