We establish necessary and sufficient conditions under which a real-valued function from \(L_p (\mathbb{T})\), 1 ≤ p < ∞, is badly approximable by the Hardy subspace Hp0:= {ƒ ∈ Hp: ƒ(0) = 0}. In a number of cases, we obtain the exact values of the best approximations in the mean of functions holomorphic in the unit disk by functions holomorphic outside this disk. We use the obtained results for finding the exact values of the best polynomial approximations and n-widths of some classes of holomorphic functions. We establish necessary and sufficient conditions under which the generalized Bernstein inequality for algebraic polynomials on the unit circle is true.