On harmonic Bloch and normal functions (Q819968)

Let \(f=h+{\overline g}\) be a harmonic univalent and sense preserving function on the unit disk \(U\), where \(h\) and \(g\) are analytic. Then \(f=h+\overline g\) is called a Bloch function if \[ \sup_{z\in U}(1-| z| ^{2})(| h'(z)| +| g'(z)| )<\infty. \] It is said to be a normal function if the normality order \[ \alpha_f=\sup_{z\neq w}\frac{\chi\bigl(f(z),f(w)\bigr)}{\rho(z,w)}<\infty, \] where \(\chi\) is the chordal metric and \(\rho\) is the hyperbolic metric on \(U\). The first result in the paper under review generalizes one of \textit{Ch. Pommerenke}'s results [J. Lond. Math. Soc., II. Ser. 2, 689--695 (1970; Zbl 0199.39803)] for harmonic functions: \(f=h+{\overline g}\) is a Bloch function if and only if there exist \(H\), \(G\in S\) and constants \(c_1\), \(c_2>0\) such that \(f(z)=c_1\log H'(z)+c_2\log{\overline G'(z)}+f(0)\), where \(S\) is the well-known class of analytic univalent functions. The authors' second result is that, in fact, \[ \alpha_f=\sup_{z\in U}(1-| z| ^{2})\,\frac{| h'(z)+| g'(z)| }{1+| f(z)^{2}| } \] for a normal harmonic function \(f=h+\overline g\). Some other results on the normality order \(\alpha_f\) are also shown.