Taylor coefficients and mean growth of the derivative of \(Q_p\) functions (Q5940324)

Let \(\Delta\) denote the unit disk in the complex plane and let \(g(z,a)\) denote the Green's function for \(\Delta\) with logarithmic singularity at the point \(a\in\Delta\). For \(0<p<\infty\), we say that the analytic function \(f\) is in the space \(Q_p\) if NEWLINE\[NEWLINE\sup_{a\in\Delta}\int\int_\Delta\bigl|f'(z) \bigr |^2 \bigl(g(z,a) \bigr)^p dx dy<\inftyNEWLINE\]NEWLINE [see \textit{R. Aulaskari} and \textit{P. Lappan}, Pitman Res. Notes Math. Ser. 305, 136--146 (1994; Zbl 0826.30027) and \textit{R. Aulaskari}, \textit{J. Xiao}, and \textit{R. Zhao} [Analysis 15, No. 2, 101--121 (1995; Zbl 0835.30027)]. It is well known that \(Q_1= \text{BMOA}\) and for all \(p\in (0,\infty)\), \(Q_p=B\), the Bloch space. NEWLINENEWLINEIf \(f(z) =\sum a_nz^n\) and \(g(z)=\sum b_nz^n\), the Hadamard product of \(f\) and \(g\) is defined by \(f*g(z)=\sum a_nb_nz^n\). In the present paper, the authors prove that for \(0<p<1\), if \(f\in Q_p\) and \(g\in B\) then \(f*g\in Q_p\) and \(\|f*g\|_{Q_p}\leq C\|f\|_{Q_p}\|g\|_{B}\). (for \(1\leq p<\infty\) [see \textit{J. M. Anderson} et al., J. Reine Angew. Math. 270, 12--37 (1974; Zbl 0292.30030) and \textit{M. Mateljević} and \textit{M. Pavlović}, Pac. J. Math. 146, No. 1, 71--84 (1990; Zbl 0731.30029)]. They also obtain some results on mean growth of the derivative of \(Q_p\) functions.