Means, Generalized Divided Differences, and Intersections of Osculating Hyperplanes

We define means in n variables by taking the intersection point in Rn of n osculating hyperplanes to a given curve in Rn. These planes are the natural extensions of the osculating plane in R3. More precisely, let C be a curve in Rn, and let 0-a1- ??? -an-`. Let Okbe the osculating hyperplane to C at ak . Under certain assumptions on the component functions xk t. of C, the Okwill have a unique point of intersection Ps i1, . . . , in.in Rn. Furthermore, a1- xyk1 ik.-anfor ks1, 2, . . . , n. This defines n symmetric means Mk a1, . . . , an.s xyk1 ik.. If xk t.stpk, ks1, . . . , n, then the means Mk are homogeneous. In particular, if xk t.stk, ks1, . . . , n, then M1 a1, . . . , an.sarithmetic means nis1ai.rn and Mn a1, . . . , an.sgeometric means a1 ??? an.1r n. Also, if xk t. styk , ks1, . . . , n, then M1equals the harmonic mean Hsnr 1ra1q???q 1ran.. Finally, if xk t.stk, ks1, . . . , ny2,xny1 t.slog t., and xn t.s1rt, then Mn a1, . . . , an.sL a1, . . . , an.. Here L is the logarithmic mean in n vari- ables defined by A. O. Pittenger Amer. Math. Monthly 92, 1995, 99]104., given by 1 n ainy2 log ai. s ny1. , L a1 , . . . ,an. is1p 1, i , n. where p 1, i, n.s njs1, /i aiyaj.. The means Mk are generalizations of means m a, b.defined by taking intersections of tangent lines to curves C in the plane, discussed in an earlier paper by the author J. Math. Anal. Appl. 149, 1990, 220]235..