An exact order of the majorant growth in the Schwarz–Pick inequality for torsional rigidity

The beginning of Schwarz–Pick type inequalities may be found in classical papers of Pick [14], Caratheodory [13], Szasz [19], Bernstein [12] and others. In recent years this program is actively developed, a number of results on inequalities of this type can be found in articles of Ruscheweyh [16;17], Yamashita [20], Avkhadiev [7–10] etc. (see also [2–4]). These results are concerned with function $f$ holomorphic or meromorphic in a domain $\Omega$ in the extended complex plane $\overline{\mathbb{C}}$ and $f(\Omega)\subset\Pi\subset\mathbb{\overline{C}}$. In [6] we obtained Schwarz–Pick type inequalities for the torsional rigidity. As known, the Saint-Venant functional P for the torsional rigidity in an arbitrary plane $\Omega$ can be found as the solution of the generalized problem (see [1;11;15])
$$ P(\Omega)=\sup\limits_{u\in C_0^{\infty}(\Omega)}\frac{\left(2\int_{\Omega} u(x)dx \right)^2}{\int_{\Omega} |\nabla u|^2 dxdy}, $$
where $(x,y)\in\Omega,\; C_0^{\infty}(\Omega)$—the space of smooth functions with compact support in $\Omega$. Let $\Omega\in\mathbb{C}$ arbitrary simply connected domain and $0\in\mathbb{C}$. According to Riemann's theorem there exists a function $f$ such that $f:\Delta\rightarrow\Omega$ and $f(0)=0$. Let $\Omega_r$ the image of the circle $\Delta_r=\{\zeta\in\mathbb{C}:|\zeta|<r\}$ under the mapping $f$ for each $r\in(0,1)$, i.e. $\Omega_r=\{z\in\Omega: z=f(\zeta), |\zeta|<r, r\in(0,1)\}$. In [6] formulated an analogue of Schwarz–Pick theorem for the $P(\Omega)$, namely proved
Theorem. Let $P(\Omega)<\infty$ and $0<r<1$. Then the following inequalities hold
$$ \frac{dP(\Omega_r)}{dr}<\frac{4r^3}{1-r^8}P(\Omega), $$
and, for each $m\in\mathbb{N}$,
$$ \left(\frac{P(\Omega_r)}{r^4}\right)^{(2m+1)}<\frac{(2m+1)!P(\Omega)}{(1-r^2)^{2m+1}}\sum\limits_{k=0}^m{m\choose k}^2r^{2k}. $$

We see, that both inequalities are strict in this theorem. In this paper we establish the asymptotic accuracy of the estimates. We prove the next theorems:
Theorem 1. For each $r_0\in[1/2,1)$ there exists $\Omega=\Omega(r_0)$, $z=0\in\Omega$, such that
$$ \left.\frac{dP(\Omega_r(r_0))}{dr}\right|_{r=r_0}\geq\frac{c_0}{1-r_0^2}, $$
where $c_0=\frac{\pi}{2^73^5}$.
Theorem 2. For each $r_0\in[1/2,1)$ there exists $\Omega=\Omega(r_0)$, $z=0\in\Omega$, such that
$$ \left.\left(\frac{P(\Omega_r(r_0))}{r^4}\right)^{(n)}\right|_{r=r_0}\geq\frac{c}{(1-r_0^2)^n}, $$
where $c=\frac{\pi}{2^{3n+2}3^{n+5}}$, n>1.