UDC 517.532.2

MATHEMATICS

V. M. TERPIGOREVA

EXTREMAL PROBLEMS FOR SOME CLASSES OF ANALYTIC FUNCTIONS WITH BOUNDED MEAN MODULUS

(Presented by Academician V. I. Smirnov on 19 XII 1966)

This paper is devoted to the study of extremal problems in classes of analytic functions generalizing the well-known classes \(H_\delta\), \(\delta>0\). Extremal problems in the classes \(H_\delta\) with \(\delta \geqslant 1\) have been studied by many authors. We mention here the papers \((^{1-5})\), in which one can find the history of the question and an extensive bibliography. Extremal problems in the classes \(H_\delta\) with \(\delta<1\) have been less studied. In paper \((^2)\) it is indicated that S. Ya. Khavinson found the form of the extremal function in the class \(H_\delta\), \(0<\delta<1\), for the problem of

\[ \sup \left|\sum_{j=1}^{\nu}\sum_{i=0}^{n_j}\gamma_{ij}f^{(i)}(z_j)\right|, \]

where \(z_1,\ldots,z_\nu\) are prescribed points of the unit disk; \(\gamma_{ij}\) are prescribed coefficients. However, this result was not published. In papers \((^6,^7)\) V. Kabaila found the form of the function with least norm

\[ \|f\|_\delta=\sup_{0<r<1}\left\{\int_0^{2\pi}|f(re^{i\theta})|^\delta\,d\theta\right\}^{1/\delta}, \]

interpolating prescribed values at prescribed points. In the same papers the problem of \(\sup |f'(z_0)|\), \(f\in H_\delta\), \(0<\delta<1\), was solved. In a recent paper \((^8)\) S. A. Gel'fer and L. V. Kresnyakova considered a number of extremal problems in the classes \(H_\delta\), \(\delta>0\), but the form of the extremal functions found by them can in some cases be substantially refined.

Let \(m(u)\geqslant 0\) be a convex nondecreasing function, defined for \(-\infty<u<+\infty\), possessing a continuous derivative \(p(u)=m'(u)\) and satisfying the conditions

\[ \lim_{u\to-\infty}m(u)=0,\qquad \lim_{u\to+\infty}m(u)/u=+\infty. \]

The function \(p(u)\) has an inverse \(q(t)\), continuous and increasing for \(0<t<+\infty\).

We now introduce the class \(H_m\) of functions \(f(z)\), analytic in the unit disk, for which

\[ \int_0^{2\pi} m[\ln |f(re^{i\theta})|]\,d\theta \leqslant 1. \tag{1} \]

Functions of the classes \(H_m\) were first considered in the work of E. D. Solomentsev \((^9)\) from the standpoint of their boundary behavior. From the results of \((^9)\) there follows a parametric representation of V. I. Smirnov type for functions from \(H_m\):

\[ f(z)=B(z)\exp\int_0^{2\pi}\frac{e^{i\theta}+z}{e^{i\theta}-z}\,d\nu(\theta). \tag{2} \]

Here \(B(z)\) is a Blaschke product, and the Borel measure \(d\nu(\theta)\) has the following structure: \(d\nu(\theta)=u(\theta)\,d\theta+d\mu(\theta)\), where \(u(\theta)\) is a summable function satisfying the condition

\[ \int_0^{2\pi} m[u(\theta)]\,d\theta \leqslant 1, \tag{3} \]

\(d\mu \leq 0\) is a singular measure. If \(m(u)\equiv e^{\delta u}\), \(\delta>0\), then the class \(H_m\) coincides with \(H_\delta\); if \(m(u)\equiv M(e^u)\), where \(M(t)\) is an \(N\)-function (see (10)), then one obtains the classes \(H_M\), the extremal problems in which were considered by us in \((^{11,12})\).

Let us introduce some further notation. By \(T_m\) we shall denote the class of Borel measures \(d\nu(\theta)\) which occur in (2). By \(Q_m^*\) we shall denote the class of functions \(\rho(z)\) having the representation

\[ \rho(z)=\frac{1}{2\pi}\int_0^{2\pi}\frac{e^{i\theta}+z}{e^{i\theta}-z}\,d\nu(\theta),\qquad \nu\in T_m, \tag{4} \]

and by \(Q_m\) the subclass of \(Q_m^*\) consisting of functions in whose representation, according to formula (4), the singular component of the measure \(d\nu\) is equal to zero. By \(H_m^0\) we shall denote the class of functions having the representation \(\exp \rho(z)\), \(\rho(z)\in Q_m^*\). If \(z_1,\ldots,z_\nu\) are points of the unit disk, \(n_1,\ldots,n_\nu\) are integers \(\geq 1\), \(n=n_1+\cdots+n_\nu\), and \(A=\{F(z)\}\) is some class of functions analytic in the unit disk, then by \(W(A,\{z_i n_i\})\), or more briefly \(W(A)\), we shall denote the set of points \((F(z_1),\ldots,F^{(n_1-1)}(z_1),\ldots,\ldots,F(z_\nu),\ldots,F^{n_\nu-1}(z_\nu))\).

Lemma 1. Let \(\alpha(\theta)\) be a continuous function on \([0,2\pi]\). In order that the upper bound

\[ \sup_{\nu\in T_m}\int_0^{2\pi}\alpha(\theta)\,d\nu \]

be finite, it is necessary and sufficient that \(\alpha(\theta)\geq 0\).

Lemma 2. Let \(\alpha(\theta)\geq 0\) be continuous and not identically zero on \([0,2\pi]\). There exists a constant \(k^*>0\) such that

\[ \int_0^{2\pi} m\{q[k^*\alpha(\theta)]\}\,d\theta=1, \tag{5} \]

\[ \sup_{\nu\in T_m}\int_0^{2\pi}\alpha(\theta)\,d\nu = \int_0^{2\pi}\alpha(\theta)\,q[k^*\alpha(\theta)]\,d\theta . \tag{6} \]

If, moreover, \(q[k^*\alpha(\theta)]\) is an integrable function, then any measure \(d\nu^*\) extremal for (6) has the form

\[ d\nu^*=q[k^*\alpha(\theta)]\,d\theta+d\mu^*, \tag{7} \]

where the singular measure \(d\mu^*\) is concentrated on the set of roots of the equation \(\alpha(\theta)=0\). Conversely, every measure of the form (7) will be extremal in (6). If \(q(k^*\alpha(\theta))\) is not an integrable function, then the upper bound (6) in the class \(T_m\) is not attained.

In the following theorems the sets \(W\) defined above are considered in \(n\)-dimensional complex space \(C^n\) if \(0\in\{z_1,\ldots,z_\nu\}\), or in \((2n-1)\)-dimensional real space \(R^{2n-1}\) if \(z_\nu=0\) (in the latter case the coordinate \(F(z_\nu)\) is always real for us).

Theorem 1. Each of the sets \(W(Q_m)\) and \(W(Q_m^*)\) is unbounded, convex, and contains interior points. The set \(W(Q_m^*)\) is closed.

Let

\[ \Phi(\theta,z)=\frac{e^{i\theta}+z}{e^{i\theta}-z},\quad \alpha_1(\theta)=\Phi(\theta,z_1),\quad \alpha_2(\theta)=\left.\frac{\partial\Phi}{\partial z}\right|_{z=z_1},\ldots,\alpha_{n_1}(\theta)=\left.\frac{\partial^{n_1-1}\Phi}{\partial z^{n_1-1}}\right|_{z=z_1}, \]

\[ \alpha_{n_1+1}(\theta)=\Phi(\theta,z_2),\ldots,\ldots,\alpha_n(\theta)=\left.\frac{\partial^{n_\nu-1}\Phi}{\partial z^{n_\nu-1}}\right|_{z=z_\nu}. \]

We shall say that the measure \(d\nu\) corresponds to the point \((\xi)=(\xi_1,\ldots,\xi_n)\) if

\[ \xi_j=\int_0^{2\pi}\alpha_j(\theta)\,d\nu,\qquad j=1,\ldots,n. \]

Theorem 2. To each boundary point \((\xi^*)=(\xi_1^*,\ldots,\xi_n^*)\) of the body \(W(Q_m^*)\) there corresponds a unique measure \(d\nu^*\in T_m\) of the form:

\[ d\nu^*=q[\alpha(\theta)\,d\theta]+d\mu^*, \tag{8} \]

where

\[ \alpha(\theta)=\operatorname{Re}\sum_{1}^{n}\eta_k\alpha_k(\theta)\geq 0, \tag{9} \]

\[ \int_{0}^{2\pi} m[q(\alpha(\theta))]\,d\theta=1 \tag{10} \]

and the function \(q[\alpha(\theta)]\) is summable, while the measure \(d\mu^*\) is concentrated on the set of roots of the equation \(\alpha(\theta)=0\). The coefficients \(\eta_k\) are determined uniquely by the point \((\xi^*)=(\xi_1^*,\ldots,\xi_n^*)\) under condition (10).

Conversely, to every measure of the form (8), where \(\alpha(\theta)\) and \(d\mu^*\) satisfy the conditions described and \(q[\alpha(\theta)]\) is summable, there corresponds a boundary point \((\xi^*)\) of the body \(W(Q_m^*)\).

Lemma 3. Let \(r\) be the multiplicity of a root of \(\alpha(\theta)\). In order that the function \(q[\alpha(\theta)]\) be summable in a neighborhood of this root, it is necessary and sufficient that the integral

\[ \int_{-\infty}^{\cdot} u\,[m'(u)]^{1/r-1}m''(u)\,du \tag{11} \]

converge.

If \(m(u)=e^{\partial u}\), then the integral (11) converges for all \(r\geq 2\). Hence the function \(\alpha(\theta)\) in (9) may have roots of arbitrary multiplicity. If \(m(u)=(-u)^{-h}\) \((h>0,\ u<0)\), then the integral (11) converges only for \(r<h+1\). Therefore, for \(0<h<1\), the function \(\alpha(\theta)\) in (9) cannot have roots, and the singular component in (8) is absent. If \(h>1\), the admissible multiplicity \(r\) of a root is determined by the inequality \(r<h+1\).

Theorem 3. Through each boundary point \((\xi^*)\in W(Q_m^*)\) there passes only one hyperplane supporting \(W(Q_m^*)\).

1. Each hyperplane supporting \(W(Q_m^*)\) has one and only one point in common with \(W(Q_m)\).

2. a) The closure \(\overline{W(Q_m)}\) coincides with \(W(Q_m^*)\).

b) The body \(W(Q_m)\), generally speaking, is not closed. This will be the case, in particular, if the integral (11) converges for all \(r\geq 2\).

c) If, however, the integral (11) diverges for every \(r\geq 2\), then \(W(Q_m)\) is closed and coincides with \(W(Q_m^*)\).

Theorem 4. Suppose the function \(m(u)\) is not linear on any interval. Then the following conditions are equivalent:

1. The point \((\xi^*)\) is an extreme point of the body \(W(Q_m^*)\).

2. In the representation of the measure \(d\nu^*\) by formula (8), corresponding to the point \((\xi^*)\), the singular component \(d\mu\) is absent.

3. The point \((\xi^*)\in \partial W(Q_m^*)\cap W(Q_m)\), where \(\partial W(Q_m^*)\) is the boundary of the body \(W(Q_m^*)\).

Theorem 5 establishes the form of the extremal functions in the class \(Q_m^*\).

Theorem 5. Let \(0\in\{z_1,\ldots,z_\nu\}\). To each boundary point \((\xi^*)\in W(Q_m^*)\) there corresponds in \(Q_m^*\) a unique function \(\rho^*(z)\) of the form

\[ \rho^*(z)=\frac{1}{2\pi}\int_{0}^{2\pi}\frac{e^{i\theta}+z}{e^{i\theta}-z}\, q\left[ \frac{ a\prod_{1}^{n}|1-\overline{\gamma_j}e^{i\theta}|^2 }{ \prod_{1}^{\nu}|e^{i\theta}-z_j|^{2n_j} } \right]\,d\theta +\sum \mu_j\frac{z+\gamma_j}{z-\gamma_j}, \tag{12} \]

where \(\gamma_1,\ldots,\gamma_n\) are certain points in the disk \(|z|\leq 1\), \(\Sigma\) extends over those \(\gamma_j\) for which \(|\gamma_j|=1\), all \(\mu_j\leq 0\), and the constant \(a\) is chosen so that

\[ \int_{0}^{2\pi} m\left\{ q\left[ \frac{ a\prod_{1}^{n}|1-\overline{\gamma_j}e^{i\theta}|^2 }{ \prod_{1}^{\nu}|e^{i\theta}-z_j|^{2n_j} } \right] \right\}\,d\theta=1. \tag{13} \]

A factor of the form \(|1-\bar\gamma_j e^{i\theta}|\), \(|\gamma_j|=1\), may enter into the structure of \(\rho(z)\) to the power \(r\) (\(r\) an even number \(\geq 2\)) only in the case when the integral (11) converges. In the case \(z_\nu=0\), in (12) and (13) \(n\) must be replaced by \(n-1\), and \(\nu\) by \(\nu-1\).

Theorem 6. To boundary points of the set \(W(H_m^0)\) there correspond in \(H_m^0\) only functions of the form \(\exp[\rho^*(z)]\), where the construction of \(\rho^*(z)\) is described in Theorem 5.

Theorem 7. To boundary points of the set \(W(H_m)\), if \(0\notin\{z_1,\ldots,z_\nu\}\), there correspond only functions of the form

\[ f^*(z)=e^{i\lambda}\prod_1^{n-1}\frac{z-\beta_j}{1-\bar\beta_j z}\, \exp\left\{\frac{1}{2\pi}\int_0^{2\pi}\frac{e^{i\theta}+z}{e^{i\theta}-z}\, q\left[ a\,\frac{\prod_1^n |1-\bar\gamma_j e^{i\theta}|^2} {\prod_1^\nu |e^{i\theta}-z_j|^{2n_j}} \right]\,d\theta\right\}. \tag{14} \]

Here \(|\beta_j|\leq 1\), \(j=1,\ldots,n-1\), are certain points, \(\lambda\) is a real number, and, with respect to \(\{\gamma_j\}\), all the conditions of Theorem 5 are fulfilled. If \(z_\nu=0\), then under the sign \(\exp\) one must replace \(n\) by \(n-1\), and \(\nu\) by \(\nu-1\).

For the class \(H_\delta\), \(\delta>0\), using the work of Kabaila \((^6,^7)\), it is not difficult to obtain a result more precise than the one following directly from Theorem 7.

Theorem 8. To boundary points of the set \(W(H_\delta)\) there correspond only functions of the form

\[ f^*(z)=ae^{i\lambda}\prod\frac{z-\gamma_j}{1-\bar\gamma_j z}\cdot \prod_1^{n-1}(1-\gamma_j z)^{2/\delta}\cdot \prod_1^\nu(1-\bar z_j z)^{-2n_j/\delta}, \tag{15} \]

where \(|\gamma_j|\leq 1\), \(j=1,\ldots,n-1\); \(\prod\) extends over some of the \(\{\gamma_j\}\); \(a>0\), \(\lambda\) are real constants; \(\|f^*\|_\delta=1\).

Corollary. Let the function \(u(c_1,\ldots,c_n)\) be continuous on \(W(H_m)\) and unable to attain its supremum at interior points of this set. Then the extremal functions in the problem

\[ \sup_{f\in H_m} u\bigl(f(z_1),\ldots,f^{(n_1-1)}(z_1),\ldots,f^{(n_\nu-1)}(z_\nu)\bigr) \]

have the form (14) \((m(u)\ne e^{\delta u})\) or (16) \((m(u)=e^{\delta u})\).

In the paper \((^8)\) the problem of \(\sup \operatorname{Re}\Phi(f'(z))\), \(f\in H_\delta\), \(\delta>0\), was considered, where \(\Phi(t)\) is an entire function. The form of the extremal function indicated there is inaccurate, since, first, it is not deciphered what the Blaschke product will be, and, second, the factor with the singular component must be absent.

I take this opportunity to express my gratitude to Prof. S. Ya. Khavinson for suggesting the topic and for advice.

Moscow Institute of Electronic Engineering

Received
4 XI 1966

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