A generalization of Fejér’s principle concerning the zeros of extremal polynomials

Abstract

Fejer's principle is readily proved; if the zero al of Pn(z) lies exterior to the convex hull of E, if a is the point of the convex hull nearest a,, and if we set a' = (a+al)/2, then the polynomial qn(Z) (z-a,' )Pn(Z)/(Z-a a) is an underpolynomial of pn(z) on E, so pn(z) cannot minimize any monotonic norm on E. The object of the present note is to give what is essentially a generalization of Fejer's principle. It applies to the minimization of the difference or quotient of twc monotonic 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.