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Estimates for integrals of derivatives of A. D. Baranova, I. R. Kayumovba a Saint Petersburg State University, St. Petersburg, Russia b Kazan (Volga Region) Federal University, Kazan, Russia
Russian version:
Matematicheskii Sbornik, 2023, Volume 214, Number 12, DOI: https://doi.org/10.4213/sm9889 
Bibliographic databases:
    § 1. IntroductionInvestigations of the behaviour of integral means of analytic functions in a disc is a classical line of research in complex analysis. Integrals of the moduli of derivatives of conformal (or just holomorphic) mappings reflect quite well the geometric and analytic properties of the maps alike. For example, one of the most important characteristics in the theory of conformal mappings is the so-called integral means spectrum. One can also mention here Brennan’s conjecture, which is still open and can be stated as follows (see [1]). Let $\Omega$ be a simply connected domain in the extended complex plane whose boundary contains more than one point, and let $\varphi$ be a conformal mapping of $\Omega$ onto the unit disc $\mathbb{D}$. Brennan conjectured that $\varphi' \in L_p(\Omega)$ for $4/3<p<4$, that is,
$$
\begin{equation*}
\int_\Omega |\varphi'(w)|^p\,dA(w)<\infty\quad\text{for } \frac 43<p<4.
\end{equation*}
\notag
$$
Here and throughout, $A$ denotes the planar Lebesgue measure normalized so that the unit disc has measure 1: $dA(z)={dx\,dy}/{\pi}$, $z=x+iy$. Problems related to integral means are not limited to conformal mappings. Problems similar to Brennan’s conjecture also arise for rational functions, which are, in particular, maps of finite multiplicity. We can point out the problem related to estimates for integrals of bounded rational functions, which was considered for the first time by Dolzhenko [2] for sufficiently smooth domains. Let $G$ be a bounded domain with finite area $S$, $R$ be a rational function of order $n$, and let $0<p \leqslant 2$. If $|R| \leqslant 1$ in $G$, then using Hölder’s inequality it is easy to show that
$$
\begin{equation}
\int_{G}|R'(w)|^p\,dA(w) \leqslant \biggl(\int_{G}|R'(w)|^2\,dA(w)\biggr)^{p/2} S^{1-p/2} \leqslant n^{p/2} S^{1-p/2},
\end{equation}
\tag{1.1}
$$
where $S=A(G)$ is the area of $G$. Indeed, $\displaystyle \int_{G} |R'(w)|^2\,dA(w) \leqslant n$, because $R$ covers each point in the unit disc with multiplicity at most $n$. This bound is sharp in order in a certain sense if we consider the extremal problem of the maximization of the integral $\displaystyle\int_{G}|R'(w)|^p\,dA(w)$ over all domains $G$ of fixed area $S$ and all rational functions of order $n$ (or all $n$-valent functions in $G$) such that $|R| \leqslant 1$ in $G$. For a discussion of this extremal problem, see § 4. Inequality (1.1) holds also in the case when in place of rational functions we consider $n$-valent ones. Recall that an analytic function $R$ in a domain $G$ is said to be $n$-valent if the equation $R(z)=w$ has at most $n$ solutions in $G$ for any right-hand side $w \in \mathbb{C}$. However, it turns out that in some cases estimate (1.1) can be improved. Dolzhenko ([2], Theorem 2.2) showed that in the case of a finitely connected domain $G$ with $C^2$-boundary the following inequalities hold: given a rational function $R$ of degree at most $n$ with poles outside $\overline{G}$, if $1<p<2$, then
$$
\begin{equation}
\int_{G} |R'(w)|^p\,dA(w) \leqslant C n^{p-1} \|R\|^p_{H^\infty(G)};
\end{equation}
\tag{1.2}
$$
while if we assume additionally that $G$ is bounded, then
$$
\begin{equation}
\int_{G}|R'(w)|\,dA(w) \leqslant C \log (n+1) \|R\|_{H^\infty(G)}.
\end{equation}
\tag{1.3}
$$
The constant $C$ depends here only on $G$ and $p$, $H^\infty(G)$ denotes the set of bounded analytic functions in $G$, and $\|f\|_{H^\infty(G)}=\sup_{w\in G} |f(w)|$. Of course, inequality (1.2) also holds trivially for $p=2$ with no restrictions on the domain whatsoever. However, for $p>2$ there can be no inequality of this type with constant depending only on the degree of the rational function, but independent on the position of its poles. Note that various inequalities for the derivatives of rational functions were considered by Dolzhenko [3], Peller [4], Semmes [5], Pekarskii [6], [7], Danchenko [8], [9] and many others (for instance, see [10]–[14]). Using methods of the theory of Hardy spaces the authors managed to extend significantly in [15] the classes of domains for which (1.2) holds. The following result was proved. Theorem A. Let $G$ be a bounded simply connected Hölder domain with rectifiable boundary, and let $1<p< 2$. Then there exists a positive constant $C_p$ depending on $G$ and $p$ such that the following inequality holds for each rational function $R$ of degree at most $n$: Even in the simplest example of the disc $\mathbb{D}$ and the function $R(z)=z^n$ we see that inequality (1.2) is sharp. However, it turns out that for $p=1$ inequality (1.3) can be improved significantly. In [15] and [16] the authors showed that the sharp order of growth of the integral $\displaystyle\int_{G}|R'(w)|\,dA(w)$ with $n$ is $\sqrt{\log(n+1)}$. Theorem B. Let $G$ be a bounded simply connected Hölder domain with rectifiable boundary. Then there exists a positive constant $C$ depending on $G$ such that for each rational function $R$ of degree at most $n$ As concerns inequality (1.5), it is sharp in order even in a disc, where we can take a polynomial or a Blaschke product of degree $n$ as $R$. The proof of sharpness in [15] was based on subtle results due to Makarov [17] and Bañuelos and Moore [18] on the boundary behaviour of functions in the Bloch space. In this paper we show that all above inequalities also hold for an arbitrary $n$-valent function in a domain with rectifiable boundary (no further regularity conditions are required). The following three theorems are the main results of our paper. Theorem 1. Let $p \in (1,2)$, let $G$ be a bounded simply connected domain with rectifiable boundary $\gamma$, and let $L=\ell(\gamma)$ be its length. Then there exists a positive constant $C_p $ depending only on $p$ such that for each bounded $n$-valent function $f$ in $G$ Theorem 1 was announced in [19] (with a sketch of the proof). As already mentioned above, the case $p=1$ is special. Let $\operatorname{diam}(G)$ denote the diameter of the domain $g$. Theorem 2. Let $G$ be a bounded simply connected domain with rectifiable boundary. Then there exists a positive absolute number constant $0$ such that for each bounded $n$-valent function $f$ in $G$ Clearly, similar estimates, with some constant $C=C(G,p)$, also hold for finitely connected domains with rectifiable boundaries, but the way in which $C$ depends on the length $L$ of the boundary can be different. In § 4 we also look at an example of an annulus showing that the bound in Theorem 1 can be improved in some cases, in particular, for ‘narrow’ domains. This example shows that the area of the domain can also be significant, although the metric properties of the boundary of the domain certainly play a central role. A natural question arising here is as follows: what if we abandon the condition of rectifiability? In this case the growth can be more rapid than $n^{p-1}$. However, then we can also improve estimate (1.1) for domains whose boundaries have Minkowski dimension strictly less than 2. Recall the definition of the Minkowski dimension of the boundary. For $\varepsilon>0$ let $N(\varepsilon)$ denote the least number of discs of radius $\varepsilon$ with centres on $\gamma$ that are required to cover $\gamma$, and we set
$$
\begin{equation*}
\alpha =\operatorname{Mdim}(\gamma)=\limsup_{\varepsilon\to 0} \frac{\log N(\varepsilon)}{\log ({1}/{\varepsilon})}.
\end{equation*}
\notag
$$
Note that for each $\delta>0$ we have $N(\varepsilon) \leqslant C \varepsilon^{-\alpha-\delta}$ for all $\varepsilon\in(0, 1/2)$, for some positive constant $C=C(G, \delta)$. The following result is proved in § 5. Theorem 3. Let $p \in (0,2)$, let $G$ be a bounded simply connected domain, and let $\gamma=\partial G$ and $\alpha =\operatorname{Mdim}(\gamma) \in [1, 2)$. 1. If $2-\alpha \leqslant p<2$, then for each $\delta>0$ there exists a positive constant $C= C(G, \alpha, p, \delta)$ such that for each bounded $n$-valent function $R$ in $G$
$$
\begin{equation}
\int_{G} |R'(w)|^p\,dA(w) \leqslant C n^{(p-2)/\alpha+1+\delta} \|R\|^p_{H^\infty(G)}.
\end{equation}
\tag{1.8}
$$
2. If $0<p<2-\alpha$, then for each bounded $n$-valent function $R$ in $G$ where $C=C(G, \alpha, p)$.In § 6 we obtain lower bounds for integrals of the moduli of the derivatives of bounded $n$-valent functions in terms of the integral means spectra of conformal mappings. These estimates show that in Theorem 3 the upper bound in (1.8) describes the actual situation quite accurately, especially in the case when the Minkowski dimension is close to 1. We also discuss the connections of the problems under consideration with Littlewood’s well-known conjecture on the growth of the spherical derivatives of polynomials and rational functions (see [20] and [21]). Another group of results is related to estimates for integrals of the form where $d_G(w)$ is the distance of $w$ to the boundary of $G$. Under certain conditions on the parameters $p$ and $\beta$ we obtain estimates for suitable Besov-type norms (see § 7).§ 2. Preliminary estimatesGiven an open set $G\subset \mathbb{C}$ with boundary $\gamma$ and given $\varepsilon>0$, set
$$
\begin{equation}
G_\varepsilon =\{w\in G\colon \operatorname{dist} (w, \gamma)>\varepsilon\}\quad\text{and} \quad H_\varepsilon =\{w\in G\colon \operatorname{dist} (w, \gamma) \leqslant \varepsilon\}.
\end{equation}
\tag{2.1}
$$
We require several well-known properties of the sets $G_\varepsilon$ and $H_\varepsilon$. For completeness we present short proofs of these properties. In what follows we write $X\lesssim Y$ when there exists a positive constant $C$ such that $X\leqslant C Y$ for all possible values of the parameters determining $X$ and $Y$. We also let $S(E)$ denote the area (normalized two-dimensional Lebesgue measure $A(E)$) of the set $E$, and let $B(w, \varepsilon)$ and $\overline{B}(w, \varepsilon)$ denote the open and closed discs with centre $w$ and radius $\varepsilon$. Lemma 1. Let $G$ be a simply connected domain with rectifiable boundary $\gamma=\partial G$ and let $\varepsilon>0$. Then where $\ell(\gamma)$ is the length of $\gamma$, and $C>0$ is an absolute scalar constant. Proof. Let $\operatorname{diam}(G)$ denote the diameter of $G$. If $\varepsilon \geqslant \operatorname{diam}(G)/10$, then we have the obvious estimates $S(H_\varepsilon) \leqslant S(G) \leqslant 4 \operatorname{diam}^2(G) \lesssim \varepsilon \ell(\gamma)$. So we assume it what follows that $\varepsilon<\operatorname{diam}(G)/10$.
Without loss of generality we can assume that ${H_\varepsilon =\{w\in G\colon \operatorname{dist} (w, \gamma)<\varepsilon\}}$; the case of nonstrict inequality follows by continuity. Then the discs $B(w, \varepsilon)$, $w\in\gamma$, cover the set $H_\varepsilon$. Using Vitali’s covering theorem we select a (clearly, finite) sequence of discs $\{B(w_k, \varepsilon)\}_{k=1}^m$ such that the $B(w_k, \varepsilon)$ are pairwise disjoint and $H_\varepsilon \subset \bigcup_{k=1}^m B(w_k, 5 \varepsilon)$. Then $S(H_\varepsilon) \leqslant 25\varepsilon^2 m$. As $\gamma$ is connected, we have $\gamma \cap \{|w-w_k|=\varepsilon\} \ne\varnothing$ (the case when the whole of $\gamma$ lies in one disc is ruled out because $\varepsilon<\operatorname{diam}(G)/10$), so that $\ell(\gamma \cap B(w_k, \varepsilon)) \geqslant \varepsilon$. Thus,
$$
\begin{equation*}
m \varepsilon \leqslant \sum_{k=1}^m \ell(\gamma \cap B(w_k, \varepsilon)) \leqslant \ell(\gamma).
\end{equation*}
\notag
$$
Lemma 1 is proved. It is well known that $\lim_{\varepsilon\to 0+} \varepsilon^{-1} S(H_\varepsilon)= \ell(\gamma)$: for instance, see [22]. Note that even when $G$ is simply connected, the open set $G_\varepsilon$ is not necessarily connected, but then each of its connected components is a simply connected domain. Set Then we have the following result.Lemma 2. Let $G$ be a simply connected domain with rectifiable boundary $\gamma$, and let $L=\ell(\gamma)$ and $\varepsilon>0$. Then the following hold. 1. For $p>1$ the estimate
$$
\begin{equation}
\int_{G_\varepsilon} d_G^{-p} (w)\,dA(w) \leqslant C_p \varepsilon^{1-p} L
\end{equation}
\tag{2.3}
$$
holds, where $C_p \leqslant K/(p-1)$ for an absolute constant $K$. 2. There exists an absolute positive constant $C$ such that for $\varepsilon < \operatorname{diam}(G)/2$
$$
\begin{equation}
\int_{G_\varepsilon} d_G^{-1} (w)\,dA(w) \leqslant C L \log\frac{\operatorname{diam}(G)}{\varepsilon},
\end{equation}
\tag{2.4}
$$
while for $\varepsilon \geqslant \operatorname{diam}(G)/2$ the trivial estimate
$$
\begin{equation*}
\int_{G_\varepsilon} d_G^{-1} (w)\,dA(w) \leqslant C \operatorname{diam}(G) \leqslant CL
\end{equation*}
\notag
$$
holds. 3. For $0<p<1$ there exists a constant $C_p$ such that Proof. We partition $G_\varepsilon$ into layers: $G_\varepsilon=\bigcup_{k\geqslant 0} F_k$, where
$$
\begin{equation*}
F_k=\{w\in G\colon 2^{k} \varepsilon \leqslant d_G(w)<2^{k+1} \varepsilon \}.
\end{equation*}
\notag
$$
Note that, as $G$ is bounded, the index $k$ takes a finite number of values, namely, $F_k=\varnothing$ for $2^{k} \varepsilon>\operatorname{diam}(G)$. Let $k_0$ be the largest integer such that $2^{k_0} \varepsilon \leqslant \operatorname{diam}(G)$. In addition, we have $F_k \subset H_{2^{k+1}\varepsilon}$, so that $S(F_k) \lesssim 2^k \varepsilon L$ by Lemma 1. Hence for ${p>1}$ we have
$$
\begin{equation*}
\begin{aligned} \, \int_{G_\varepsilon} d_G^{-p} (w)\,dA(w) &=\sum_{k=0}^{k_0} \int_{F_k} d_G^{-p} (w)\,dA(w) \leqslant \sum_{k=0}^{k_0} S(F_k) 2^{-kp}\varepsilon^{-p} \\ & \lesssim \varepsilon^{1-p} L \sum_{k=0}^{k_0} 2^{-k(p-1)} \leqslant \frac{K}{p-1}\varepsilon^{1-p} L, \end{aligned}
\end{equation*}
\notag
$$
where $K$ is an absolute constant.
Note that all but the last inequality in the previous chain also hold for $0<p\leqslant 1$. Thus, for $p=1$ and $\varepsilon<\operatorname{diam}(G)/2$ we have
$$
\begin{equation*}
\int_{G_\varepsilon} d_G^{-1} (w)\,dA(w) \lesssim L k_0 \lesssim L \log\frac{\operatorname{diam}(G)}{\varepsilon}.
\end{equation*}
\notag
$$
Now assume that $p<1$. Then
$$
\begin{equation*}
\begin{aligned} \, \int_{G_\varepsilon} d_G^{-p} (w)\,dA(w) & \leqslant C_p \varepsilon^{1-p} L 2^{k_0(1-p)} \\ & \leqslant C_p \varepsilon^{1-p} L \biggl(\frac{\operatorname{diam}(G)}{\varepsilon}\biggr)^{1-p}= C_p L (\operatorname{diam}(G))^{1-p}. \end{aligned}
\end{equation*}
\notag
$$
Since this inequality holds for all sufficiently small $\varepsilon>0 $, taking the limit we obtain (2.5). Lemma 2 is proved. Corollary 1. Let $G$ be a simply connected domain with rectifiable boundary $\gamma$, and let $L=\ell(\gamma)$, $\varepsilon>0$ and $p>1$. Then for each bounded $n$-valent function $R$ where $K>0$ is a scalar constant. Proof. The required inequality is a direct consequence of the bound and inequality (2.3) in Lemma 2.
Remark 1. For completeness we also present another (more analytic) proof of inequality (2.3). Let $\varphi$ be a conformal mapping of the disc $\{|z|<1\}$ onto $G$. Then $\varphi'$ belongs to the Hardy space $H^1$ in the disc and $\|\varphi'\|_{H^1}=\ell(\gamma)$. It is well known that $|\varphi'(z)|(1-|z|^2)/4 \leqslant d_G(\varphi(z)) \leqslant |\varphi'(z)|(1-|z|^2)$. Therefore, § 3. Proofs of Theorems 1 and 2 Proof of Theorem 1. We set $\varepsilon={L}/{n}$ and consider the partition $G=G_\varepsilon\cup H_\varepsilon$. Using estimate (2.6) we immediately obtain
$$
\begin{equation*}
\int_{G_\varepsilon} |R'(w)|^p\,dA(w) \leqslant \|R\|^p_{H^\infty(G)} \int_{G_\varepsilon} d_G^{-p} (w)\,dA(w) \leqslant \frac{K}{p-1} L^{2-p}n^{p-1}\|R\|^p_{H^\infty(G)}.
\end{equation*}
\notag
$$
On the other hand, from Hölder’s inequality with exponents $2/{p}$ and $2/(2-p)$ we obtain
$$
\begin{equation*}
\int_{H_\varepsilon} |R'(w)|^p\,dA(w) \leqslant \biggl(\int_{H_{\varepsilon}} |R'(w)|^2\,dA(w) \biggr)^{p/2} (S(H_{\varepsilon}))^{1-p/2}.
\end{equation*}
\notag
$$
Since $R$ is $n$-valent, we have $\displaystyle\int_G |R'(w)|^2\,dA(w) \leqslant n \|R\|^2_{H^\infty(G)}$. Using (2.2) for ${\varepsilon={L}/{n}}$ we obtain
$$
\begin{equation*}
\int_{H_\varepsilon} |R'(w)|^p\,dA(w) \leqslant C L^{2-p} n^{p-1} \|R\|^p_{H^\infty(G)},
\end{equation*}
\notag
$$
where $C $ is a scalar constant independent of $p$, $n$, $G$ and $R$. The proof of Theorem 1 is complete. The following lemma, which holds for an arbitrary bounded (not necessarily $n$-valent function) is the core of the proof of Theorem 2. Lemma 3. Let $G$ be a simply connected domain with rectifiable boundary. Then there exists a positive constant $C$ independent of $G$ such that for each function $R\in H^\infty(G)$ Proof. We pass to an equivalent problem in the disc. Let $\varphi$ be a conformal mapping of $\mathbb{D}$ onto $G$. Since $L=\|\varphi'\|_{H^1(\mathbb{D})}$ and $d_G(\varphi(z)) \leqslant (1-|z|^2) |\varphi'(z)|$, it is sufficient to show that Let $f=R\circ \varphi$; then $f\in H^\infty(\mathbb{D})$.
For $g\in H^s=H^s(\mathbb{D})$, $s>0$, we set
$$
\begin{equation*}
S(g)(e^{it})=\biggl(\int_0^{1}(1-r^2)|g'(re^{it})|^{2} r\,dr\biggr)^{1/2}.
\end{equation*}
\notag
$$
By the classical Littlewood-Paley theorem (see, for instance, [23], p. 332)
$$
\begin{equation*}
\|S(g) \|_{L^s(\mathbb{T})} \leqslant C(s) \|g\|_{H^s}.
\end{equation*}
\notag
$$
Set $g=f(\varphi')^{1/2}$ (for some analytic branch of the root function). Then $g\in H^2$ and $\|g\|_{H^2} \leqslant \|f\|_{H^\infty} \|\varphi'\|_{H^1}^{1/2}$. We have By the Littlewood-Paley theorem
$$
\begin{equation*}
\begin{aligned} \, &\int_0^{2\pi} \int_0^{1}(1-r^2)|g'(re^{it})|^{2} r\,dr\,dt =\int_0^{2\pi} |S(g) (e^{it})|^2\,dt\lesssim \|g\|_{H^2}^2 \leqslant \|f\|_{H^\infty}^2 \|\varphi'\|_{H^1}, \\ &\int_0^{2\pi} \int_0^{1} (1-r^2) \bigl|f(re^{it})((\varphi')^{1/2})' \bigr|^{2} r\,dr\,dt \\ &\qquad \leqslant\|f\|_{H^\infty}^2 \int_0^{2\pi} |S((\varphi')^{1/2}) (e^{it})|^2\,dt \lesssim \|f\|_{H^\infty}^2 \|\varphi'\|_{H^1}. \end{aligned}
\end{equation*}
\notag
$$
These inequalities prove the lemma. Proof of Theorem 2. As in the proof of Theorem 1, we look at the partition ${G=G_\varepsilon\cup H_\varepsilon}$ for $\varepsilon={L}/{n}$. Since $S(H_\varepsilon) \lesssim L\varepsilon=L^2/n$, it follows that
$$
\begin{equation*}
\begin{aligned} \, \int_{H_\varepsilon} |R'(w)|\,dA(w) & \leqslant \biggl(\int_{H_{\varepsilon}} |R'(w)|^2\,dA(w) \biggr)^{1/2} (S(H_{\varepsilon}))^{1/2} \\ &\lesssim n^{1/2}\|R\|_{H^\infty(G)} Ln^{-1/2}=L\|R\|_{H^\infty(G)}. \end{aligned}
\end{equation*}
\notag
$$
Note that nothing depends on $n$ in this bound. On the other hand
$$
\begin{equation*}
\begin{aligned} \, \int_{G_\varepsilon} |R'(w)|\,dA(w) & \leqslant \biggl(\int_{G} |R'(w)|^2\,d_G(w)\,dA(w) \biggr)^{1/2} \biggl(\int_{G_\varepsilon} d_G^{-1} (w)\,dA(w)\biggr)^{1/2} \\ & \leqslant L^{1/2} \|R\|_{H^\infty(G)} \biggl(\int_{G_\varepsilon} d_G^{-1} (w)\, dA(w)\biggr)^{1/2} \end{aligned}
\end{equation*}
\notag
$$
by Lemma 3. It remains to note that by inequality (2.4) in Lemma 2 we have
for $(\operatorname{diam}(G)/L) n>2$ and $\displaystyle\int_{G_\varepsilon} d_G^{-1} (w)\,dA(w) \leqslant CL$ for $(\operatorname{diam}(G)/L) n \leqslant 2$, where $C$ is an absolute constant. From this inequality we conclude that
$$
\begin{equation*}
\int_{G} |R'(w)|\,dA(w) \leqslant C L \sqrt{\log\biggl(2+\frac{\operatorname{diam}(G)}{L} n\biggr)}\|R\|_{H^\infty(G)}.
\end{equation*}
\notag
$$
Theorem 2 is proved. Using the same estimates we can prove the following more general result. Corollary 2. Let $G$ be a simply connected domain with rectifiable boundary $\gamma$, and let $L=\ell(\gamma)$. Then the following inequalities hold for each bounded analytic function $R$ in $G$ which has a finite Dirichlet integral $\displaystyle\mathcal{D}=\int_G |R'|^2\,dA$: and Proof. Set $\varepsilon=L \|R\|_\infty^2 \mathcal{D}^{-1}$. Repeating the argument in the proof of Theorem 1 we obtain
$$
\begin{equation*}
\begin{aligned} \, \int_{G} |R'(w)|^p\,dA(w) & \leqslant C(p)\biggl((L\, \varepsilon)^{1-p/2}\biggl(\int_G |R'|^2\,dA \biggr)^{p/2}+L \, \varepsilon^{1-p}\|R\|^p_\infty\biggr) \\ & \leqslant C(p)L^{2-p}\biggl(\int_G |R'|^2\,dA\biggr)^{p-1}\|R\|^{2-p}_\infty. \end{aligned}
\end{equation*}
\notag
$$
The result for $p=1$ can be derived similarly to Theorem 2. § 4. Estimates for integrals in an annulusNow we examine an important special case when $G$ is an annulus. For $r>1$ and $0<l<r$ consider the annulus $K_{l,r}=\{z\colon r-l<|z|<r\}$ of radius $r$ and width $l$. The problem under consideration is interesting from the standpoint of the effect that the area of the ring has on estimates for our integrals. It turns out that the area affects the result, but this influence fades with the increase of the radius of the annulus. The same can be said about the influence of the length of the boundary. Proposition 1. Let $p \in (1,2)$. Then there exists a positive constant $C=C(p)$ such that for any $l$ and $r$ satisfying the conditions $r\geqslant 1$ and $0<l \leqslant r/2$, and for each rational function $R$ of degree $n$ such that $\|R\|_{H^\infty(K_{l,r})} \leqslant 1$ the following inequality holds: Proof. We show that $\displaystyle\int_{K_{l,r}} |R'(w)|^p\,dA(w)$ does not exceed any of the expressions on the right-hand side of (4.1) for all admissible values of $l$ and $r$. The bound $\displaystyle\int_{K_{l,r}} |R'(w)|^p\,dA(w) \leqslant C(p) r^{2-p} n^{p-1}$ is a special case of Theorem 1.
By Dolzhenko’s well-known inequality (see [3])
$$
\begin{equation}
\int_0^{2\pi} |Q'(e^{it })|\,dt \leqslant 2\pi n\|Q\|_{L^\infty(\mathbb{T})}
\end{equation}
\tag{4.2}
$$
for each rational function $Q$ of degree $n$ with poles outside the unit circle $\mathbb{T}$. Inequality (4.2) (re-proved subsequently by many authors) is a quite special case of the results in [3], which hold for algebraic functions and arbitrary measurable subsets of the circle (for a short proof of Dolzhenko’s inequalities and the history of the problem the reader can consult [24]).
It follows from (4.2) that
$$
\begin{equation*}
\int_0^{2\pi} \rho |R'(\rho e^{it })|\,dt \leqslant 2\pi n\|R\|_{H^\infty(K_{l,r})} \lesssim n, \qquad r-l\leqslant \rho\leqslant r.
\end{equation*}
\notag
$$
Since $|R'(w)| \leqslant d_{K_{l,r}} (w)$, we have
$$
\begin{equation*}
\begin{aligned} \, \int_{K_{l,r}} |R'(w)|^p\,dA(w) & \lesssim \int_{r-l}^r \int_0^{2\pi} \frac{\rho |R'(\rho e^{it })|}{d^{p-1}_{K_{l,r}} (\rho e^{it })}\,dt\,d\rho \\ & \lesssim n \int_{r-l}^r \frac{d\rho}{\min^{p-1}(r-\rho, \rho -r+l)} \lesssim nl^{2-p}, \end{aligned}
\end{equation*}
\notag
$$
where the constants in inequalities depend only on $p$. The proof is complete. It is easy to see that the fact that (4.1) is sharp in order follows from the standard example of $R=z^n/r^n$ and some simple calculations. A similar inequality for $p=1$ was established in [16]. In Proposition 1 the function $R$ is assumed to be rational. Dolzhenko’s inequality (4.2) fails to hold for arbitrary bounded $n$-valent functions. Combining Theorem 1 with the trivial bound (1.1), for a bounded $n$-valent function $R$ such that $\|R\|_{H^\infty(K_{l,r})} \leqslant 1$ we obtain the inequalities
$$
\begin{equation*}
\int_{K_{l,r}} |R'(w)|^p\,dA(w) \leqslant \begin{cases} C r^{2-p} n^{p-1}, & r\leqslant nl , \\ C r^{1-p/2} l^{1-p/2} n^{p/2}, & r\geqslant nl. \end{cases}
\end{equation*}
\notag
$$
Again, here the second inequality is sharper for $nl\ll r$. Consider the problem of the maximization of the integral $\displaystyle\int_{G}|R'(w)|^p\, dA(w)$, where $1 \leqslant p < 2$, over the domains $G$ with fixed area or with fixed length of the boundary:
$$
\begin{equation*}
\mathcal{S} :=\sup \biggl\{\int_{G}|R'(w)|^p\,dA(w)\colon S(G)=S, \, R \text{ is }n\text{-valent in }G, \, \|R\|_{H^\infty(G)} \leqslant 1\biggr\}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\mathcal{L} :=\sup \biggl\{\int_{G}|R'(w)|^p\,dA(w)\colon\ell(\partial G)=L, \,R \text{ is }n\text{-valent in }G,\,\|R\|_{H^\infty(G)} \leqslant 1\biggr\}.
\end{equation*}
\notag
$$
In both cases we consider the suprema only over the class of simply connected bounded domains $G$. We can also consider the supremum over all rational functions of order at most $n$. We will see below that then the order of growth remains the same. We know that $\mathcal{S} \leqslant S^{1-p/2}n^{p/2}$ (inequality (1.1)) and $\mathcal{L} \leqslant C(p)L^{2-p}n^{p-1}$ for ${1<p<2}$ (Theorem 1). The example of the disc $B(0,r)$ and the function $R(z)=z^n/r^n$ shows that $\mathcal{L} \geqslant C(p) L^{2-p}n^{p-1}$ for some positive constant $C(p)$. The sharpness of the estimate for $\mathcal{S}$ can be seen from the example of an annulus considered above. Let $G=K_{l,r}$, where $l=r/n$, and let $R(z)=z^n/r^n$. It is easy to see that then $C_1 r^2/n \leqslant S \leqslant C_2 r^2/n$, while
$$
\begin{equation*}
\int_{G} |R'(w)|^p\,dA(w) \geqslant C_3 r^{2-p} n^{p-1} \geqslant C_4 S^{1-p/2} n^{p/2},
\end{equation*}
\notag
$$
where $C_1,C_2, C_3, C_4>0$ are scalar constants independent of $r$ and $n$.
§ 5. The case of fractal boundary Proof of Theorem 3. Throughout this section we write $X\lesssim Y$ if $X\leqslant C Y$ for some positive constant $C$, which can depend on $G$, $\alpha$, $p$ and $\delta$, but is independent of $R$ and $n$. Without loss of generality we assume below that $\|R\|_\infty \leqslant 1$. Then $|R'(w)| \leqslant d_G^{-1}(w)$ and $\displaystyle\int_{G} |R'(w)|^2\,dA(w) \leqslant n$.
Recall that we let $N(\varepsilon)$ denote the minimum number of discs of radius $\varepsilon$ with centres on $\gamma$ that are required to cover $\gamma$. We fix $\delta>0$ and choose ${\varepsilon_0=\varepsilon(G, \alpha, \delta)\!>\!0}$ so that $N(\varepsilon) \leqslant \varepsilon^{-\alpha-\delta}$ for all $\varepsilon\in(0, \varepsilon_0)$. Setting $\varepsilon=n^{-1/\alpha}$, we let $H_\varepsilon$ and $G_\varepsilon$ be the sets defined by (2.1), as before. Let $\{B_m\}_{m=1}^{N(\varepsilon)}$, $B_m=B(x_m, \varepsilon)$, be the family of discs covering $\gamma$. It is easy to see that $H_\varepsilon \subset \bigcup_{m=1}^{N(\varepsilon)} B(x_m, 2\varepsilon)$. In fact, if $w\in H_\varepsilon$, then there exist $\zeta\in\gamma$ and $m$ such that $|\zeta-w|\leqslant \varepsilon$ and $\zeta\in B_m$. Then for $\varepsilon<\varepsilon_0$ we obtain
$$
\begin{equation*}
S(H_\varepsilon) \leqslant 4N(\varepsilon) \varepsilon^2 \lesssim n^{1-{2}/{\alpha}+{\delta}/{\alpha}}.
\end{equation*}
\notag
$$
Therefore,
$$
\begin{equation*}
\begin{aligned} \, \int_{H_\varepsilon} |R'(w)|^p\,dA(w) & \leqslant \biggl(\int_{H_\varepsilon} |R'(w)|^2\, dA(w)\biggr)^{p/2} (S(H_\varepsilon))^{1-p/2} \\ & \lesssim n^{p/2} n^{(1-{2}/{\alpha}+{\delta}/{\alpha})(1-{p}/{2})} =n^{(p-2)/{\alpha}+1+{\delta}/{\alpha}(1-{p}/{2})}. \end{aligned}
\end{equation*}
\notag
$$
It is obvious that if $\varepsilon\geqslant \varepsilon_0$ (so that $n\leqslant \varepsilon_0^{-\alpha}$), then $\displaystyle\int_{H_\varepsilon} |R'(w)|^p\,dA(w)\leqslant C(G,\alpha, \delta)$.
The integral over $G_{\varepsilon_0}$ has the obvious estimate
$$
\begin{equation*}
\int_{G_{\varepsilon_0}} |R'(w)|^p\,dA(w) \leqslant \int_{G_{\varepsilon_0}} d_G^{-p}(w)\,dA(w) \leqslant \varepsilon_0^{-p} S(G).
\end{equation*}
\notag
$$
Thus, it remains to estimate the integral over $G_\varepsilon \setminus G_{\varepsilon_0}$.
We select the largest $k_0$ such that $2^{k_0-1}\varepsilon<\varepsilon_0$. Then
$$
\begin{equation*}
G_\varepsilon \setminus G_{\varepsilon_0} \subset \bigcup_{k=1}^{k_0} F_k, \qquad F_k=\{w\in G\colon 2^{k-1}\varepsilon<d_G(w) \leqslant 2^k \varepsilon\}.
\end{equation*}
\notag
$$
We have
$$
\begin{equation*}
\int_{G_\varepsilon \setminus G_{\varepsilon_0}} |R'(w)|^p\,dA(w) \leqslant \sum_{k=1}^{k_0} \int_{F_k} d_G^{-p}(w)\,dA(w) \lesssim \sum_{k=1}^{k_0} 2^{-kp} n^{{p}/{\alpha}} S(F_k).
\end{equation*}
\notag
$$
For an estimate of $S(F_k)$ we cover $\gamma$ by the discs $B(x_m, 2^{k-1}\varepsilon)$, $m\!=\!1,\dots, N(2^{k-1}\varepsilon)$. Since $2^{k-1}\varepsilon<\varepsilon_0$, $k\leqslant k_0$, we have the inequality $N(2^{k-1}\varepsilon) \lesssim (2^{k}\varepsilon)^{-\alpha-\delta}$. Clearly, $F_k \subset \bigcup_{m=1}^{N(2^{k-1}\varepsilon)} B (x_m, 2^{k}\varepsilon)$, so that
$$
\begin{equation*}
S(F_k) \lesssim N(2^{k-1}\varepsilon) 2^{2k} \varepsilon^2 \lesssim (2^k \varepsilon)^{2-\alpha-\delta}=2^{-k(\alpha-2+\delta)} n^{1- 2/\alpha+\delta/{\alpha}}.
\end{equation*}
\notag
$$
Therefore,
$$
\begin{equation*}
\int_{G_\varepsilon \setminus G_{\varepsilon_0}} |R'(w)|^p\,dA(w) \lesssim n^{1+ (p-2)/\alpha+\delta/{\alpha}} \sum_{k=1}^{k_0} 2^{-k(p+\alpha-2+\delta)}.
\end{equation*}
\notag
$$
If $p\geqslant 2-\alpha$, then the last sum is bounded by a constant that only depends on $\alpha$, $p$ and $\delta$. This completes the proof for $p\geqslant 2-\alpha$.
Now let $0<p<2-\alpha$. We choose $\delta>0$ so that $p+\alpha-2+\delta<0$. Then, taking the inequality $2^{k_0-1}<\varepsilon_0 \varepsilon^{-1}=\varepsilon_0 n^{{1}/{\alpha}}$ into account we obtain
$$
\begin{equation*}
\begin{aligned} \, \int_{G_\varepsilon \setminus G_{\varepsilon_0}} |R'(w)|^p\,dA(w) &\lesssim n^{1+(p-2)/{\alpha}+{\delta}/{\alpha}} 2^{k_0(2-p-\alpha -\delta)} \\ &\lesssim n^{1+(p-2)/{\alpha}+{\delta}/{\alpha}} n^{(2-p-\alpha -\delta)/\alpha}=1. \end{aligned}
\end{equation*}
\notag
$$
Thus, for $0\ < p < 2 - \alpha$ we have $\displaystyle\int_{G} |R'(w)|^p\,dA(w) \leqslant C(G, \alpha, p, \delta)$, where the constant is independent of $R$ and $n$. Theorem 3 is proved. § 6. Lower bounds and the integral means spectrumIn this section we obtain lower estimates for integrals of derivatives of $n$-valent functions using the integral means spectrum. A review of the theory of integral means can be found in Makarov [25] (also see the introduction to [26]). Given a univalent function $\varphi$ in the disc and $s\in \mathbb{R}$, set
$$
\begin{equation*}
\beta_{\varphi}(s):=\limsup_{r \to 1} \frac{1}{|\log(1-r)|}\int_{0}^{2\pi} |\varphi'(re^{it})|^s\,dt.
\end{equation*}
\notag
$$
The universal integral means spectrum is defined by $B(s) = \sup \beta_\varphi(s)$, where the supremum is taken over all univalent function $\varphi$ in the disc. In a similar way the spectrum for bounded functions $B_b(s)$ is defined to be the supremum of the quantities $\beta_\varphi(s)$ over all bounded univalent functions $\varphi$ in the disc. Makarov [25] showed that $B(s)=\max (B_b(s), 3s-1)$. Brennan’s conjecture is equivalent to the relation $B(-2)=1$. There also is a conjecture of Kraetzer that $B_b(s)={s^2}/{4}$, $|s| \leqslant 2$. The best upper estimates so far for the integral means spectrum (including in the context of Brennan’s conjecture) are due to Hedenmalm and Shimorin [26]. We can also consider the universal integral means spectrum under a constraint on the dimension of the boundary of the domain (see [27] and [25]). For $1\leqslant \alpha\leqslant 2$ set
$$
\begin{equation*}
B_\alpha(s):=\sup \{ \beta_\varphi(s)\colon \varphi \text{ is bounded and }\operatorname{Mdim}(\partial \varphi(\mathbb{D}))=\alpha\}.
\end{equation*}
\notag
$$
The definition of the integral means spectrum implies directly that
$$
\begin{equation*}
\int_{0}^{2\pi} |\varphi'(re^{it})|^s\,dt \geqslant C \biggl(\frac{1}{1-r}\biggr)^{\beta_\varphi(s)-\varepsilon}
\end{equation*}
\notag
$$
for some infinite sequence of real numbers $r$ converging to 1 and some positive constant $C $ that only depends on $\varphi$ and $\varepsilon$. It is easy to show that there exist an infinite number of positive integers $n$ such that
$$
\begin{equation}
\int_{0}^{2\pi} \biggl|\varphi'\biggl(\biggl(1-\frac 1n\biggr)e^{it}\biggr)\biggr|^s\,dt \geqslant C n^{\beta_\varphi(s)-\varepsilon}.
\end{equation}
\tag{6.1}
$$
Using (6.1) we can construct examples of $n$-valent functions that ensure good lower bounds for Theorem 3. Set $R(w)=(\varphi^{-1}(w))^n$, where $\varphi$ is a conformal mapping of the disc $\mathbb{D}$ onto the simply connected domain $G$. It is clear that $R$ is at worst $n$-valent in $G = \varphi(\mathbb{D})$. For this function we have
$$
\begin{equation*}
\begin{aligned} \, \int_{G} |R'(w)|^p\,dA(w) &=\int_{0}^{2\pi} \int_{0}^1n^pr^{p(n-1)} |\varphi'(z)|^{2-p}r\,dr\,dt \\ &\geqslant \int_{0}^{2\pi}\int_{1-1/n}^{1-1/(2n)} n^pr^{p(n-1)} |\varphi'(z)|^{2-p}r\,dr\,dt \,{\geqslant}\, Cn^{\beta_\varphi(2-p)+p-1-\varepsilon}. \end{aligned}
\end{equation*}
\notag
$$
Thus, for an arbitrary domain $G=\varphi(\mathbb{D})$ and $n$ in some infinite sequence of positive integers there exists an $n$-valent function $R$ such that $\|R\|_{H^\infty(G)} \leqslant 1$ and
$$
\begin{equation}
\int_{G} |R'(w)|^p\,dA(w) \geqslant Cn^{\beta_\varphi(2-p)+p-1-\varepsilon},
\end{equation}
\tag{6.2}
$$
where $c>0$ depends on $\varphi$ and $\varepsilon$, but is independent of $n$ and $R$. By taking an appropriate domain $G=\varphi(\mathbb{D})$ for which the value calculated for $\varphi$ is close to the maximum one we obtain the following result. Proposition 2. For any $p\in [1, 2)$ and $\varepsilon>0$ there exists a bounded domain $G$ such that for $n$ in some infinite sequence of positive integers there exists an $n$-valent function $R$ such that $\|R\|_{H^\infty(G)} \leqslant 1$ and where $c>0$ depends on $G$ and $\varepsilon$, but is independent of $n$ and $R$. Moreover, for each $\alpha\in [1,2)$ there is a domain $G$ satisfying $\operatorname{Mdim}(\partial G)=\alpha$ and such that for $n$ in some infinite sequence of positive integers there exists an $n$-valent function $R$ such that $\|R\|_{H^\infty(G)} \leqslant 1$ and Since $B_b(s)>0$ and $B_\alpha(s)>0$ for $s>0$ and $\alpha>1$, we see that in the general case there is no estimate with growth order $n^{p-1}$. Now consider the question of the gap between the lower bound (6.2) and the upper bound from Theorem 3. It turns out that for $p=1$ these bounds come close when the dimension $\alpha$ is close to 1. Makarov and Pommerenke [27] (also see [25], § 5) showed that
$$
\begin{equation*}
B_\alpha(1)=\alpha-1 -(\alpha-1)^2+O((\alpha-1)^3), \qquad \alpha \to 1.
\end{equation*}
\notag
$$
Hence it immediately follows from (6.4) that there exists a simply connected domain $G$ and an $n$-valent function $R$ bounded by 1 such that
$$
\begin{equation*}
\int_{G} |R'(w)|\,dA(w) \geqslant Cn^{\alpha-1 -(\alpha-1)^2+O((\alpha-1)^3)-\varepsilon}.
\end{equation*}
\notag
$$
Set $\varepsilon=\delta=(\alpha-1)^3$. Then it is easy to see that our lower bound matches the upper bound in Theorem 3 to within $(\alpha-1)^3$. The problem of estimates for the derivatives of bounded rational functions is closely connected with a well-known problem going back to Littlewood. Given an analytic function $f$ in the disc and $p>0$, let $S_p(f)$ denote the integral of the spherical derivative of $f$:
$$
\begin{equation*}
S_p(f)=\int_{\mathbb{D}} \biggl(\frac{|f'(z)|}{1+|f(z)|^2}\biggr)^p\,dA(z).
\end{equation*}
\notag
$$
Arguing as in the proof of inequality (1.1) (and considering the sets $\{|f| \leqslant 1\}$ and $\{|f|>1\}$ separately), it is easy to see that $S_p(f) \leqslant Cn^{p/2}$ for all rational functions $f$ of degree at most $n$ and for $0<p\leqslant 2$. Littlewood [28] posed the question of the growth with $n$ of the quantities
$$
\begin{equation*}
\varphi(n)=\sup \{S_1(f)\colon f \text{ is a polynomial of degree at most }n\}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\psi(n)=\sup \{S_1(f)\colon f \text{ is a rational function of degree at most }n\}
\end{equation*}
\notag
$$
and conjectured that $\varphi(n) \leqslant C n^{1/2-\delta}$ and even $\psi(n) \leqslant C n^{1/2-\delta}$ for some $\delta>0$. The latter conjecture fails in fact: Chen and Liu [20] showed that for any $n$ and $\delta>0$ there are rational functions of degree $n$ such that $S_1(f)>c n^{1/2-\delta}$, so that $S_p(f)>c n^{p/2-\delta}$ for $1\leqslant p\leqslant 2$ (here the positive constant $c$ depends only on $\delta$). Littlewood’s conjecture on $\varphi(n)$ turns out to be true: after a number of intermediate results, Beliaev and Smirnov [21] showed that the sharp growth exponent of $\varphi(n)$ is $B_b(1)$. Moreover, for $0<p<2$, $\varepsilon>0$ and any polynomial $f$ of degree at most $n$ the inequality is true. Note that the exponent in this inequality is the same as in the lower bound for means of $n$-valent functions in Proposition 2. Results due to Chen and Liu support the conjecture that the exponent $p/2$ in inequality (1.1) can be sharp for the class of all rational function in the case when the boundary of the domain $G$ has no regularity. However, formally, from the inequality $S_p(f)>n^{p/2-\delta}$ it only follows that for each $n$ and $\delta>0$ there exist a bounded (not necessarily simply connected) domain $G$ depending on $n$, and a rational function $f$ of degree at most $n$ and bounded in $G$ such that $\displaystyle\int_G |f'|^p\,dA>c n^{p/2-\delta}$. It can be interesting to compare the above results with growth estimates for the lengths of the images of circles under the action of univalent polynomials and rational functions. Let $\mathcal{S}$ denote the class of univalent functions $f$ in $\mathbb{D}$ such that $f(0)=0$ and $f'(0)=1$. Set
$$
\begin{equation*}
\Phi_p=\sup \biggl\{\frac{1}{\log n} \log \int_0^{2\pi} |f'(e^{it})|^p\,dt\colon f\in \mathcal{S} \text{ is a polynomial},\,\operatorname{deg}f\leqslant n\biggr\},
\end{equation*}
\notag
$$
and let $\Psi_p$ denote the corresponding supremum over all univalent rational functions $f$ of degree at most $n$ such that $|f|\leqslant 1$ in $\mathbb{D}$. It was shown in [29] that $\Phi_p=B(p)$. On the other hand $\Psi_1=1/2$: the upper bound for $\Psi_1$ was established in [24] and the lower bound was proved in [30]. Thus, similarly to Littlewood’s conjecture, in the polynomial case the order of growth is determined by the integral means spectrum, while for rational functions it is not.
§ 7. Weighted inequalitiesConsider a more general inequality, in which the growth of the derivative near the boundary is compensated for by multiplying by the distance to the boundary raised to some power. Inequalities of this type are well known for rational functions in the disc. For instance, a well-known inequality due to Peller (see [4]) states that for a rational function $R$ of degree $n$ with poles outside $\overline{\mathbb{D}}$
$$
\begin{equation*}
\|R\|_{B_p^{1/p}} \leqslant C n^{1/p} \|R\|_{\mathrm{BMOA}},
\end{equation*}
\notag
$$
where $B_p^{1/p}$ is the Besov space, $p>0$, $C=C(p)$, and $\mathrm{BMOA}$ is the space of analytic functions with bounded mean oscillation. In particular, for $1<p<\infty$
$$
\begin{equation*}
\int_\mathbb{D} |R'(z)|^p (1-|z|)^{p-2}\,dA(z) \leqslant C n \|R\|^p_{H^\infty}.
\end{equation*}
\notag
$$
The reader can find various proofs and generalizations of this inequality in [5]–[7], [9]–[11], [31] and [13]. Let $p\geqslant 1$ and $\beta \in \mathbb{R}$. We are interested in estimates of the form
$$
\begin{equation}
I_{p,\beta} (R) :=\int_G |R'(w)|^p\,d_G^\beta(w)\,dA(w) \leqslant C \Psi(n) \|R\|^p_{H^\infty(G)},
\end{equation}
\tag{7.1}
$$
where the constant $C$ can depend on $G, p$ and $ \beta$, but is independent of $n$ and $R$, while the dependence of $n$ is included in $\Psi(n)$. The parameters satisfy the natural restriction $p-2\leqslant \beta \leqslant p-1$. If $\beta<p-2$, then even in the case of rational functions in the disc we do not have inequality (7.1) with a constant $C$ independent of $R$, because then $I_{p,\beta} (R)$ depends on the distance of the poles to the boundary. If $\beta>p-1$ and the boundary of $G$ is rectifiable, then by Lemma 2 (assertion 3)
$$
\begin{equation*}
I_{p,\beta} (R) \leqslant \|R\|_\infty^p \int_G d_G^{\beta -p}(w)\,dA(w) \leqslant C(L, p, \beta) \|R\|_\infty^p.
\end{equation*}
\notag
$$
For rational functions in domains with sufficiently regular boundaries inequalities of the type of (7.1) were established in Theorem 5 in [15]. The following corollaries generalize the results in [15] to $n$-valent functions and arbitrary domains with rectifiable boundaries. For $\beta=p-1$ we obtain the following result straight away. Corollary 3. Let $G$ be a bounded simply connected domain with rectifiable boundary, and let $L=\ell (\partial G)$. Then there exists a positive scalar constant $C$ such that for each bounded $n$-valent function $R$ in $G$ The case $p\geqslant 2$ is a direct consequence of Lemma 3, while the case $1<p<2$ follows from Theorem 2 and Hölder’s inequality. Corollary 4. Let $G$ be a bounded simply connected domain with rectifiable boundary, and let $L=\ell(\partial G)$. If $p>1$, $p-2\leqslant \beta<p-1$ and $\beta\geqslant 0$ (this holds automatically for $p\geqslant 2$), then there exists a constant $C=C(p, \beta)$ such that for each bounded $n$-valent function $R$ in $G$ This is a consequence of Theorem 1 because $I_{p,\beta} (R) \lesssim \|R\|^\beta_{\infty} I_{p-\beta, 0} (R)$ and $1<p-\beta\leqslant 2$. The case when $1\leqslant p<2$ and $\beta<0$ is more complicated: here our result covers only some values of $\beta$, and the question of an inequality of the type of (7.1) for $\beta$ in some subinterval of $(p-2, p-1)$ remains open. Theorem 4. Let $G$ be a bounded simply connected domain with rectifiable boundary, and let $L=\ell(\partial G)$. If $1\leqslant p<2$ and then there exists a constant $C=C(p, \beta)$ such that for each bounded $n$-valent function $R$ in $G$ Proof. As before, we set $\varepsilon={L}/{n}$ and $G=G_\varepsilon \cup H_\varepsilon$, where $H_\varepsilon$ and $G_\varepsilon$ are defined by (2.1). Then by Lemma 2 (assertion 1)
$$
\begin{equation*}
\begin{aligned} \, \int_{G_\varepsilon} |f'(w)|^p\,d_G^\beta(w)\,dA(w) & \leqslant \|R\|_{\infty}^p \int_{G_\varepsilon} \frac{dA(w)}{d_G^{p-\beta}(w)} \\ & \leqslant C L\, \varepsilon^{1-p+\beta} \|R\|_{\infty}^p=C L^{2-p+\beta} n^{p-\beta-1}\|R\|_{\infty}^p, \end{aligned}
\end{equation*}
\notag
$$
where $C$ depends only on $p$ and $\beta$. By Hölder’s inequality
$$
\begin{equation*}
\begin{aligned} \, &\int_{H_\varepsilon} |f'(w)|^p\,d_G^\beta(w)\,dA(w) \\ &\qquad \leqslant \biggl(\int_{H_\varepsilon} |R'(w)|^2\,dA(w)\biggr)^{p/2} \biggl(\int_{H_\varepsilon} d_G^{2\beta/(2-p)} (w)\,dA(w) \biggr)^{1-p/2}. \end{aligned}
\end{equation*}
\notag
$$
It follows from the condition $\beta>{p}/{2} -1$ that $s={2\beta}/(2-p)>-1$. Subdividing $H_\varepsilon$ into layers: $H_\varepsilon=\bigcup_{k=0}^\infty \{w\in G\colon 2^{-k-1}\varepsilon<d_G(w) \leqslant 2^{-k}\varepsilon\}$, and using Lemma 1 we obtain
$$
\begin{equation*}
\int_{H_\varepsilon} d_G^s (w)\,dA(w) \leqslant C_s \varepsilon^{s} \sum_{k=0}^\infty 2^{-ks} S(F_k) \leqslant C_s \varepsilon^{s+1} L \sum_{k=0}^\infty 2^{-k(1+s)} \leqslant C_s \varepsilon^{s+1} L,
\end{equation*}
\notag
$$
because $s>-1$. Here $C_s$ can denote different constants depending on $s$ alone. Thus,
$$
\begin{equation*}
\begin{aligned} \, \int_{H_\varepsilon} |f'(w)|^p\,d_G^\beta(w)\,dA(w) & \leqslant C n^{{p}/{2}} n^{(-1- {2\beta}/(2-p))(2-p)/{2}} L^{(2+ 2\beta/(2-p))(2-p)/{2}}\|R\|_{\infty}^p \\ &=C L^{2-p+\beta} n^{p-\beta-1}\|R\|_{\infty}^p. \end{aligned}
\end{equation*}
\notag
$$
Theorem 4 is proved. Now we show that, by contrast with rational functions (Peller’s inequality) for $n$-valent functions an estimate of the form (7.1) cannot hold for all $\beta\in [p-2, p-1]$; moreover, the integral on the left-hand side can be divergent. Recall that we let $B_b(p)$ denote the universal integral means spectrum for bounded univalent functions. It is known that Proposition 3. For all $p\in [1,2)$ and $\beta<B_b(p)-1$ there exists a bounded univalent function $\varphi$ in the disc $\mathbb{D}$ such that Proof. Let $\delta>0$ and $\beta+\delta<B_b(p)-1$. Then by the definition of the integral means spectrum there exist a bounded univalent function $\varphi$ and a sequence $\varepsilon_n\to 0$ such that Then The proof is complete. Thus, for $1\leqslant p<2$ an inequality of the type (7.1) holds for $\beta>{p}/{2} -1$ and, generally speaking, fails for $\beta<B_b(p)-1$. The interval $\beta\in [B_b(p)-1,\, {p}/{2} -1]$ escapes investigation. Note that $B_b(p) \to 1$ as $p\to 2$, so that the upper and lower bounds agree for $p\to2$. Recall that Kraetzer’s conjecture states that $B_b(p)={p^2}/{4}$ for $|p| \leqslant 2$. So far, the best estimates for the integral means spectrum are those in [26], but even they provide an upper bound of 0.46 for $B_b(1)$, which is quite far from 0.25 predicted by Kraetzer’s conjecture (on the other hand the rather close lower estimate of $B_b(1)$ by 0.23 is known; see [32]). § 8. ConclusionFor domains with nonrectifiable boundaries the interesting questions of whether or not the upper bounds in Theorem 3 and the lower bounds in Proposition 2 are sharp are still open. Another natural problem is the description of a possibly wider class of domains such that an estimate with growth order $n^{p-1}$ holds. By Theorem 1 this holds for domains $G$ with rectifiable boundaries, that is, when the conformal mapping $\varphi$ of the disc $\mathbb{D}$ onto the domain $G$ satisfies $\varphi' \in H^1$. On the other hand there is an obvious necessary condition. Let $p \in (1,2)$, and let $G= \varphi(\mathbb{D})$ be a bounded simply connected domain. If there exists a positive constant $C =C(G,p)$ such that for each bounded $n$-valent function $R$ in $G$ we have
$$
\begin{equation}
\int_{G} |R'(w)|^p\,dA(w) \leqslant C n^{p-1} \|R\|^p_{H^\infty(G)},
\end{equation}
\tag{8.1}
$$
then $\varphi' \in H^{2-p}$. For a proof we look at the $n$-valent function $R(w)=(\varphi^{-1}(w))^n$. In this case
$$
\begin{equation*}
\begin{aligned} \, &\int_{G} |R'(w)|^p\,dA(w) =\int_{\mathbb{D}} |(R\circ \varphi)'(z)|^p |\varphi'(z)|^{2-p}\,dA(z) \\ &\quad =n^p \int_{\mathbb{D}} |z|^{p(n-1)} |\varphi'(z)|^{2-p}\,dA(z) \geqslant n^p \int_{1-1/n<|z|<1} |z|^{p(n-1)} |\varphi'(z)|^{2-p}\,dA(z) \\ &\quad \gtrsim n^{p-1} \int_0^{2\pi} \biggl|\varphi'\biggl(\biggl(1-\frac 1n\biggr)e^{it}\biggr)\biggr|^{2-p}\, dt. \end{aligned}
\end{equation*}
\notag
$$
Since $R$ satisfies (8.1), it follows directly that $\varphi' \in H^{2-p}$. We can conjecture that $\varphi' \in H^{2-p}$ is also a sufficient condition for (8.1) to hold. 
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