Extreme Values of Certain Integrals

Abstract

Let the zeros of a polynomial be prescribed in magnitude but not in phase. It is a familiar fact that certain integrals involving the polynomial are maximized when the zeros all lie on a ray through the origin. Theorems of this type were first proved by Gol’ dberg in 1954 and were used by him in the study of the deficiencies of meromorphic functions of genus zero. In 19 83 a theorem of the same type was encountered by Kolesnik and Straus in their investigation of the Tur án inequalities. Here we give a proof which depends on ideas similar to Gol’ dberg’s but is simpler in detail. This simplicity enables us to complete the original result of Gol’ dberg, showing that a certain condition of convexity is necessary as well as sufficient, and it leads to generalizations that have not been noted hitherto.

loving memory of Ed Beckenbach and Ernst Straus