The following result is proved: Theorem. Let S be an inner function |S|≤1, Spec S⊂E, and assume that the set E satisfies the conditions[Figure not available: see fulltext.] and[Figure not available: see fulltext.] where ω is an arbitrary modulus of continuity. Then there exists a function[Figure not available: see fulltext.][Figure not available: see fulltext.]such that[Figure not available: see fulltext.].