In a classical paper L6wner [-5] characterized those real functions f(2) which are monotonic for symmetric matrices. Let us call f(2), defined on the interval [a, b], n-monotonic if for arbitrary n by n symmetric matrices A, B satisfying a(x, x) < (Ax, x) < (Bx, x) < b(x, x) for all xe~" (with the usual scalar product), the inequality (f(A)x,x) < (f(B)x x) holds. L6wner showed that f is n-monotonic for all n if and only if f is analytic on (a, b) with limits at a and b (provided -oo < a and b < oo) and has an analytic extension to the upper half plane, with nonnegative imaginary part. We take this last characterization of f as a definition of"[-a, b]-monotonic" and prove that if a < A < B _< b (0 < a < b < ~) for operators in a Hilbert space .~ and f is [_a, hi-monotonic, then f(A)