Various Remarks on Univalent Functions

Let F„ denote the wth coefficient region for this class of functions [S, §1.2]. Let F=Fia2, ä2, ■ • ■ , a„, a«) be a real-valued function. Write x,= (l/2)(a,+á,), y,= (l/2í)(a,-á,), F,= (l/2)(r3F/r)x, — idF/dy,). Let F satisfy the further conditions (a) F is defined in an open set O containing V„, (b) F and its derivatives F, are continuous in 0, (c) |grad F\ = ( E?-a | F,\ 2)1/2>0 in 0. Let/(z)* = E^-* o?V. Then the following is one of the basic results of Schaeffer and Spencer [5, Lemma VII]. I. Every function/(z) of class 5 belonging to a point (a2,a3, • • • , an) where F attains its maximum in F„ must satisfy the differential equation