Forward and Reverse Carleson Inequalities for Functions in Bergman Spaces and Their Derivatives

is satisfied. In the process, a simple proof of certain inequalities on derivatives of functions in HP is obtained (Section 3) as well as some information on interpolation sequences for LP (Section 5). Inequalities like (1.1) have already been used to provide information on Bergman-Toeplitz operators L1 and are useful for obtaining representations of functions in Lp (Section 4, Theorem 4.6). The inequality which reverses the roles of A and m in (1.1) is relatively easy-even if an appropriate weighting factor is included with m (Theorem A below). It was obtained by Oleinik and Pavlov OP and independently (when p = q > 1) by Stegenga S and (with c = 0) Hastings Ha. The results of Oleinik and Pavlov (which are more general than Theorem A) were extended somewhat by Oleinik in 0. Cima and Wogen CW, using Stegenga's methods, proved the analogue of Theorem A in several variables. The methods of L3 more or less supercede all these results.