A new characterization of \(Q_p^{\#}\) functions (Q812457)

The class \(Q_p^{\#}\), \(0 < p < \infty\), consists of all functions \(f\), meromorphic in the unit disk \(\Delta\), such that \[ \sup_{a \in \Delta} \iint_\Delta \bigl( f^{\#}(z) \bigr)^2 \bigl( g(z, a) \bigr)^p \, \mathrm dm(z) < \infty . \] Here \(f^{\#}\) is the spherical derivative of \(f\), \(g\) is the Green function and \(\mathrm dm\) is the two-dimensional Lebesgue measure. Define \[ T_p(1, f) = p \int_0^1 t^{-1} \bigl( \log \tfrac1t \bigr)^{p-1} \iint_{| z| < t} \bigl( f^{\#}(z) \bigr)^2 \, \mathrm dm(z) \, \mathrm dt \] and \(\phi_a(z) = \frac{a - z}{1 - \bar a z}\). The authors prove that a meromorphic function \(f\) is in \(Q_p^{\#}\), \(0 < p < \infty\) if and only if \[ \limsup_{a \rightarrow \partial \Delta} T_p(1, f \circ \phi_a) < \infty . \] This new characterization gives easy proofs of some known results. Also, a similar characterization of the class \(M_p^{\#}\), \(0 < p < \infty\), consisting of meromorphic functions \(f\) with \[ \sup_{a \in \Delta} \iint_\Delta \bigl( f^{\#}(z) \bigr)^2 \bigl( 1 - | \phi_a(z)| ^2 \bigr)^p \, \mathrm dm(z) < \infty , \] is given.