Let m and n be given integers, 0 < m<n. Let f(x) be a real- or complex-valued function of a real variable x on an interval I such that f(n − 1)(x) is absolutely continuous and f(n)(x) is bounded.

The Landau problem is estimating an intermediate derivative f(m)(x) when bounds for f(x) and f(n)(x) are given. In this paper we present uniform bounds for f(m)(x) in terms of uniform bounds of f(x) and f(n)(x). This improves earlier bounds given by H. Cartan by, roughly, a factor of 1(e4m).

Our method is based on the approximation of f(m)(x) by the mth derivative of a polynomial interpolating f(x) at n points in I. A technique to study the sign variations of the Peano kernel earlier used by us, Schönhage, and Schneider is developed further. We also use results by Gusev and by Rivlin.

Our method enables us to get estimates of the truncation error and of the magnification of errors in the values employed for f(x) in such approximations.