On the spherical norm of Doob and Seidel's class

AbstractLet D(ϱ) be the Doob's class containing all functions f(z) analytic in the unit disk Δ such that f(0) = 0 and lim inf¦f(z) ¦ ⩾ 1 on an arc A of ∂Δ with length ¦A ¦⩾ ϱ. It is first proved that if f ϵ D(ϱ) then the spherical norm ∥ f ∥ = supzϵΔ(1 − ¦z¦2)¦f′(z)¦(1 + ¦f(z)¦2) ⩾ C1 sin(π − (ϱ2))/ (π − (g92)), where C1 = limn→∞∥ zn∥ and 12 < C1 < 2e. Next, U represents the Seidel's class containing all non-constant functions f(z) bounded analytic in Δ such that ¦tf(ei0)¦ = 1 almost everywhere. It is proved that inff ϵ U∥f∥ = 0, and if f has either no singularities or only isolated singularities on ∂Δ, then ∥f∥ ⩾ C1. Finally, it is proved that if f is a function normal in Δ, namely, the norm ∥f∥< ∞, then we have the sharp estimate ∥fp∥ ⩽ p∥f∥, for any positive integer p