Extremal problems in the class σ(Τ)

Let L_f(r)={w=f(x), |z|=r}, 1 < r < ∞, be a level line of the function f(z) ∃ σ. Sharp upper bounds are obtained for the diameter of the curve L_f(r) in the class σ(Τ) for functions f(z)=z+α₀+α₁z^-1+... ∃ σ for which there exists a domain δ_f that complements the exterior of the unit disk and has conformal radius at the point w=0 satisfying the condition R(δ_f,0) ≥ Τ, 0 < Τ < 1. Also, a set of values is found for the coefficient α₁ in the class σ(Τ).