The Schwarz-Pick theorem and its applications (Q2890389)

The authors derive estimates on the derivatives of functions of exponential type in a half-plane. They prove that if \(f\) is a function of exponential type in the upper-half plane whose Phragmén-Lindelöf indicator function satisfies \(h_f(\pi /2)\leq c \neq 0\) and if \(f\) is continuous in the closed upper-half plane with the property that \(|f(x)|\leq M\) for all real \(x\), then NEWLINE\[NEWLINE |f^{(k)}(z)|\leq \begin{cases} M|c|^ke^{cy}\qquad&\text{if}\enskip y:= \Im z\geq k/|c|,\\ M\big( \frac{k^2+c^2y^2}{2ky}\big)^ke^{cy} \qquad &\text{if} \enskip 0<y<k/|c| \end{cases}NEWLINE\]NEWLINE for \(k=1,2,3,\dots\) They also derive estimates on two differential operators acting on functions of exponential type in a half-plane. With the help of a preliminary transformation of an angle \(\mathcal{A}(\theta_1, \theta_2):=\{z=re^{i\theta}: r>0,\, \theta_1<\theta<\theta_2,\, \theta_2-\theta_1<2\pi\}\) onto a half-plane, the authors derive related results for functions analytic in an angle.NEWLINENEWLINETheir main tool is the Schwarz-Pick theorem: if \(\phi \) is holomorphic with \(|\phi (z)|\leq 1\) for \(|z|<1\), then \((1-|z|^2)|\phi '(z)|+|\phi (z)|^2\leq 1\) for \(|z|<1\). In an earlier paper [Complex Var. Elliptic Equ. 58, No. 8, 1071--1084 (2013; Zbl 1291.30006)], the authors proved an analogue of the Schwarz-Pick theorem for functions analytic in the upper-half plane. Here the authors continue the study of such problems. They show that if \(f\) is analytic in \(\mathcal{A}(-\theta, \theta)\) and \(|f(\zeta )|\leq 1\) therein, then NEWLINE\[NEWLINE\frac{4\theta}{\pi}\rho\,\big(\cos \frac{\pi \phi}{2\theta}\big)|f'(\zeta )|+|f(\zeta )|^2\leq 1\quad (\zeta=\rho e^{i\phi}\in \mathcal{A}(-\theta, \theta))NEWLINE\]NEWLINE They demonstrate that this inequality holds true under other conditions on \(f\).NEWLINENEWLINEThe authors also derive analogues of the Schwarz-Pick theorem for functions analytic in a strip \(S(-b,b):=\{ \zeta=\xi+i\eta:\, -b<\xi <b\}\).