1. In a joint paper with A. C. Schaeffer1 we discussed the following question: Let D be a closed domaina in the complex z-plane and z0 a fixed pointt of D. Let its consider all polynomtials f(z) of givez degree n forwhich f Jf(z) ? 1 in D and f(z0) is real. What is the largest possible value of !f (z) if f(z) is an arbitrary polynomial of the hinzd mentioned and z is arbitrary in D? Under certain conditions on D and zo the inequality I sf(z)! 0 clepends only oi 1D and zo. This theorem was well-known in the special case when D is a circle anld zO is ani in-terior point of D. fHowever Mr. P. Erd6s has called my attention to the fact that the best possible bound of j 3:tf(z) j is nlot known in this case. In the present paper I wish to determine the best possible bounid in case D is a circle ancd zo is the center of D. I prove the following