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This article is cited in 2 scientific papers (total in 2 papers) An upper bound for the least critical values of finite Blaschke products V. N. Dubininab a Far Eastern Center for Research and Education in Mathematics, Far Eastern Federal University, Vladivostok, Russia b Institute for Applied Mathematics, Far Eastern Branch, Russian Academy of Sciences, Vladivostok, Russia
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     § 1. Introduction and the statement of the resultA finite Blaschke product of degree $n \geqslant 2$ is a rational function of the form
$$
\begin{equation*}
B(z)=\alpha \prod^n_{k=1} \frac{z-z_k}{1- \overline{z}_k z},
\end{equation*}
\notag
$$
where $|\alpha|=1$ and the complex numbers $z_1,\dots, z_n$ lie in the unit disc $|z|<1$. Such products and their applications have been investigated by many authors (for instance, see [1]–[4] and the bibliography there). On the other hand problems related to the critical values of Blaschke products are poorly studied (see [3] and [5]). Apart from other results, in the recent paper [6] we found the greatest lower bound for the maximum modulus of the critical values of products $B$ normalized by the conditions $B(0)=0$ and $B'(0) \neq 0$ (see [6], Corollary 5.3). In this case the extremal function coincides with the so-called Chebyshev-Blaschke product (see [2]) up to a linear fractional change of independent variable. The question of an upper estimate for the minimum modulus of critical values was previously raised in [7], § 5.4. We prove the following result. Theorem. Let $B$ be a finite Blaschke product of degree $n\geqslant 2$ such that $B(0)= 0$, and $|B'(0)|=c$, $0<c<1$. Then where It is easy to see that $B_c$ is a Blaschke product. Critical points of $B_c$ are roots of the equation
$$
\begin{equation*}
\zeta^{2n-2} + \biggl((n-2)c-\frac nc\biggr) \zeta^{n-1} +1=0,
\end{equation*}
\notag
$$
and the critical values of $B_c(\zeta)$ that deliver the minimum on the right-hand side of (1.1) lie symmetrically at the same distance from the origin. From the above theorem we can obtain the corresponding result on critical values of complex polynomials (see [8]). In fact, let
$$
\begin{equation*}
P(z)=z^n +\dots +cz=z \prod^{n-1}_{k=1} (z- \alpha_k), \qquad c>0,
\end{equation*}
\notag
$$
and let $P^*(z)=z^n -cz$. Comparing the critical values of the Blaschke products
$$
\begin{equation*}
\begin{gathered} \, B(z)=\frac{t^n P(z/t)}{\prod^{n-1}_{k=1} (1- \overline{\alpha}_k tz)}=z \prod^{n-1}_{k=1} \frac{z- \alpha_k t}{1- \overline{\alpha}_k tz} \\ \text{and}\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
\\
B_{\widetilde{c}} (z)=\frac{t^n P^* (z/t)}{1-ct^{n-1} z^{n-1}}=z \frac{z^{n-1} - ct^{n-1}}{1-ct^{n-1} z^{n-1}}, \qquad \widetilde{c}=ct^{n-1}, \end{gathered}
\end{equation*}
\notag
$$
with the help of the theorem for small values of $t>0$, we conclude that
$$
\begin{equation*}
\min \lbrace |P(\zeta)|\colon P'(\zeta)=0 \rbrace \leqslant \min \lbrace |P^*(\zeta)|\colon P^{*'} (\zeta)=0 \rbrace=(n-1) \biggl( \frac{c}{n}\biggr)^{n/(n-1)}.
\end{equation*}
\notag
$$
This inequality answers a question of Smale (see [9] and [8]). Note that $P^*$ is also an extremal polynomial in Smale’s famous mean value conjecture of the existence of a critical point $\zeta$ such that
$$
\begin{equation}
\biggl|\frac{P (\zeta)}{c\zeta }\biggr| \leqslant \frac{n-1}{n}
\end{equation}
\tag{1.3}
$$
(see [9]). Sheil-Small raised the question of finding an inequality similar to (1.3) in which $P$ is replaced by a finite Blaschke product $B$ such that $B(0)=0$ and $B'(0)=c$ (see [5], § 10.4.8). Some progress in this direction was due to Ng and Zhang [3]. In view of the above, it is natural to conjecture that the function $B_c$ in the theorem is an extremal Blaschke product in the conjecture analogous to (1.3). We present the proof of the theorem in § 3. Using methods from [8], we rely significantly on the structure of the Riemann surface of the inverse function of $B_c$ and on the dissymmetrization of real-valued functions (see [10]).
§ 2. The surface $ {\mathscr{R}}(B_c)$In this section we describe the Riemann surface $\mathscr{R}(B_c)$ to which $B_c$, $0<c<1$, maps the disc $|z|<1$. Throughout what follows a Riemann surface is a surface $\mathscr{R}$ glued of a finite or countable number of domains in the closed complex plane so that: the projection of each point on $\mathscr{R}$ is a point in some domain participating in gluing; a neighbourhood of each point on $\mathscr{R}$ is a single-sheeted disc or a multi-sheeted disc with a single ramification point at the centre (see [11], Ch. 3, for details). When this cannot lead to confusion, we do not distinguish between plane domains before gluing them (identifying some parts of the boundaries of these domains) and after that (when they are subdomains of $\mathscr{R}$). Boundary points of domains that are not involved in gluing give rise to boundary points of the surface $\mathscr{R}$. We denote the set of boundary points by $\partial \mathscr{R}$. Points $W \in \partial \mathscr{R}$ can be regarded as boundary points of the set $\mathscr{R}$ on the surface $ \widetilde{\mathscr{R}} \supset \mathscr{R}$ obtained by gluing together suitable extensions of the domains forming $\mathscr{R}$. We start by looking at the function $f$ taking the sector
$$
\begin{equation*}
S=\biggl\{ z\colon|z| < 1,\ 0 < \arg z < \frac{\pi}{n-1}\biggr\}
\end{equation*}
\notag
$$
conformally and univalently to the sector
$$
\begin{equation*}
\biggl\{ w \colon|w | < 1,\ 0 < \arg w < \pi + \frac{\pi}{n-1}\biggr\}
\end{equation*}
\notag
$$
cut along an interval $[A_c, 0]$, $ -1 < A_c <0$, so that
$$
\begin{equation*}
f(0)=0, \qquad f(1)=1, \qquad f\biggl(\exp\biggl(\frac{i \pi}{n-1}\biggr)\biggr)=- \exp \biggl(\frac{i \pi}{n-1}\biggr).
\end{equation*}
\notag
$$
We determine $A_c$ from the condition $f(a)=0$, where $a=c^{1/(n-1)} >0$. That we can find such $A_c$ follows, for instance, from the following argument. As $A_c$ varies from $-1$ to zero, the modulus of the family of curves in the domain $f(S)$ that connect $[A_c, 0]$ with the arc $\{w \colon|w |=1,\ 0< \arg w < \pi + \pi /(n-1) \}$ of a circle changes continuously and strictly monotonically from $\infty$ to zero1. Hence there exists $A_c$ such that this modulus is equal to the modulus of the family of curves in $S$ that connect the interval $[0,a]$ with the arc $\{ z\colon|z|=1,\ 0< \arg z< \pi /(n-1) \}$. By conformal invariance this is just the required value of $A_c$. The function $f$ has an analytic extension from the sector $S$ to the whole of the plane $\overline{\mathbb C}$ by means of the Riemann-Schwarz symmetry principle applied to arcs of the circle $|z|=1$ and the rays $\arg z^{2(n-1)}=0$. We keep the notation $f$ for this extension. Note that $A_c$ is a critical value of $f$. The part of the $f$-preimage of the interval $[1/A_c, A_c]$ that lies in the sector $|\arg z|< \pi /(n-1)$ is a closed Jordan curve $\gamma_1 $, which is symmetric with respect to the circle $|z|=1$ and the real axis. We denote by $\gamma_k$, $k=2,\dots, n-1$, the curves obtained from $\gamma_1 $ by the anticlockwise rotations by $2 \pi (k-1)/(n-1)$. Let $G_k$ be a finite domain bounded by $\gamma_k$, $k=1,\dots, n-1$, and let
$$
\begin{equation*}
G_0=\overline{\mathbb C}\setminus\bigcup^{n-1}_{k=1} \overline{G}_k.
\end{equation*}
\notag
$$
It follows from the definition of $f$ that the $f$-image of $G_0$ is the $w$-plane cut along the line segments
$$
\begin{equation*}
\lambda_k :=\biggl[\frac{\exp (2\pi i(k-1)/(n-1))}{A_c},\, A_c \exp\biggl(\frac{2 \pi i (k-1)}{n-1}\biggr)\biggr], \qquad k=1, \dots, n-1.
\end{equation*}
\notag
$$
We denote this image by $D_0$. Now, $f(G_k):=D_k$ is the $w$-plane cut along the segment $\lambda_k$, $k=1, \dots, n-1$. We obtain the Riemann surface $\mathscr{R}$ onto which $f$ maps the plane $\overline{\mathbb C}_z$ by gluing the domains (sheets) $D_k$, $k=1, \dots, n-1$, to $D_0$ ‘crosswise’ along the sides of the corresponding cut. The compact surface $\mathscr{R}$ consists of $n$ sheets, and is similar to a single-sheeted surface. Hence $f$ is a rational function (see [11], Pt. 3, Ch. 8, § 10). By construction $f$ is holomorphic in the disc ${|z|<1}$, takes the circle $|z|=1$ to $|w|=1$, and has the same zeros as the Blaschke product (1.2). Hence the real part of the holomorphic branch of $\log (f/B_c)$ in the disc $|z|<1$ vanishes on the boundary of the disc. By the maximum principle $\log (f/B_c)\equiv i \theta$, where $\theta$ is a real constant. Taking the normalization $f(1)=B_c(1)=1$ into account we conclude that $f\equiv B_c $. Thus, $\mathscr{R}$ coincides with the Riemann surface of the inverse function of $B_c$. Let $\mathscr{R}(B_c)$ denote the part of this surface lying over the disc $|w|<1$. It follows from the above that $\mathscr{R}(B_c)$ can be obtained by gluing the domains (sheets)
$$
\begin{equation*}
U^*_k=\lbrace w\colon|w|<1,\, w \notin \lambda_k \rbrace, \qquad k=1, \dots, n-1,
\end{equation*}
\notag
$$
‘crosswise’ to the sheet
$$
\begin{equation*}
U^*_0=\biggl\lbrace w\colon|w|<1,\, w \notin\bigcup^{n-1}_{k=1} \lambda_k \biggr\rbrace
\end{equation*}
\notag
$$
along the sides of the corresponding cuts of the unit disc.
§ 3. Proof of the theoremLet $B$ be a function satisfying the assumptions of the theorem and let $|B'(0)|=c$, $0<c<1$. We can assume that the critical points of $B$ are first-order zeros of its derivative $B'$ and the critical values of $B$ have distinct arguments. (Cases not satisfying these conditions can be treated by means of a limit transition.) Assume that the theorem fails for our Blaschke product $B$. Then there exists $d$, $c<d<1$, such that
$$
\begin{equation}
\min \lbrace |B(\zeta)|\colon B'(\zeta)=0 \rbrace > |A_d|,
\end{equation}
\tag{3.1}
$$
where $A_d$ is a critical value of $B_d$ (see § 2). Let $\mathscr{R}$ denote the Riemann surface of the inverse function $\mathscr {B}$ of $w=B(z)$, $|z|<1$. We regard $\mathscr {B}$ as a single-valued function on $\mathscr{R}$. By the projection of a point $W \in \mathscr{R}$ we mean the point $B(\mathscr {B}(W))$ in the disc $|w|< 1$. The curves on $\mathscr{R}$ lying with multiplicity one over rays of the form $\arg w=\mathrm{const}$, $|w| \geqslant \lambda $, and connecting ramification points of $\mathscr{R}$ with the boundary $\partial \mathscr{R}$ partition $\mathscr{R}$ into $n$ sheets. We number these sheets as follows. Let $U_0$ denote the sheet containing the $\mathscr {B}$-preimage of $z=0$. The sheets $U_1, \dots, U_m$, $ m \leqslant n-1$, glued to $U_0$ (that is, having common sides of cuts with $U_0$) follow it in an arbitrary order. After that we number (arbitrarily) the sheets glued to $U_1$ (if they exist and have not received numbers yet), then the sheets glued to $U_2$ (if they exist and have not received numbers yet) and so on, until we arrive at the last sheet $U_{n-1}$. Since $\mathscr{R}$ is connected, each sheet obtains some number. Note that for each sheet $U_k$, $k \geqslant 1$, there exists a unique sheet $U_{k'}$ with $k' <k$ that is glued to it because $\mathscr{R}$ contains no noncontractible cycles. Now we look at the surface $\mathscr{R}(B_d)$ and the function $\mathscr {B}_d$ inverse to the Blaschke product $w=B_d (z)$, $|z|<1$; $\mathscr {B}_d\colon \mathscr{R}(B_d)\to \lbrace z\colon|z|<1 \rbrace$. On the set
$$
\begin{equation*}
\mathscr{R}^*_0 :=\lbrace W^* \in \mathscr{R}(B_d)\colon\mathscr {B}_d(W^*) \neq 0 \rbrace
\end{equation*}
\notag
$$
we define a real-valued function $\omega^*$ by This function has the following symmetries. On $U^*_0$ it is symmetric with respect to the straight lines
$$
\begin{equation*}
\biggl\lbrace w=\rho \exp \biggl(\pi i + \frac{\pi ij}{n-1}\biggr)\colon- \infty < \rho < + \infty \biggr\rbrace , \qquad j=1, \dots, 2n-2
\end{equation*}
\notag
$$
(that is, it takes the same values at points symmetric across these lines). On each sheet $U^*_k$ it is symmetric with respect to the straight line
$$
\begin{equation*}
\biggl\lbrace w=\rho \exp \biggl(\pi i + \frac{2\pi i(k-1)}{n-1}\biggr)\colon- \infty < \rho < + \infty \biggr\rbrace, \qquad k=1, \dots, n-1.
\end{equation*}
\notag
$$
Moreover, if a sheet $U^*_{k'}$ is obtained from $U^*_{k''}$ by a rotation $\varphi$ about the origin: $\varphi \colon U^*_{k''} \to U^*_{k'}$, $k',k'' \geqslant 1$, then $\omega^* \equiv \omega^* \circ \varphi$ on $U^*_{k''}$. The rest of the proof reduces to constructing a real-valued function $\omega$ on the surface
$$
\begin{equation*}
\mathscr{R}_0 :=\lbrace W \in \mathscr{R} \colon\mathscr{B} (W) \neq 0 \rbrace
\end{equation*}
\notag
$$
that can be treated as the dissymmetrization of $\omega^*$. Since $\omega^*$ is symmetric on $U^*_0$ it can be extended by continuity to the disc $|w|<1$ punctured at $w=0$ and then dissymmetrization can be used (see [10], § 4.4). The idea of this dissymmetrization is as follows. Let $l_1 ,\dots, l_m$ be the rays through the origin that contain the cuts of $U_0$, and let
$$
\begin{equation*}
l^*_k=\biggl\lbrace w\colon\arg w=\pi +\frac{2 \pi (k-1)}{n-1}\biggr \rbrace, \qquad k=1, \dots, m,
\end{equation*}
\notag
$$
be the rays containing the cuts of $U^*_0$. Then there exist open sectors $P_j$, $j=1,\dots,N$, with vertices at the origin and there exist rotations $\lambda_j$, $j=1,\dots,N$, about the origin such that
$$
\begin{equation*}
\begin{gathered} \, P_j\cap P_{j'}=\varnothing, \quad \lambda_j(P_j)\cap\lambda_{j'} (P_{j'})=\varnothing, \qquad j\neq j', \quad j,j'=1,\dots,N, \\ \bigcup_{j=1}^{N}\overline{P}_j=\overline{\mathbb C}_w, \quad \bigcup_{j=1}^{N}\lambda_j(l^*_k\cap\overline{P}_j)=l_k, \qquad k=1,\dots,m, \end{gathered}
\end{equation*}
\notag
$$
and the function
$$
\begin{equation*}
\omega(w)\equiv\operatorname{Dis}\omega^*(w):=\omega^*(\lambda^{-1}_j(w)), \qquad w\in\lambda_j(\overline{P}_j), \quad j=1,\dots,N,
\end{equation*}
\notag
$$
is well defined on $\overline{\mathbb C}_w$. We call $\omega$ the result of the dissymmetrization of $\omega^*$ (see details in [10], § 4.4). We regard $\omega$ as a function on $U_0\cap\mathscr{R}_0$. More precisely, the composition with $\omega^*$ of the inverse map of the projection from $\mathscr{R}^*_0$ is extended to ${\{w\colon 0<|w|<1\}}$ by continuity and dissymmetrized, after which we consider the composition of the projection of $\mathscr{R}$ and the result of this dissymmetrization. This composition is the function $\omega$ on $U_0\cap \mathscr{R}_0$. The values of $\omega$ on the other sheets are defined by
$$
\begin{equation*}
\omega=\omega^*\circ\varphi_k \quad\text{on } U_k, \quad k=1,\dots,n-1,
\end{equation*}
\notag
$$
where $\varphi_k\colon U_k\to U^*_k$ is the rotation (or more precisely, the projection, a rotation about the origin, and the lift) defined as follows. For the sheets $U_k$, $1\leqslant k\leqslant m$, the map $\varphi_k$ is the rotation taking the ray $l_k$ to $l^*_k$, $k=1,\dots,m$. Now, if the sheet $U_k$, $k>m$, is glued to $U_{k'}$, $1<k'<k$, then the inverse images of the cuts on $U^*_k$ and $U^*_{k'}$ under the rotations $\varphi_k$ and $\varphi_{k'}$ have the same projection. Since the sheet $U_{k'}$ is uniquely defined for such $k$, the maps $\varphi_k$, $k=m+1,\dots,n-1$, are well defined. We can extend $\omega$ to the whole of $\mathscr{R}_0$ by continuity. In fact, by the symmetry of $\omega^*$ and the properties of dissymmetrization, $\omega$ extends continuously to the closure (relative to $\mathscr{R}_0$) of each sheet $U_{k}$, $k=0,\dots,n-1$. Now, taking symmetry into account again, in view of condition (3.1), $\omega$ takes the same values on the ‘upper’ and ‘lower’ sides of the cuts along which $U_0$ if glued to the sheets $U_1,\dots,U_m$. Hence $\omega$ extends continuously to the set $\bigl(\bigcup_{k=0}^m\overline{U}_k\bigr)\,\cap\,\mathscr{R}_0$. Finally, if for $k>m$ the sheet $U_{k}$ is glued to some $U_{k'}$, $1<k'<k$, then since $\omega^*$ has symmetry of order $(n-1)$, by the construction of $\omega$ it takes equal values on the ‘upper’ and ‘lower’ sides of the cuts along which $U_k$ and $U_{k'}$ are glued the values of $\omega$. Hence $\omega$ extends continuously to $(\overline{U}_k\cup \overline{U}_{k'})\cap\mathscr{R}_0$ and therefore to the whole of $\mathscr{R}_0$. Below we let $\omega$ denote the result of this extension to $\mathscr{R}_0$. We complete the proof by comparing the Dirichlet integrals of $\omega^*$ and $\omega$. For open sets $B$ in the plane, or on the surface $\mathscr{R}_0$, or on $\mathscr{R}^*_0$ and for sufficiently smooth functions $v$ on $B$ we set Let $s>0$ be sufficiently small, and let $\mathscr{R}_s$ denote the set of points on $\mathscr{R}_0$, other than the points $W$ on $U_0$, such that their projections satisfy $|\operatorname{pr}W|\leqslant s$. The set $\mathscr{R}^*_s$ is defined in a similar way. Since dissymmetrization and the rotations $\varphi_k$ do not change the Dirichlet integral, we have
$$
\begin{equation}
I(\omega^*,\mathscr{R}^*_s)=\sum_{k=0}^{n-1} I(\omega^*,U^*_k\cap \mathscr{R}^*_s)=\sum_{k=0}^{n-1} I(\omega,U_k\cap \mathscr{R}_s)=I(\omega,\mathscr{R}_s).
\end{equation}
\tag{3.2}
$$
Bearing in mind that the Dirichlet integral is invariant we obtain
$$
\begin{equation*}
I(\omega^*,\mathscr{R}^*_s)\leqslant I\biggl(\log|z|,\biggl\{z\colon \frac sd(1+o(1))<|z|<1\biggr\}\biggr)=-2\pi\log\frac sd+o(1), \qquad s\to 0.
\end{equation*}
\notag
$$
Note that the limit boundary values of $\omega$ over the circle $|w|=1$ are zero. Using the conformal invariance of the Dirichlet integral and the Dirichlet principle we obtain
$$
\begin{equation*}
\begin{aligned} \, I(\omega,\mathscr{R}_s) &\geqslant I\biggl(\omega\circ\mathscr{B}^{-1},\,\biggl\{z\colon \frac sc(1+o(1))<|z|<1\biggr\}\biggr) \\ &\geqslant I\biggl(\frac{\log(s/d)}{\log(s/c)}\log|z|,\ \biggl\{z\colon \frac sc(1+o(1))<|z|<1\biggr\}\biggr) \\ &=-2\pi\log\frac sc\biggl(\frac{\log(s/d)}{\log(s/c)}\biggr)^2+o(1), \qquad s\to 0. \end{aligned}
\end{equation*}
\notag
$$
Comparing these inequalities with (3.2) we arrive at a contradiction: $c \geqslant d$. The proof is complete.
Citation in format AMSBIB
\Bibitem{Dub22} |
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