On strong asymptotic spots of functions holomorphic in the circle

For entire functions of finite order the Ahlfors' classical theorem about finiteness of the set of asymptotic values is well known. In 1999 one of the authors has introduced the concept of the strong asymptotic value of entire functions and has obtained an analogue of the Ahlfors' theorem for distinct strong asymptotic spots of entire functions of infinite order. In this article the concept of a strong asymptotic spot for functions holomorphic in the circle has been introdused. The sharp estimate of the number of strong asymptotic spots related to a point $z_0$ has been obtained by the magnitude of the deviation $b(\infty,f)$. The magnitude $b(\infty,f) $ was introduced by A. Eremenko for functions meromorphic in the whole plane. In particular, if $b(\infty,f)>0$ then the number of strong asymptotic spots is finite.