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This article is cited in 4 scientific papers (total in 4 papers) Estimates for the integrals of derivatives of rational functions in multiply connected domains in the plane A. D. Baranova, I. R. Kayumovb a Saint Petersburg State University b Kazan (Volga Region) Federal University
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     § 1. Introduction70 years ago Mergelyan [1] showed that there exists a bounded analytic function $f$ in the disc $\mathbb{D}=\{|z|<1\}$ such that where $dA(z) = (1/\pi)\, dx\,dy$, $z=x+iy$. This problem was further investigated by Rudin [2] who constructed an infinite Blaschke product
$$
\begin{equation*}
B(z)=\prod_{k=1}^\infty \frac{|z_k|}{z_k}\, \frac{z_k-z}{1-\overline{z_k} z}
\end{equation*}
\notag
$$
such that $I(B)=\infty$ and, moreover, $\int_{0}^1 |B'(re^{i\theta})|\, dr = \infty$ for a.e. $\theta\in[0, 2\pi]$. A similar but more explicit example was given by Piranian [3]. It is then natural to ask what happens when $B$ is a finite Blaschke product of degree $n$. The quantity $I(B)$ is clearly bounded for every fixed $n$, but it cannot be uniformly bounded with respect to $n$ since any bounded function in $\mathbb{D}$ is a locally uniform limit of finite Blaschke products. We find the sharp order of growth for such integrals. Namely, the following theorem holds. Theorem 1. Let $B$ be a finite Blaschke product of degree $n$. Then On the other hand, there exists an absolute constant $c>0 $ such that for any $n \in \mathbb{N}$ there exists a finite Blaschke product of degree $n$, satisfying $I(B) \geqslant c(1+\sqrt{\log n}\,)$. The proof of sharpness of this inequality is based on subtle results of Makarov [4] and Bañuelos and Moore [5] on the boundary behaviour of functions in the Bloch space. It should be noted that there exists a vast literature studying whether the derivatives of Blaschke products belong to various functional spaces, for example, Bergman-type spaces (see [6]–[8] and the references therein). However, most of these results concern infinite products and the conditions are formulated in terms of their zeros. Since a Blaschke product is a bounded rational function in the unit disc, the problem about the estimates of the derivatives of Blaschke products is related to a more general question about the integrals of bounded rational functions. The latter was studied for the first time by Dolzhenko [9] for sufficiently smooth domains. We will say that a curve belongs to the class $\mathrm{K}$ if it is a closed Jordan curve whose curvature $k(s)$ is a Hölder continuous function of the arc length $s$. Let $G$ be a finitely connected domain whose boundary curves belong to the class $\mathrm{K}$. Assume that $1\leqslant p \leqslant 2$, and let $R$ be a rational function of degree at most $n$ with poles outside $\overline{G}$. Dolzhenko ([9], Theorem 2.2) showed that there exists a constant $C$, depending only on the domain $G$ and on $p$, such that
$$
\begin{equation}
\int_{G}|R'(w)|^p\, dA(w) \leqslant C n^{p-1} \|R\|_{H^\infty(G)}^p, \qquad p \in (1,2],
\end{equation}
\tag{1.2}
$$
$$
\begin{equation}
\int_{G}|R'(w)|\, dA(w) \leqslant C \ln (n+1) \|R\|_{H^\infty(G)}.
\end{equation}
\tag{1.3}
$$
Here we write $H^\infty(G)$ for the space of all bounded analytic functions in $G$, and $\|f\|_{H^\infty(G)} = \sup_{w\in G} |f(w)|$. Later, inequalities for the derivatives of rational functions (mainly in the disc) were studied by Peller [10], Semmes [11], Pekarskii [12], [13], Danchenko [14], [15] and many other authors (see, e.g., [16]–[20]). A short proof of the Dolzhenko inequalities in the case of the disc when the $H^\infty$-norm is replaced by the weaker $\mathrm{BMOA}$-norm can be found in [19]. In the present article the inequalities (1.2) and (1.3) are proved under substantially weaker restrictions on the domain, namely, under the condition that the domain has no zero interior angles (more precisely, for domains in the John class; see the definition in § 3). Theorem 2. Let $G$ be a finitely connected John domain with rectifiable boundary and let $1\leqslant p \leqslant 2$. When $p=1$, we also assume that $G$ is bounded. Then there exists a constant $C> 0$, depending on the domain $G$ and on $p$, such that the inequalities (1.2) and (1.3) hold for any rational function $R$ of degree at most $n$. The sharpness of (1.2) is seen already in the simplest example of the function $R(z) = z^n$ in the disc (polynomials can clearly be regarded as a special case of rational functions with pole at infinity). The question of sharpness of (1.3) under the hypotheses of Theorem 2 remains open. However, it turns out that (1.3) can be improved under some additional regularity of the domain $G$. Theorem 3. Let $G$ be a simply connected domain with rectifiable boundary such that $\varphi' \in H^2$, where $\varphi$ is the conformal map of the disc $\mathbb{D}$ onto $G$. Then there exists a constant $C> 0$, depending on the domain $G$, such that for any rational function $R$ of degree at most $n$ one has It follows from Theorem 1 that the dependence on $n$ in this inequality is sharp. Finally, we give a statement in the case $p> 2$. Here $q$ is the conjugate exponent, i.e., $1/p+1/q= 1$. Theorem 4. Let $G$ be a bounded simply connected domain. We put $G_\rho = \{ z\in G$: $\operatorname{dist}(z, \partial G) >\rho\}$. Then for any rational function $R$ of degree at most $n$ and $p> 2$ one has The inequality (1.5) was established in [9] for domains of class $\mathrm{K}$, but our result shows that no restrictions on the regularity of the domain are required. Another generalization of the Dolzhenko inequality for rational functions in the disc to the case $p> 2$ was obtained in [20]. Let $R$ be a rational function of degree at most $n$ whose poles lie in the complement of the disc $\{|z| <1+\rho\}$. It follows directly from Theorem 8.2 in [19] that
$$
\begin{equation*}
\|R'\|_{A^p(\mathbb{D})} \leqslant C(p) n^{1/q} \rho^{1/p-1/q} \|R\|_{\mathrm{BMOA}},
\end{equation*}
\notag
$$
where $\mathrm{BMOA}$ denotes the analytic space of functions of bounded mean oscillation in the disc. It is interesting to note that the dependence on $n$ in Theorem 4 is substantially weaker (since in this case the function $R$ is assumed to be bounded on a larger set). In § 5 we obtain more general inequalities for weighted norms of the derivatives of rational functions, where the weight is a power of the distance to the boundary of the domain. A suitable toolbox for the study of such inequalities is provided by the theory of Hardy spaces. For $p> 0$ the Hardy space $H^p$ is the set of all analytic functions in $\mathbb{D}$ with $\|f\|_{H^p}< \infty$, where
$$
\begin{equation*}
\|f\|^p_{H^p}:=\sup_{0<r<1}\frac{1}{2\pi}\int_0^{2\pi}|f(re^{it})|^p\, dt.
\end{equation*}
\notag
$$
Note that for $p \geqslant 1$ this is a norm with respect to which $H^p$ is a Banach space.
§ 2. An estimate for the integral of the modulus of the derivative of a finite Blaschke productIn the proof of Theorem 1 we will use the following simple lemma. Lemma 1. Assume that the function $g(z)\,{=}\sum_{k=0}^\infty b_k z^k$ is analytic in the disc $\mathbb{D}$. If $\|g\|_\infty \leqslant 1$ and $p(z) = \sum_{k=0}^n b_k z^k$, $n\geqslant 2$, then there exists an absolute constant $C_0$ such that and Proof. Since $|b_k| \leqslant 1$ and $|z|^k \leqslant 1/n^2$ for $|z|\leqslant 1-2\log n/ n$ and $k\geqslant n$, the function $|g(z) - p(z)|$ admits a uniform estimate for $|z|\leqslant 1-2\log n/ n$, and the first estimate follows.
Clearly, and the second inequality is proved. $\Box$ Proof of Theorem 1. We use the following well-known facts: for any finite Blaschke product of degree at most $n$ and for any function $f(z) = \sum_{n\geqslant 0} a_n z^n$ in the Hardy space $H^2$.
Let $s \in [0,1]$. We have
$$
\begin{equation*}
\int_{\{s<|z| <1\}} |B'(z)|\, dA(z)= \frac{1}{\pi} \int_0^{2\pi} \int_s^1 |B'(re^{it})| r\, dr\, dt \leqslant 2n \int_s^1 r\, dr= n (1-s^2).
\end{equation*}
\notag
$$
In the remaining part, we apply the Cauchy–Schwarz inequality:
$$
\begin{equation*}
\begin{aligned} \, &\int_{\{0 <|z| \leqslant s\}} |B'(z)|\, dA(z) \\ &\qquad\leqslant \biggl( \int_{\{0 <|z| \leqslant s\}} |B'(z)|^2 (1-|z|^2)\, dA(z) \biggr)^{1/2} \biggl( \int_{\{0 <|z| \leqslant s\}} \frac{dA(z)}{1-|z|^2} \biggr)^{1/2} \\ &\qquad\leqslant \sqrt{2\int_0^s \frac{r\, dr}{1-r^2}} = \sqrt{\log\frac{1}{1-s^2}}. \end{aligned}
\end{equation*}
\notag
$$
Thus,
$$
\begin{equation*}
I(B) \leqslant n (1-s^2)+ \sqrt{ \log\frac{1}{1-s^2}}.
\end{equation*}
\notag
$$
Taking $s^2=1-1/n$, we obtain (1.1).
The estimate from below can be obtained by the methods based on the Makarov law of the iterated logarithm [4]. Recall that the Bloch class $\mathcal{B}$ consists of functions analytic in $\mathbb{D}$ with finite seminorm $ \|f\|_{\mathcal{B}} = \sup_{z\in \mathbb{D}} (1-|z|^2) |f'(z)|$. In [5], Bañuelos and Moore, answering a question of Makarov and Przytycki, constructed a function $f(z) = \sum_{k=1}^\infty a_k z^k$ in the Bloch class such that its asymptotic entropy admits the lower bound
$$
\begin{equation*}
\liminf_{r\to 1-} \frac{\sum_{k=1}^\infty |a_k|^2 r^{2k}}{\log(1/(1-r))} >0,
\end{equation*}
\notag
$$
whereas for all $\zeta$ with $|\zeta| = 1$, one has
$$
\begin{equation*}
\limsup_{r\to 1-} \frac{f(r \zeta)}{\sqrt{\log(1/(1-r)) \log\log\log(1/(1-r))}} =0.
\end{equation*}
\notag
$$
Moreover, in [5], p. 482, they constructed a sequence of polynomials
$$
\begin{equation*}
p_n(z)=\sum_{k=4}^{4^{n+1} -1} a_k z^k=\sum_{j=1}^{n} b_j(z), \quad \text{where} \quad b_j(z)=\sum_{k=4^j}^{4^{j+1} -1} a_k z^k,
\end{equation*}
\notag
$$
such that $\|b_j\|_\infty \leqslant 1$,
$$
\begin{equation*}
\sum_{k=1}^{4^{n+1} -1} |a_k|^2 \geqslant c \log m, \qquad \|p_n\|_{H^\infty} \leqslant C\sqrt{\log m},
\end{equation*}
\notag
$$
where $m = \operatorname{deg} p_n = 4^{n+1} -1$, and $C,c> 0$ are some absolute positive constants.
It is not difficult to deduce from $\|b_j\|_\infty \leqslant 1$ that $\sup_n\|p_n\|_{\mathcal{B}} < \infty$. Indeed, we have $|b_j(rz)| \leqslant r^{4^j}$ by the Schwarz lemma, whence $|b_j' (rz)| \leqslant 4^j r^{4^j}$ by the classical Bernstein inequality. It is well known (and easy to prove) that $\sum_{j\geqslant 1} 4^j r^{4^j} \leqslant C_1/(1-r^2)$ for some constant $C_1> 0$ and, thus, $\sup_n \|p_n\|_{\mathcal{B}} < \infty$. Without loss of generality we may assume that $\|p_n\|_{\mathcal{B}} \leqslant 1$. Let $r=1-1/m$. Then, for some absolute constants $C', c'> 0$,
$$
\begin{equation*}
c' \log\frac{1}{1-r} \,{\leqslant} \sum_{k=1}^m |a_k|^2 r^{2k} \,{\leqslant}\, 2 \int_{|z|<r}|p_n'(z)|^2(1-|z|^2)\,dA(z) \leqslant C' \! \int_{|z|<r}|p_n'(z)|\,dA(z).
\end{equation*}
\notag
$$
Now put $q_n = p_n/(C\sqrt{\log m}\,)$. Then $\|q_n\|_\infty \leqslant 1$ and
$$
\begin{equation*}
\int_{|z|<1-1/m}|q_n'(z)|\, dA(z) \geqslant c_1\sqrt{\log m}
\end{equation*}
\notag
$$
for some absolute constant $c_1> 0$. Since
$$
\begin{equation*}
\begin{aligned} \, &\int_{1-2\log m/m < |z|<1-1/m}|q_n'(z)| \, dA(z) \\ &\qquad \leqslant \int_{1-2\log m /m < |z|<1-1/m} \frac{dA(z)}{1-|z|^2}=O(\log\log m), \end{aligned}
\end{equation*}
\notag
$$
we have, for some $c_2> 0$,
$$
\begin{equation}
\int_{|z|<1-2\log m/m}|q_n'(z)|\,dA(z) \geqslant c_2 \sqrt{\log m}.
\end{equation}
\tag{2.2}
$$
Let $B$ be a Blaschke product of degree at most $m+1$ such that its first $m$ Taylor coefficients coincide with the corresponding coefficients of the polynomial $q_n$. By Lemma 1,
$$
\begin{equation*}
|q_n'(z)-B'(z)| \leqslant 2C_0, \qquad |z| \leqslant 1-2\frac{\log m}{m}.
\end{equation*}
\notag
$$
Hence, it follows from (2.2) that for some absolute constant $c_3$. $\Box$ § 3. Estimates for the integrals of rational functionsRecall that a finitely connected domain $\Omega$ is called a John domain if there exists a constant $C> 0$ such that any points $a,b\in \Omega$ can be connected by a curve $\gamma$ in $\Omega$ with the following property. For every $x\in\gamma$ we have
$$
\begin{equation*}
\min \bigl( \operatorname{diam}\gamma(a,x), \operatorname{diam}\gamma(x,b) \bigr) \leqslant C \operatorname{dist} (x, \partial \Omega),
\end{equation*}
\notag
$$
where $\gamma (a,x)$ and $\gamma (x,b)$ are the subarcs into which $x$ divides $\gamma$. For equivalent definitions and properties of John domains see, e.g., [21], [22]. Essentially, this definition means that the domain has no zero interior angles. In particular, a domain is a John domain if it satisfies the cone condition: one can touch each boundary point from inside of the domain by some sufficiently small triangle with fixed angles. In what follows we will essentially use the following property of simply connected John domains. If $\varphi$ is a conformal map of $\mathbb{D}$ onto a simply connected John domain, then for some $\alpha\in(0,1)$ and $C> 0$ (see [22], pp. 96–100).In the proof of Theorem 2 we will use the following simple lemma: Proof. Let $g(z) = \sum_{k=0}^\infty a_kz^k$. Then Here we used the elementary inequality $kr^{k-1}(1-r) \leqslant 1$, $k\geqslant 1$, $r \in [0,1)$. Since $g$ is at most $n$-valent in $\mathbb{D}$, the classical area theorem yields that Proof of Theorem 2. We first assume that $G$ is bounded. There is no loss of generality in assuming that $G$ is a simply connected domain with rectifiable boundary. Indeed, making smooth cuts (and controlling the angles), one can easily represent $G$ as a finite union of simply connected John domains.
Let $w=\varphi(z)$ be a conformal map of $\mathbb{D}$ onto $G$. Since the boundary of $G$ is rectifiable, we have $\varphi' \in H^1$. Also, $\varphi'$ satisfies the inequality (3.1). By changing the variable, we have
$$
\begin{equation*}
\begin{aligned} \, \int_{G}|R'(w)|^p\, dA(w) &= \int_{\mathbb{D}}|R'(\varphi(z))|^p|\varphi'(z)|^2\, dA(z) \\ &=\int_{\mathbb{D}} |(R\circ \varphi)'(z)|^p |\varphi'(z)|^{2-p}\, dA(z). \end{aligned}
\end{equation*}
\notag
$$
When $p= 2$, the last integral clearly does not exceed $nM^2$ since the function $R\circ\varphi$ is at most $n$-valent in the disc $\mathbb{D}$.
We split the last integral into the integrals over $\{|z| \leqslant r_n\}$ and $\{r_n < |z| < 1\}$, where $r_n = 1- 1/(n+1)^{K}$ and $K> 0$ is to be chosen later. Estimate of the integral over $\{|z| \leqslant r_n\}$. Let $M = \|R\|_{H^\infty(G)}$. Set
$$
\begin{equation*}
J := \int_{\{|z|\leqslant r_n\}} |(R\circ \varphi)'(z)|^p\, |\varphi'(z)|^{2-p}\, dA(z).
\end{equation*}
\notag
$$
When $p= 1$, we use the estimate $(1-|z|^2) |(R\circ \varphi)'(z)| \leqslant M$. We have
$$
\begin{equation*}
\begin{aligned} \, J &\leqslant \frac{M}{\pi} \int_0^{r_n} \frac{1}{1-r}\int_0^{2\pi} |\varphi'(re^{it})|\, dt\, dr \\ &\leqslant 2 \|\varphi'\|_{H^1} M \int_0^{r_n} \frac{dr}{1-r} = 2K \|\varphi'\|_{H^1} \log(n+1) M. \end{aligned}
\end{equation*}
\notag
$$
When $1< p< 2$, we separately consider the integrals over $\{|z| \leqslant 1-1/(n+1)\}$ and $\{1-1/(n+1) <|z| \leqslant r_n\}$. Since $\varphi' \in H^1$, we have $\varphi'\in H^{2-p}$ and $\|\varphi'\|_{H^{2-p}} \leqslant \|\varphi'\|_{H^1}$. Hence,
$$
\begin{equation*}
\begin{aligned} \, \int_{\{|z| \leqslant 1- 1/(n+1)\}} |(R\circ \varphi)'(z)|^p\, |\varphi'(z)|^{2-p} \, dA(z) & \leqslant 2\|\varphi'\|_{H^1}^{2-p} \int_0^{1-1/(n+1)} \frac{M^p}{(1-r)^p}\, dr \\ &=2\|\varphi'\|_{H^1}^{2-p} (p-1)^{-1} (n+1)^{p-1} M^p. \end{aligned}
\end{equation*}
\notag
$$
To estimate the integral over $\{1-1/(n+1)<|z| \leqslant r_n\}$, we use the Hölder inequality with exponents $(2-p)^{-1}$ and $(p-1)^{-1}$:
$$
\begin{equation*}
J \,{\leqslant}\, 2 \int_{1-1/(n+1)}^{r_n} \biggl(\frac{1}{2\pi} \int_0^{2\pi} |(R\circ \varphi)'(re^{it})|^{p/(p-1)}\, dt\biggr)^{p-1} \!\biggl(\frac{1}{2\pi} \int_0^{2\pi} |\varphi'(re^{it})|\, dt\biggr)^{2-p} dr.
\end{equation*}
\notag
$$
Using successively the inequality $(1-|z|^2) |(R\circ \varphi)'(z)| \leqslant M$ and Lemma 2, we get
$$
\begin{equation*}
\begin{aligned} \, J &\leqslant 2 \|\varphi'\|_{H^1}^{2-p} M^{p-2(p-1)} \int_{1-1/(n+1)}^{r_n} \frac{1}{(1-r)^{p-2(p-1)}} \\ &\qquad\times \biggl(\frac{1}{2\pi} \int_0^{2\pi} |(R\circ \varphi)'(re^{it})|^2 \, dt\biggr)^{p-1}\, dr \\ &\leqslant 2 \|\varphi'\|_{H^1}^{2-p} M^p n^{p-1} \int_{1-1/(n+1)}^{r_n} \frac{dr}{(1-r)^{2-p +p-1}} \\ &=2 \|\varphi'\|_{H^1}^{2-p} M^p n^{p-1} \int_{1-1/(n+1)}^{1-1/(n+1)^K} \frac{dr}{1-r} = 2 \|\varphi'\|_{H^1}^{2-p} K n^{p-1} M^p. \end{aligned}
\end{equation*}
\notag
$$
Estimate of the integral over $\{r_n<|z| <1\}$In this case the argument applies to all $p\in [1, 2)$. Choose any $\delta$ with $0< \delta<1-p/2$. Then $2-p-\delta \in (0,1)$. Let $\beta$ be the exponent conjugate to $(2-p-\delta)^{-1}$. By (3.1),
$$
\begin{equation*}
\begin{aligned} \, I &:= \int_{r_n<|z|<1} |(R\circ \varphi)'(z)|^p \, |\varphi'(z)|^{2-p} \, dA(z) \\ &\leqslant C^\delta \int_{r_n<|z|<1} \frac{|(R\circ \varphi)'(z)|^p \, |\varphi'(z)|^{2-p-\delta}}{(1-|z|)^{\alpha\delta}}\, dA(z) \\ &\leqslant 2C^\delta \int_{r_n}^1 \frac{1}{(1-r)^{\alpha\delta}} \biggl(\frac{1}{2\pi} \int_0^{2\pi} |(R\circ \varphi)'(re^{it})|^{p\beta}\, dt\biggr)^{1/\beta} \\ &\qquad\times \biggl(\frac{1}{2\pi} \int_0^{2\pi} |\varphi'(re^{it})|\, dt\biggr)^{2-p-\delta} \, dr. \end{aligned}
\end{equation*}
\notag
$$
Note that it follows from $\delta<1-p/2$ that $p\beta> 2$. Applying the estimate $(1- |z|^2) |(R\circ \varphi)'(z)| \leqslant M$ and Lemma 2, we get
$$
\begin{equation*}
\begin{aligned} \, I &\leqslant 2 C^\delta\|\varphi'\|_{H^1}^{2-p-\delta} M^{(p\beta - 2)/\beta} \int_{r_n}^1 \frac{1}{(1-r)^{\alpha\delta + (p\beta - 2)/\beta}} \\ &\qquad\times \biggl(\frac{1}{2\pi} \int_0^{2\pi} |(R\circ \varphi)'(re^{it})|^2\, dt\biggr)^{1/\beta}\, dr \\ &\leqslant 2 C^\delta\|\varphi'\|_{H^1}^{2-p-\delta} M^p n^{1/\beta} \int_{r_n}^1 \frac{dr}{(1-r)^{\alpha\delta + (p\beta-2)/\beta +1/\beta}} \\ &=2 C^\delta\|\varphi'\|_{H^1}^{2-p-\delta} M^p n^{1/\beta} \int_{r_n}^1 \frac{dr}{(1-r)^{\alpha\delta + p -1/\beta}}. \end{aligned}
\end{equation*}
\notag
$$
It remains to notice that $\alpha\delta +p- 1/\beta = \alpha\delta +p- (1- (2-p-\delta)) = 1-(1-\alpha)\delta$, whence
$$
\begin{equation}
I \leqslant 2(1-\alpha)^{-1}\delta^{-1}C^\delta\|\varphi'\|_{H^1}^{2-p-\delta} M^p n^{1/\beta}.
\end{equation}
\tag{3.2}
$$
If we fix $\delta\in (0, 1-p/2)$ and choose a sufficiently large $K$ in $r_n = 1- 1/(n+1)^K$, we conclude that $I \leqslant C^\delta \|\varphi'\|_{H^1}^{2-p-\delta} M^p$ (and even $o(1)$ as $n\to \infty$). We now consider the case when $\infty \in G$. It is clear that such a domain can be represented as a union (not necessarily disjoint) of the complement of a disc of sufficiently large radius and a simply connected bounded John domain. The statement is already proved for bounded simply connected domains, while for the complement of a disc it follows from the results of Dolzhenko cited above. $\Box$ § 4. Proofs of Theorems 3 and 4 Proof of Theorem 3. As in the proof of Theorem 2, we set $r_n=1-1/(n+1)^K$, where $K> 0$. Since $\varphi' \in H^2 \subset H^1$ and (3.1) holds with $\alpha=1/2$, one can use the estimate (3.2) for the integral over $\{r_n<|z|<1\}$, which was established in the proof of Theorem 2. For a sufficiently large $K$ this integral is uniformly bounded with respect to $n$ (and even tends to zero as $n\to \infty$).
Thus, it suffices to estimate
$$
\begin{equation*}
J:=\int_{0<|z| \leqslant r_n} |(R\circ \varphi)'(z)| |\varphi'(z)|\, dA(z).
\end{equation*}
\notag
$$
By the Cauchy–Schwarz inequality,
$$
\begin{equation*}
\begin{aligned} \, J & \leqslant \biggl(\int_{0<|z|\leqslant r_n} (1-|z|) |(R\circ \varphi)'(z)|^2\, dA(z) \biggr)^{1/2} \biggl(\int_{0<|z| \leqslant r_n} \frac{|\varphi'(z)|^2}{1-|z|}\, dA(z) \biggr)^{1/2} \\ & \leqslant M \biggl( \int_{0<|z| \leqslant r_n} \frac{|\varphi'(z)|^2}{1-|z|}\, dA(z) \biggr)^{1/2} \leqslant \sqrt{2} M \|\varphi'\|_{H^2} \biggl( \int_0^{r_n} \frac{dr}{1-r} \biggr)^{1/2} \\ & =\sqrt{2} M \|\varphi'\|_{H^2} \sqrt{K\ln (n+1)}.\qquad\Box \end{aligned}
\end{equation*}
\notag
$$
Proof of Theorem 4. Let $\varphi$ be a conformal map of $\mathbb{D}$ onto $G$ and let $D_\rho = \varphi^{-1}(G_\rho)$. Since $\rho \leqslant (1-|z|^2)|\varphi'(z)|$, $z\in D_\rho$, we have
$$
\begin{equation*}
\begin{aligned} \, \int_{G_\rho} |R'(\zeta)|^p\, dA(\zeta) & = \int_{D_\rho} |(R\circ \varphi)'(z)|^p |\varphi'(z)|^{2-p}\, dA(z) \\ & \leqslant \rho^{2-p} \int_{D_\rho} |(R\circ \varphi)'(z)|^p (1-|z|^2)^{p-2}\, dA(z) \\ & \leqslant \rho^{2-p} M^{p-2} \int_{D_\rho} |(R\circ \varphi)'(z)|^2\, dA(z) \leqslant \rho^{2-p} n M^p. \end{aligned}
\end{equation*}
\notag
$$
At the last step we used the fact that $R\circ \varphi$ covers each point of the disc of radius $M$ with multiplicity at most $n$. $\Box$ § 5. Weighted inequalities of Dolzhenko and Peller typeAs a natural generalization of the Dolzhenko inequalities, one can consider weighted integrals of derivatives of rational functions. Similar inequalities were studied extensively in the setting of Bergman (or Besov) spaces. For example, a well-known inequality by Peller [10] states that for a rational function $R$ of degree $n$ with poles outside $\overline{\mathbb{D}}$ one has
$$
\begin{equation*}
\|R\|_{B_p^{1/p}} \leqslant C n^{1/p} \|R\|_{\mathrm{BMOA}},
\end{equation*}
\notag
$$
where $B_p^{1/p}$ is the Besov space, $p> 0$, $C=C(p)$. In particular, for $1< p< \infty$,
$$
\begin{equation*}
\int_\mathbb{D} |R'(z)|^p (1-|z|)^{p-2}\, dA(z) \leqslant C n \|R\|^p_{H^\infty}.
\end{equation*}
\notag
$$
Various proofs and generalizations of this inequality can be found in [11]–[13], [19]. Using the methods of § 3, one can obtain more general weighted estimates, where the weight is a power of the distance to the boundary. We now state the result. Given a bounded domain $G\subset \mathbb{C}$ and a point $z\in G$, we put Given any $p\geqslant 1$, $\beta \in \mathbb{R}$ and an analytic function $f$ in $G$, we put
$$
\begin{equation*}
I_{p,\beta} (f) := \int_G |f'(\zeta)|^p \, d_G^\beta(\zeta)\, dA(\zeta)
\end{equation*}
\notag
$$
(in general, $I_{p,\beta} (f)$ can be infinite). We are interested in estimates of the form
$$
\begin{equation*}
I_{p,\beta}(R) \leqslant C\Psi(n) \|R\|^p_{H^\infty(G)},
\end{equation*}
\notag
$$
which hold for all rational functions $R$ of degree at most $n$ with poles outside $\overline{G}$ and the constant $C$ depends on $G$, $p$ and $\beta$, but not on $n$ and $R$. Here $\Psi$ is a function depending only on $n$. It is easy to see that such estimates are possible only for $\beta \geqslant p-2$. Indeed, putting $G=\mathbb{D}$ and considering the rational fractions $R(\zeta) = 1/(\zeta-\lambda)$, we see that the integral $I_{p,\beta} (f)$ with $\beta< p-2$ admits no estimate depending only on $n$, but the distance from the poles of $R$ to $\partial G$ must be taken into account. To simplify the notation, we write $X(R,n) \lesssim Y(R,n)$ if $X(R, n)\leqslant CY(R,n)$ with a constant $C$ depending only on $G$, $p$ and $\beta$, but not on $n$ and $R$. Theorem 5. Suppose that $G$ is a simply connected bounded domain, $\varphi$ is a conformal map of $\mathbb{D}$ onto $G$, $p\geqslant 1$, $\beta \geqslant p-2$. Then the following estimates hold. 1) If $\beta > p-1$ and $\varphi'\in H^{\gamma}$ for some $\gamma> 1$, then $I_{p,\beta}(R) \lesssim \|R\|^p_{H^\infty(G)}$, i.e., the dependence on $n$ disappears. 2) If $\beta = p-1$, $1\leqslant p < 2$ and $\varphi'\in H^{2/(2-p)}$, then
$$
\begin{equation*}
I_{p,\beta}(R) \lesssim (\log n)^{1- p/2} \|R\|^p_{H^\infty(G)}.
\end{equation*}
\notag
$$
If $\beta = p-1$, $p \geqslant 2$ and $\varphi'\in H^\infty$, then $I_{p,\beta}(R) \lesssim \|R\|^p_{H^\infty(G)}$. 3) If $p-2 \leqslant \beta <p-1$, $p\geqslant 2$, $\varphi' \in H^1$ and $G$ is a John domain, then The dependence on $n$ in the inequalities of Theorem 5 is sharp already in the case of the unit disc. The optimal growth in part 3) is attained at $R(z) = z^n$, while the sharpness of the inequality in part 2) can be shown by considering the Bañuelos–Moore construction (and taking a polynomial or a Blaschke product for $R$). Note that part 3) does not cover the case when $p-2 \leqslant \beta <p-1$ and $1< p< 2$. In this case it would be sufficient to prove the following analogue of Peller’s inequality:
$$
\begin{equation*}
\int_\mathbb{D} |(R\circ\varphi)'(z)|^p (1-|z|)^{p-2}\, dA(z) \lesssim n\|R\|_{H^\infty(G)}^p,
\end{equation*}
\notag
$$
where $G = \varphi(\mathbb{D})$ is a John domain and $R$ is a rational function of degree at most $n$ with poles outside $\overline{G}$. However, we do not know whether this inequality holds true. Proof of Theorem 5. Put $M = \|R\|_{H^\infty(G)}$. We make a change of the variable $\zeta = \varphi(z)$. Since $d_G(\zeta) \leqslant |\varphi'(z)|(1-|z|^2)$, we obtain
Part 3) follows from Theorem 2. Indeed, $1< p-\beta \leqslant 2$ and, using the inequality $|(R\circ\varphi)'(z)| (1-|z|) \leqslant M$, we obtain
$$
\begin{equation*}
I_{p,\beta}(R) \lesssim M^\beta \int_{\mathbb{D}} |(R\circ\varphi)'(z)|^{p-\beta} |\varphi'(z)|^{2-(p-\beta)}\, dA(z) \lesssim n^{p-\beta-1}M^p.
\end{equation*}
\notag
$$
We now prove part 1). Since $d_G(z)$ is bounded, it suffices to prove the statement for $\beta \in (p-1, p-2 +\gamma]$. For these $\beta$, it follows from the inequality $p-\beta < 1$ and the inclusion $\varphi' \in H^\gamma \subset H^{2-p+\beta}$ that
$$
\begin{equation*}
I_{p,\beta}(R) \lesssim M^p \int_0^1 \biggl(\int_0^{2\pi}|\varphi'(re^{it})|^{2-p+\beta} \, dt\biggr)\, \frac{r\, dr}{(1-r)^{p-\beta}} \lesssim M^p.
\end{equation*}
\notag
$$
At the first step we used the inequality $|(R\circ\varphi)' (z)|(1-|z|) \leqslant M$.
Consider the most interesting part 2): $\beta = p-1$. Suppose that $1\leqslant p < 2$. Put $s=1-1/n$. Applying the Hölder inequality with exponents $2/p$ and $2/(2-p)$ and the inequality (2.1), we get
$$
\begin{equation*}
\begin{aligned} \, &\int_{\{|z|\leqslant s\}} |(R\circ\varphi)'(z)|^p\, |\varphi'(z)| (1-|z|)^{p-1}\, dA(z) \\ &\ \leqslant \biggl( \int_{\{|z|\leqslant s\}} |(R\circ\varphi)'(z)|^2 (1-|z|)\, dA(z) \biggr)^{p/2} \biggl( \int_{\{|z|\leqslant s\}} \frac{|\varphi'(z)|^{2/(2-p)}}{1-|z|}\, dA(z) \biggr)^{1-p/2} \\ &\ \lesssim M^p \biggl( \int_0^s \biggl(\int_0^{2\pi} |\varphi'(re^{it})|^{2/(2-p)}\, dt \biggr) \frac{r\, dr}{1-r} \biggr)^{1-p/2} \lesssim (\log n)^{1- p/2} M^p. \end{aligned}
\end{equation*}
\notag
$$
It remains to estimate the integral over $\{s<|z| <1\}$:
$$
\begin{equation*}
\begin{aligned} \, &\int_{\{s<|z| <1\}} |(R\circ\varphi)'(z)|^p\, |\varphi'(z)| (1-|z|)^{p-1}\, dA(z) \\ &\ \leqslant M^{p-1} \int_{\{s<|z| <1\}} |(R\circ\varphi)'(z)| \, |\varphi'(z)|\, dA(z) \\ &\ \lesssim M^{p-1} \biggl(\int_{\{s<|z| <1\}} |(R\circ\varphi)'(z)|^2\, dA(z)\biggr)^{1/2} \biggl(\int_{\{s<|z| <1\}} \! |\varphi'(z)|^2\, dA(z)\biggr)^{1/2} \,{\lesssim}\, M^p. \end{aligned}
\end{equation*}
\notag
$$
In the last inequality we used the inclusion $\varphi'\in H^2$ and the fact that
$$
\begin{equation*}
\int_{\{s<|z| <1\}} |(R\circ\varphi)'(z)|^2\, dA(z) \lesssim n M^2
\end{equation*}
\notag
$$
since $R\circ\varphi$ covers the disc of radius $M$ with multiplicity at most $n$.
The case $p\geqslant 2$ is trivial: i.e., the quantity $I_{p, p-1}(R)$ is uniformly bounded with respect to $R$ and $n$. $\Box$
Citation in format AMSBIB
\Bibitem{BarKay22} |
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