Means and Hermite Interpolation

Let m2 < m1 be two given nonnegative integers with n = m1+m2+1. For suitably differentiable f , we let P,Q ∈ πn be the Hermite polynomial interpolants to f which satisfy P (a) = f (a), j = 0, 1, ..., m1 and P (b) = f (b), j = 0, 1, ..., m2, Q (a) = f (a), j = 0, 1, ..., m2 and Q(b) = f (b), j = 0, 1, ..., m1. Suppose that f ∈ C (I) with f (x) 6= 0 for x ∈ (a, b). If m1 − m2 is even, then there is a unique x0, a < x0 < b, such that P (x0) = Q(x0). If m1 − m2 is odd, then there is a unique x0, a < x0 < b, such that f(x0) = 1 2 (P (x0) +Q(x0)). x0 defines a strict, symmetric mean, which we denote by Mf,m1 ,m2(a, b). We prove various properties of these means. In particular, we show that f(x) = x12 yields the arithmetic mean, f(x) = x yields the harmonic mean, and f(x) = x12 yields the geometric mean.