Let p(ϛ) be a positive function defined on the unit circle (Formula presented) and satisfying the condition (Formula presented) (Formula presented). Futhermore, let 0 <α< 1, r ≥ 0, (Formula presented), and assume that (Formula presented). Define a class of analytic functions in the unit disk (Formula presented) as follows: (Formula presented). The following main results are proved. Theorem 1. Let (Formula presented), and let I be an inner function, f/I ∈ H¹. Then (Formula presented). Theorem 2. Let (Formula presented), and let I be an inner function, f/I ∈∞. Assume that the multiplicity of every zero of f in (Formula presented) is at least r + 1.Then (Formula presented).