Inner functions in Cn

In this paper we prove that if f is a holomorphic function in a strictly pseudoconvex region D in Cⁿ, n > 1, with radial limit equal to 1 in modulus at each point of some nonempty open subset S of the boundary of D, then f const in D.Translated from Matematicheskii Zametki, Vol. 19, No. 1, pp. 63–66, January, 1976.I wish to thank B. V. Shabat for a discussion of the result and for critical remarks.     

Fatou's example

Examples due to P. Fatou and L. Biberbach show that Picard's little theorem cannot be extended to apply to holomorphic mappings of the complex space C_{n, n>1}. These examples are used to construct a pair of functionally independent entire functions of two complex variables such that the image of C² by the mapping realized by this pair does not contain any sphere. Here we give an example of the same type which implies a stronger assertion concerning the set of values not taken.Translated from Matematicheskie Zametki, Vol. 6, No. 4, pp. 437–441, October, 1969.We thank B. V. Shabat for suggesting this subject and for his valuable help in the work.