TWO-PARAMETER GENERALIZED WEIGHTED FUNCTIONAL MEAN ZHEN-GANG XIAO, ZHI-HUA ZHANG, V. LOKESHA, AND K. M. NAGARAJA Abstract. In this paper, a generalized weighted functional mean is defined. This includes as special cases various generalizations of the two-parameters means. Some elementary properties are listed. An explicit form is given for the special case when all variables have the special weights. 1. Introduction The generalized weighted means of the function f with weight p and two parameters r and s are defined in [1] by (1.1) M r,s (f ; p; u, v)=                    v u p(x)f r (x)dx  v u p(x)f s (x)dx  1 r-s , (r − s)(u − v) = 0; exp   v u p(x)f r (x) ln f (x)dx  v u p(x)f r (x)dx  , r = s, u − v = 0; f (x), r = s, u = v; where u, v, r, s ∈ R, p ≥ 0,f > 0 integrable functions on the interval [u, v] ⊂ R. The basic properties of M r,s (f ; p; u, v) were studied in [2]-[9]. In this paper, we will define a generalized weighted functional mean for two parameters and prove its monotonicity. An explicit form is given for the special case when all variables have the special weights. 2. Definition And Properties Throughout the paper we assume R be a set of real numbers and R + a set of strictly positive real numbers. Let E ⊂ R n + represent the simplex E = {(x 1 ,x 2 , ··· ,x n ):  n i=1 x i  1,x i  0,i =1, 2, ··· ,n}, and x =(x 0 ,x 1 ,x 2 , ··· ,x n ), where x 0 =1 − ∑ n i=1 x i . Let dx = dx 1 dx 2 ··· dx n denote the differential of the volume in E. Definition 2.1. Let f be a positive real function and p a nonnegative integrable function on E, then the generalized weighted functional means of f with weight function p and two parameters r, s are defined by (2.1) M r,s (f ; p)=            E p(x)f r (x)dx  E p(x)f s (x)dx  1 r-s , r = s; exp   E p(x)f r (x) ln f (x)dx  E p(x)f r (x)dx  , r = s. Lemma 2.1. (see [6]) Suppose f,g ≥ 0 and p are integrable, and f/g is continuous on E. Then there exists at least one point v ∈ E such that (2.2)  E p(x)f (x)dx  E p(x)g(x)dx = lim x→v f (x) g(x) . We call Lemma 2.1 the revised Cauchy’s mean value theorem in the integral form. 2000 Mathematics Subject Classification. Primary 26D15. Key words and phrases. Weighted mean, two-parameter, inequality, Van der Monde determinant. This paper was typeset using A M S-L A T E X. 1