| Russian Mathematical Surveys |
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This article is cited in 1 scientific paper (total in 1 paper) Brief Communications Dolzhenko's inequality for A. D. Baranova, I. R. Kayumovba a Saint Petersburg State University b Kazan Federal University
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     Estimates of the norms of rational functions and their derivatives in different function spaces can be considered as classical problems in function theory. A natural generalization of rational functions is the concept of an $n$-valent function. Recall that a function $f$ is called $n$-valent in a domain $G$ if the equation $f(z)=w$ has at most $n$ solutions in $G$ for any $w\in\mathbb{C}$. The purpose of this note is to prove the following theorem. Theorem 1. Let $p \in (1,2]$ and let $G$ be a simply connected domain with rectifiable boundary $\gamma$ of length $L=\ell(\gamma)$. Then there exists a positive constant $C_p$ such that for every $n$-valent bounded function $R$ in $G$ the following inequality holds: where the symbol $dA$ denotes the Lebesgue measure on the plane. Moreover, $C_p \leqslant K/(p-1)$, where $K$ is an absolute constant. In particular, inequality (1) is true for every rational function $R$ of degree at most $n$ with poles outside $\overline{G}$. One of the first results in this direction was obtained by Dolzhenko [1], who showed that under certain conditions on the regularity of the boundary (the existence of a curvature satisfying the Hölder condition), for any rational function of order $n$ an inequality of the form (1) holds with the quantity $C_p(G) n^{p-1}\|R\|^p_{H^\infty(G)}$ on the right-hand side. Subsequently, integral inequalities for derivatives of rational functions (mainly in a disc) were considered by Peller [2], Semmes [3], Pekarskii [4], Mardvilko and Pekarskii [5], Danchenko [6], [7], Dyn’kin [8], [9], and many others. In our recent work [10] Dolzhenko’s result was extended to Hölder domains, and also to finitely connected John domains. In the proof of the theorem we use the following (well-known) geometric lemma. Lemma. Let $G$ be a simply connected domain with rectifiable boundary $\gamma$ of length $L$. For $\varepsilon>0$ set $H_\varepsilon=\{w\in G\colon \operatorname{dist}(w,\gamma)<\varepsilon\}$. Then where $S(H_\varepsilon)$ is the area of the set $H_\varepsilon$, and $C>0$ is an absolute constant. Proof of Theorem 1. Let $d_G(w)=\operatorname{dist}(w,\gamma)$. We set $\varepsilon=L/n$ and consider the sets $G_{\varepsilon}=\{w\in G\colon d_G(w) \geqslant \varepsilon\}$ and $H_{\varepsilon}=\{w\in G\colon d_G(w)<\varepsilon\}$. Then
$$
\begin{equation*}
\int_{H_{\varepsilon}}|R'(w)|^p\,dA(w) \leqslant \biggl(\int_{H_{\varepsilon}}|R'(w)|^2\,dA(w)\biggr)^{p/2} (S(H_{\varepsilon}))^{1-p/2}.
\end{equation*}
\notag
$$
Since the function $R$ is $n$-valent, we have
$$
\begin{equation*}
\int_G |R'(w)|^2\,dA(w) \leqslant n\|R\|^2_{H^\infty(G)}.
\end{equation*}
\notag
$$
Using (2) we easily obtain where $C>0$ is a constant that does not depend on $p$, $n$, $G$, or $R$.
Now we estimate the integral with respect to $G_{\varepsilon}$. Using the inequality $|R'(w)| \leqslant d_G^{-1} (w)$ we have
$$
\begin{equation*}
\int_{G_\varepsilon}|R'(w)|^p\,dA(w) \leqslant \int_{G_\varepsilon} d_G^{-p}(w)\,dA(w).
\end{equation*}
\notag
$$
Let $\varphi$ be the conformal mapping of the disc $\{|z|<1\}$ onto $G$. Then $\varphi'$ belongs to the Hardy space $H^1$ in the disc and $\|\varphi'\|_{H^1}=L$. It is well known that $2|\varphi'(z)|(1-|z|) \geqslant d_G(\varphi(z))\geqslant |\varphi'(z)|(1-|z|^2)/4$. Hence
$$
\begin{equation*}
\int_{G_{\varepsilon}}d_G^{-p}(w)\,dA(w) \leqslant 4^p \iint_{2(1-r)|\varphi'(re^{it})| \geqslant\varepsilon} \frac{|\varphi'(re^{it})|^{2-p}}{(1-r)^p}\,r\,dr\,dt.
\end{equation*}
\notag
$$
To estimate the last integral it will be convenient to use the Hardy–Littlewood maximal function $\varphi'_*(t):=\sup\{|\varphi'(re^{it})|\colon r \in [0,1)\}$. We have
$$
\begin{equation*}
\begin{aligned} \, &\iint_{2(1-r)|\varphi'(re^{it})| \geqslant \varepsilon} \frac{|\varphi'(re^{it})|^{2-p}}{(1-r)^p}\,r\,dr\,dt \\ &\qquad\leqslant \int_{0}^{2\pi} dt \int_{0}^{1-\varepsilon/(2\varphi'_*(t))} \frac{(\varphi'_*(t) )^{2-p}}{(1-r)^p}\,dr \\ &\qquad\leqslant \frac{2^{p-1}}{p-1}\int_0^{2\pi} (\varphi'_*(t))^{2-p}\varepsilon^{1-p}(\varphi'_*(t))^{p-1}\,dt \\ &\qquad\leqslant \frac{2^{p-1} L^{1-p}n^{p-1}}{p-1}\int_0^{2\pi}\varphi'_*(t)\,dt \leqslant C\,\frac{L^{2-p} n^{p-1}}{p-1} \end{aligned}
\end{equation*}
\notag
$$
due to the classical Hardy–Littlewood inequality. $\Box$ The condition of rectifiability is essential. In the general case there is no estimate of the form (1) with the dependence on the order of valency of the form $n^{p-1}$. However, Theorem 1 can also be extended to domains with fractal boundaries, and one can get estimates in terms of the (upper) Minkowski dimension $\operatorname{Mdim}(\gamma)\kern-1pt= \limsup_{\varepsilon\to 0}(\log N(\varepsilon)/\log(1/\varepsilon))$ (here $N(\varepsilon)$ is the minimum number of discs of radius $\varepsilon$ covering $\gamma$). Theorem 2. Let $p \in (1,2]$, let $G$ be a bounded simply connected domain, and let $\gamma=\partial G$ and $\alpha=\operatorname{Mdim}(\gamma) \in [1,2]$. Then for any $\delta>0$ there exists a positive constant $C=C(G,p,\delta)$ such that the following inequality holds for every $n$-valent bounded function $R$ in $G$: As an application of Theorem 2, using the estimate for the dimension of quasicircles established in [11] we have obtained a similar result for domains with quasiconformal boundaries. 
Citation in format AMSBIB
\Bibitem{BarKay22} |
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