Means, Generalized Divided Differences, and Intersections of Osculating Hyperplanes

Abstract

We define means innvariables by taking the intersection point inRnofnosculating hyperplanes to a given curve inRn. These planes are the natural extensions of the osculating plane inR3. More precisely, letCbe a curve inRn, and let 0<a1<···<an<∞. LetOkbe the osculating hyperplane toCatak. Under certain assumptions on the component functionsxk(t) ofC, theOkwill have a unique point of intersectionP=(i1,…,in) inRn. Furthermore,a1<xk−1(ik)<anfork=1,2,…,n. This definesnsymmetric meansMk(a1,…,an)=xk−1(ik). Ifxk(t)=tpk,k=1,…,n, then the meansMkare homogeneous. In particular, ifxk(t)=tk,k=1,…,n, thenM1(a1,…,an)=arithmetic mean=(∑i=1nai)/nandMn(a1,…,an)=geometric mean=(a1···an)1/n. Also, ifxk(t)=t−k,k=1,…,n, thenM1equals the harmonic meanH=n/(1/a1+···+1/an). Finally, ifxk(t)=tk,k=1,…,n−2,xn−1(t)=log(t), andxn(t)=1/t, thenMn(a1,…,an)=L(a1,…,an). HereLis the logarithmic mean innvariables defined by A. O. Pittenger (Amer. Math. Monthly92, 1995, 99–104), given by[formula]wherep(1,i,n)=∏nj=1,≠i(ai−aj). The meansMkare generalizations of meansm(a,b) defined by taking intersections of tangent lines to curvesCin the plane, discussed in an earlier paper by the author (J. Math. Anal. Appl.149, 1990, 220–235).