On a pro � Lem of I . S � Hur

1 . Let n >= 3, and let Q,1 denote the class of polynomials f (x) of degree n satisfying the condition I f(x) 1 <_ 1 in the interval -1 <_ x _<_ +1 . Let Q~(xo) denote the subclass of Qn characterized by the further restriction f"(x o ) = 0 . A well-known theorem of A. Markoff1 states that I f' (x) I < n 2 for -1 _< x 5 +1 provided that f(x) e Qn ; here I f'(x) I = n2 holds if and only if x = ±1 and f(x) = =LT.(x), where T. (x) denotes the nth Tchebycheff polynomial . We observe that T0,(x) does not belong to the classes Q, (.±1) . Some years ago I. Schur2 proved the following interesting theorem : Let -1 5 xo <_ +1, and let f(x) belong to Q,(xo) . Then f'(x o) < 2n 2 . Moreover he showed : Let m„ be the least positive constant (depending only on n) such that f'(xo) 1 5 Mn •n2 for all f(x) e Q n(xo ), and x 0 in -1 < x < +1. If μ = lim sup n_ mn , then