A refinement of an inequality of the brothers Markoff

Abstract

1.3*5 ... (2p 1) We show in this paper that the condition lf(x) j < 1 in the whole interval is unnecessarily restrictive. Instead it is only necessary to assume that if(x) I < I at the n + I points x = cos (kr/n); k=O, 1, 2,, n, and the conclusions of Markoffs' theorem are unchanged. These are exactly the points at which Tn(x) attains its maximum in the interval (-1, 1). In a previous paper(3) we showed that under the assumptions of Markoffs' theorem all derivatives of f(x) must satisfy