Radial and transverse variations of analytic functions (Q5926430)

If \(\Gamma\) is a curve given by \(w=w(t)\), \(a\leq t\leq b\), then the length of \(\Gamma\) is given by \(\int^b_a|w'(t)|dt\). If, instead of \(w'(t)\), the projection of the vector \(w'(t)\) onto the vector \(w(t)\) is used, the corresponding integral is \(\int^b_a|{\text{Re} \{\overline ww'\}\over w}|dt=\int^b_a |{d\over dt} |w||dt\) and is called the radial variation of the curve \(\Gamma\). If one projects the vector \(w'(t)\) onto \(iw(t)\), the analogous integral is \(\int^b_a |{\text{Im}\{\overline ww'\} \over w}|dt=\int^b_a |w{d\over dt} \text{arg} w|dt\) and is called the transverse variation of \(\Gamma\). If \(f\) is an analytic function in the unit disk, then the radial variation of the curve \(w=f(re^{it})\), \(0\leq t\leq 2\pi\), is \(\int_0^{2\pi} |f(re^{it}) \text{Im}\{{re^{it} f'(re^{it})\over f (re^{it})}\} |dt\). Let \(R^1\) denote the family of functions for which these integrals are bounded for \(0< r <1\). More generally, \(R^p(p>0)\) denotes the family of functions \(f\) that are analytic in the unit disk and satisfy the condition \[ \sup_{0<r<1} \int_0^{2\pi} \left|f(re^{it}) \text{Im} \left\{ {re^{it} f'(re^{it}) \over f(re^{it})}\right\} \right|^p dt<\infty. \] Similarly, \(T^p(p>0)\) denotes the family of functions \(f\) that are analytic in the unit disk and satisfy the condition \[ \sup_{0<r<1} \int^{2\pi}_0 \left|f(re^{it}) \text{Re}\left \{{re^{it}f' (re^{it})\over f(re^{it})} \right\} \right |^pdt <\infty. \] The author proves several theorems about functions in \(R^p\) and \(T^p\) (including statements about the membership in the Hardy classes) and ends with two conjectures: Conjecture 1. If \(f\in R^p\) for some \(p>0\) and \(\varphi\) is a conformal automorphism of the unit disk, then \(f \circ \varphi\in R^p\). Conjecture 2. If \(f\in T^p\) for some \(p>0\) and \(\varphi\) is a conformal automorphism of the unit disk, then \(f\circ \varphi\in T^p\).