THE SCHUR CONVEXITY FOR THE GENERALIZED MUIRHEAD MEAN

For x, y > 0, a, b is an element of R with a+ b not equal 0, the generalized Muirhead mean is defined by M( a, b; x, y) = (x(a)y(b) + x(b)y(a)/2) . In this paper, we prove that M( a, b; x, y) is Schur convex with respect to ( x, y) is an element of ( 0,infinity) x( 0,infinity) if and only if ( a, b) is an element of{( a, b)is an element of R-2 : ( a- b) (2) >= a+ b > 0& ab <= 0} and Schur concave with respect to ( x, y). ( 0,infinity) x( 0,infinity) if and only if ( a, b)is an element of{( a, b)is an element of R-2 + : ( a- b)(2) not equal a+ b & ( a, b) = ( 0,0)}.{( a, b). R2 : a+ b < 0}, where R+ : = [ 0,8)