Integral means of derivatives of locally univalent Bloch functions (Q2769209)

Let \(f(z)=\log(1-z)\), \(\omega(z)= \exp(-\pi {1+z\over 1-z})\), \(F_0=f \circ\omega\), \(F_k=F_{k-1} \circ\omega\), \(k=1,2,\dots\) The functions \(F_k\), \(k=1,2,\dots\), are locally univalent Bloch functions in the unit disk \(\Delta\) and for \(p>{1\over 2}\) the integral means NEWLINE\[NEWLINEI_p(r,F_k')= {1 \over 2\pi} \int^{2\pi}_0 \bigl|F_k'(re^{i\theta}) \bigr|^p d\theta,\;0<r<1NEWLINE\]NEWLINE satisfy NEWLINE\[NEWLINEI_p(r,F_k') \geq{c(k,p) \over(1-r^2)^{p-\frac 12}}\log^k {1 \over 1-r^2},NEWLINE\]NEWLINE where \(0\leq c(k,p)<1\).