On an estimate for Bloch and Doob norm and covering problem in Doob's class

For any fixed 0 < π ? 2π, let D(π) be the family of all holomorphic functions in the unit disk Δ which satisfy (i)f(0) = 0 and (ii) $\text{lim inf}{}_{\mathrm{z\; \to \; \pi ¦f(z)¦\; ?\; 1}}$, for all π lying on some arc A_f ? ?Δ with arclength $\text{¦A}{}_{\mathrm{f}}\text{¦ ? π}$. We show that for each 0 < ε < 1, there is a π₀ > 0 such that for any f?D(π) with π < π₀, the Bloch and Doob norm respectively satisfy

$\text{6f6}{}_{\mathrm{B}}\text{=}\text{sup}\text{z?Δ}\text{|f′(z)| (1?|z|}{}^2\text{) > 2(1 ? ε)}\text{log}\text{1+}\text{cos}\text{(}\text{p}\text{2}\text{1?}\text{cos}\text{(}\text{p}\text{2}{}^{\mathrm{?1}}$

$\text{6f6}{}_{\mathrm{D}}\text{=}\text{sup}\text{z?Δ}\text{|f′(z)| (1?|z|) > (1 ? ε)}\text{log}\text{1}\text{1?}\text{cos}\text{(}\text{p}\text{2}{}^{\mathrm{?1}}$

These two estimates do not hold with ε = 0.