Three Extremal Problems in the Hardy and Bergman Spaces of Functions Analytic in a Disk

Let a nonnegativemeasurable function γ(ρ) be nonzero almost everywhere on (0, 1), and let the product ργ(ρ) be summable on (0, 1). Denote by B = B γ p, q , 1 ≤ p≤ ∞, 1 ≤ q < ∞, the space of functions f analytic in the unit disk for which the function M p q (f, ρ)ργ(ρ) is summable on (0, 1), where Mp(f, ρ) is the p-mean of f on the circle of radius ρ; this space is equipped with the norm $$||f||_{B_\gamma ^{p,q}} = ||{M_P}(f,.)||_{L_{\rho \gamma (p)}^q(0,1)}.$$ In the case q = ∞, the space B = B γ p, q is identified with the Hardy space Hp. Using an operator L given by the equality \(Lf(z) = \sum\nolimits_{k = 0}^\infty {{l_k}{c_k}{z^k}} \) on functions \(f(z) = \sum\nolimits_{k = 0}^\infty {{c_k}{z^k}} \) analytic in the unit disk, we define the class $$LB_\gamma^{p,q}(N) := \{f:||Lf||_{B_{\gamma}^{p,q}}\leq N \}, N > 0.$$