A Generalization of Fejér's Principle concerning the Zeros of Extremal Polynomials1

Fejér's principle is readily proved; if the zero ai of pn(z) lies exterior to the convex hull of E, if a is the point of the convex hull nearest ai, and if we set a{ =(a+ai)/2, then the polynomial qn(z) = (z — ce{ )pn(z)/(z — ái) is an underpolynomial of pn(z) on E, so pn(z) cannot minimize any monotonie norm on E. The object of the present note is to give what is essentially a generalization of Fejér's principle. It applies to the minimization of the difference or quotient of twe monotonie norms of a polynomial on two disjoint point sets: It is especially appropriate that this paper should be dedicated to Professor Einar Hille, in view of his now classical work on the complex zeros of solutions of differential equations.