Mean modulus and the fractional derivative of an inner function

It is known that if is an inner function and if or if , then ϕ must be a Blaschke product. In this paper we show that if ϕ is an inner function and if then ϕ is a Blaschke product. (Here Dαϕ denotes the fractional derivative of ϕ of order α.) The cases q = ½ and q = 2 improve the two results mentioned above. It is known that if B is a Blaschke product and if then the zero sequence of B is a finite union of sequences that go exponentially to the boundary, and conversely. We extend this to the case 0 < ρ > 1 and also show that for each ρ, 0 < ρ < ∞. There is an ∊ρ > 0 such that if then B is a finite Blaschke product.