Reports of the Academy of Sciences of the USSR

1962, Volume 145, No. 6

MATHEMATICS

I. I. Bavrin

On the Uniqueness of Extremal Functions in Estimates of Taylor Coefficients of Bounded Functions of Two Complex Variables

(Presented by Academician M. A. Lavrent'ev, 29 III 1962)

In the case of two complex variables I have given exact estimates of Taylor coefficients for bounded functions \((^1)\). However, in these estimates the question of the uniqueness of the extremal functions was not clarified; this question is considered in the present note.

Let the function

\[ F(w,z)=\sum_{m,n=0}^{\infty} a_{mn} w^m z^n \tag{1} \]

satisfy the following conditions: a) it is regular in the bicylinder \(E\{|w|<R_1,\ |z|<R_2\}\); b) for almost every fixed \(t_0\) from the segment \([0,2\pi]\), for \(0<\rho<1\) the integral * is bounded

\[ \int_0^{2\pi} \left|\psi(\rho e^{i\varphi},t_0)\right|\,d\varphi, \tag{2} \]

where \(\psi(\rho e^{i\varphi},t)\equiv F(R_1\rho e^{i\varphi},R_2\rho e^{i(t-\varphi)})\). We shall denote this class of functions by \(\lambda\) **.

As is known, a function \(F(w,z)\), regular in \(E\), in addition to its representation by the series (1), can be represented in \(E\) by the diagonal series

\[ F(w,z)=\sum_{k=0}^{\infty}\left(\sum_{l=0}^{k} a_{k-l,l} w^{k-l} z^l\right). \tag{3} \]

It follows from conditions a) and b) that for almost all \(t\) in \([0,2\pi]\) the function \(\psi(\zeta,t)\in H_1\), and therefore for almost all \(t\) in \([0,2\pi]\) it has, almost everywhere on \(|\zeta|=1\), definite limiting values along nontangential paths, which form the boundary function \(\psi(e^{i\varphi},t)\), and for almost all \(t\) in \([0,2\pi]\), according to Cauchy’s formula ***

\[ \sum_{l=0}^{m+n} a_{m+n-l,l} R_1^{m+n-l} R_2^l e^{-ilt} =(2\pi)^{-1}\int_0^{2\pi}\psi(e^{i\varphi},t)e^{-i(m+n)\varphi}\,d\varphi, \tag{4} \]

since, by virtue of the representation (3),

\[ [(m+n)!]^{-1}\psi_{\zeta}^{(m+n)}(0,t) =\sum_{l=0}^{m+n} a_{m+n-l,l} R_1^{m+n-l} R_2^l e^{-ilt}. \tag{5} \]

Multiplying both sides of equality (4) by \(e^{int}\), from the resulting equality, integrating with respect to \(t\) from \(0\) to \(2\pi\), we find

\[ a_{mn}=(4\pi^2 R_1^m R_2^n)^{-1} \int_0^{2\pi} dt \int_0^{2\pi}\psi(e^{i\varphi},t)e^{-i[(m+n)\varphi-nt]}\,d\varphi, \tag{6} \]

where the integrals are understood in the Lebesgue sense.

* That is, the set of those points \(t_0\) in \([0,2\pi]\) at which the boundedness of the integral (2) fails for \(0<\rho<1\) has measure zero.
* Obviously, the class \(q\) considered in \((^2)\) is contained in \(\lambda\).
** Here the integral is understood in the Lebesgue sense.

Since for almost all \(t\) in \([0,2\pi]\) we have \(\psi(\zeta,t)\in H_1\), it follows that for almost all \(t\) in \([0,2\pi]\),
\(\psi(\zeta,t)\zeta^{m+n-1}e^{-int}\in H_1\) \((m+n=1,2,\ldots)\), and hence, by Riesz’ theorem \((^3)\), for almost all \(t\) in \([0,2\pi]\),

\[ \int_0^{2\pi}\psi(e^{i\varphi},t)e^{i[(m+n-1)\varphi-nt]}\,de^{i\varphi}=0, \]

where the integral is understood in the Lebesgue sense; whence

\[ \int_0^{2\pi}dt\int_0^{2\pi}\psi(e^{i\varphi},t)e^{i[(m+n-1)\varphi-nt]}\,de^{i\varphi}=0, \tag{7} \]

where the integrals are understood in the Lebesgue sense. Thus we have proved:

Theorem 1. For functions
\[ F(w,z)=\sum_{m,n=0}^{\infty}a_{mn}w^m z^n\in\lambda \]
the formulas (6), (7) hold.

We shall say that a certain property \(L\) holds for almost all points \((t,\varphi)\) of the square \(S\{0\leq t\leq 2\pi,\ 0\leq\varphi\leq 2\pi\}\), or, briefly, almost everywhere on \(S\), if almost all \(t\) with \(0\leq t\leq 2\pi\) have the property that for them almost all \(\varphi\) with \(0\leq\varphi\leq 2\pi\) have property \(L\).

Corollary 1. If a function \(F(w,z)\in\lambda\) is such that \(\psi(e^{i\varphi},t)=0\) almost everywhere on \(S\), then \(F(w,z)=0\) in \(E\).

Remark 1. Let the function \(F(w,z)\) be regular in \(E\). Then the function \(\psi(\rho\zeta,t)\) \((0<\rho<1)\), for any fixed \(t,\ 0\leq t\leq 2\pi\), is regular in \(|\zeta|\leq 1\), and therefore, by Cauchy’s formula,

\[ [(m+n)!]^{-1}\rho^{m+n}\psi_{\zeta^{m+n}}^{(m+n)}(0,t) =(2\pi)^{-1}\int_0^{2\pi}\psi(\rho e^{i\varphi},t)e^{-i(m+n)\varphi}\,d\varphi. \]

Taking (5) into account, we conclude that the expression
\([(m+n)!]^{-1}\left|\psi_{\zeta^{m+n}}^{(m+n)}(0,t)\right|\)
is bounded as a function of \(t\), continuous on the interval \([0,2\pi]\); this implies that the modulus of the integral

\[ \int_0^{2\pi}\psi(\rho e^{i\varphi},t)e^{-i[(m+n)\varphi-nt]}\,d\varphi,\qquad 0\leq t\leq 2\pi, \]

is bounded for \(0<\rho<1\). This property of functions regular in \(E\) was noted without proof in the note \((^2)\).

Theorem 2. If in the bicylinder \(E\) the function
\[ F(w,z)=\sum_{m,n=0}^{\infty}a_{mn}w^m z^n, \]
where \(a_{00}\) is prescribed, is regular and \(|F(w,z)|\leq 1\), then for \(m+n>0\)

\[ |a_{mn}|\leq (1-|a_{00}|^2)R_1^{-m}R_2^{-n}, \tag{8} \]

and equality holds only for the function

\[ F(w,z)=\bigl(a_{00}R_1^mR_2^n+\eta_{mn}w^m z^n\bigr) \bigl(R_1^mR_2^n+\bar a_{00}\eta_{mn}w^m z^n\bigr)^{-1}, \qquad |\eta_{mn}|=1. \tag{9} \]

Proof. By the hypothesis of Theorem 2, in the disk \(|\zeta|<1\), for any fixed \(t\) with \(0\leq t\leq 2\pi\), the function \(\psi(\zeta,t)\) is regular and \(|\psi(\zeta,t)|\leq 1\). Therefore the function \(\psi(\zeta,t)\), for any fixed \(t\) with \(0\leq t\leq 2\pi\), has almost everywhere on \(|\zeta|=1\) definite limiting values along nontangential paths, forming the boundary function \(\psi(e^{i\varphi},t)\); moreover, for any \(t\) with \(0\leq t\leq 2\pi\), almost everywhere on \(|\zeta|=1\),

\[ |\psi(e^{i\varphi},t)|\leq 1. \tag{10} \]

Obviously \(F(w,z)\in\lambda\), and therefore, by Theorem 1, formulas (6), (7) hold. Using these formulas, for \(m+n>0\) we have \(*\)

\[ a_{mn}R_1^mR_2^n+2a_{00}\bar a_{00}\eta_{mn} =(4\pi^2)^{-1}\int_0^{2\pi}dt\int_0^{2\pi} \psi(e^{i\varphi},t)e^{-i[(m+n)\varphi-nt]}K(\varphi,t)\,d\varphi, \]

where

\[ K(\varphi,t)=\bigl(1+\bar a_{00}e^{i[(m+n)\varphi-nt]}\eta_{mn}\bigr)^2. \]

\[ \text{* In the proof of Theorem 2 below, the author follows the method of G. M. Goluzin } ({}^4). \]

Hence, for \(\eta_{mn}=e^{i\arg a_{mn}}\) (if \(a_{00}=0\), then \(\eta_{mn}\) is arbitrary with \(|\eta_{mn}|=1\)),

\[ \begin{aligned} |a_{mn}|R_1^mR_2^n+2|a_{00}|^2 &=(4\pi^2)^{-1}\left|\int_0^{2\pi}dt\int_0^{2\pi}\psi(e^{i\varphi},t)e^{-i[(m+n)\varphi-nt]}K(\varphi,t)\,d\varphi\right| \\ &\le (4\pi^2)^{-1}\int_0^{2\pi}\left|\int_0^{2\pi}\psi(e^{i\varphi},t)e^{-i[(m+n)\varphi-nt]}K(\varphi,t)\,d\varphi\right|dt \\ &\le (4\pi^2)^{-1}\int_0^{2\pi}dt\int_0^{2\pi}|\psi(e^{i\varphi},t)|\,|K(\varphi,t)|\,d\varphi \\ &\le (4\pi^2)^{-1}\int_0^{2\pi}dt\int_0^{2\pi}|K(\varphi,t)|\,d\varphi =1+|a_{00}|^2, \end{aligned} \tag{11} \]

which leads to estimate (8). Using the proposition for one complex variable* and the fact that equality in

\[ \int_0^{2\pi}dt\int_0^{2\pi}|\psi(e^{i\varphi},t)|\,|K(\varphi,t)|\,d\varphi \le \int_0^{2\pi}dt\int_0^{2\pi}|K(\varphi,t)|\,d\varphi \tag{12} \]

holds only when \(|\psi(e^{i\varphi},t)|=1\) almost everywhere on \(S\),** we conclude that the equality signs in (11) occur only in the case when

\[ \arg \psi(e^{i\varphi},t)e^{-i[(m+n)\varphi-nt]}K(\varphi,t)=\operatorname{const}\pmod{2\pi} \]

and \(|\psi(e^{i\varphi},t)|=1\) almost everywhere on \(S\). But these conditions can be written otherwise:

\[ \arg \psi(e^{i\varphi},t)R(\varphi,t)=\operatorname{const}\pmod{2\pi}, \qquad |\psi(e^{i\varphi},t)R(\varphi,t)|=1, \]

where

\[ R(\varphi,t)= \frac{R_1^mR_2^n+\bar a_{00}\eta_{mn}(R_1e^{i\varphi})^m(R_2e^{i(\varphi-t)})^n} {a_{00}R_1^mR_2^n+\eta_{mn}(R_1e^{i\varphi})^m(R_2e^{i(\varphi-t)})^n}, \]

which is equivalent to the single condition

\[ \psi(e^{i\varphi},t)R(\varphi,t)=\operatorname{const}=e^{i\alpha} \]

almost everywhere on \(S\), which, by virtue of Corollary 1 and the fact that \(F(0,0)=a_{00}\), is fulfilled only in the case of the function (9), for which equality holds in (8).

Theorem 3. If in the domain \(D\)*** the function

\[ F(w,z)=\sum_{m,n=0}^{\infty}a_{mn}w^mz^n, \]

* The equality sign in
\[ \left|\int_0^{2\pi} f(e^{i\varphi})\,d\varphi\right| \le \int_0^{2\pi}|f(e^{i\varphi})|\,d\varphi, \]
where \(f(e^{i\varphi})\) and \(|f(e^{i\varphi})|\) are summable on \([0,2\pi]\), is attained if and only if
\[ \arg f(e^{i\varphi})=\operatorname{const}\pmod{2\pi} \]
for almost all \(\varphi\), \(0\le\varphi\le2\pi\). Indeed, let
\[ \theta=\arg\int_0^{2\pi}f(e^{i\varphi})\,d\varphi. \]
Then
\[ \left|\int_0^{2\pi}f(e^{i\varphi})\,d\varphi\right| =e^{-i\theta}\int_0^{2\pi}f(e^{i\varphi})\,d\varphi = \int_0^{2\pi}|f(e^{i\varphi})|\cos[\arg f(e^{i\varphi})-\theta]\,d\varphi \le \int_0^{2\pi}|f(e^{i\varphi})|\,d\varphi, \]
from which the assertion easily follows.

** In fact, bearing in mind (10), equality in (12), as is known, is obtained only when
\[ \int_0^{2\pi}|\psi(e^{i\varphi},t)|\,|K(\varphi,t)|\,d\varphi = \int_0^{2\pi}|K(\varphi,t)|\,d\varphi \]
for almost all \(t\in[0,2\pi]\). The latter equality, taking (10) into account, holds for each of the indicated almost all \(t\) only when
\[ |\psi(e^{i\varphi},t)|=1 \]
for almost all \(\varphi\in[0,2\pi]\). Thus, equality in (12) holds only when
\[ |\psi(e^{i\varphi},t)|=1 \]
almost everywhere on \(S\).

*** \(D\) is a bounded complete bicircular domain (containing its center \((0,0)\)), whose boundary is twice continuously differentiable and analytically convex from the outside. As A. A. Temlyakov proved (5), the boundary of this domain can be parametrically given in the form
\[ |w|=r_1(\tau),\quad |z|=r_2(\tau),\quad 0\le\tau\le1, \]
where \(r_1(0)=0\), \(r_1(1)<\infty\), \(r_1'(\tau)>0\) on \((0,1]\), and
\[ r_2(\tau)=\exp\left[-\int\frac{\tau}{1-\tau}\,d\ln r_1(\tau)\right]\quad (r_2(1)=0). \]

where \(a_{00}\) is given, is regular, and \(|F(w,z)|\leqslant 1\), then for \(m+n>0\)

\[ |a_{mn}|\leqslant (1-|a_{00}|^2)M^{-1}(m,n), \tag{13} \]

where

\[ M(m,n)\equiv r_1^m\left(\frac{m}{m+n}\right)r_2^n\left(\frac{m}{m+n}\right) \]

(taking \(0^0=1\)); equality for \(m>0,\ n>0\) occurs only for the function

\[ F(w,z)=\bigl(a_{00}M(m,n)+\eta_{mn}w^m z^n\bigr) \bigl(M(m,n)+\overline{a}_{00}\eta_{mn}w^m z^n\bigr)^{-1},\quad |\eta_{mn}|=1. \tag{14} \]

Proof. Since the hypersurface \(|w|=r_1(\tau)\), \(|z|=r_2(\tau)\), \(0\leqslant \tau\leqslant 1\), is composed of the surfaces \(|w|=r_1(\tau)\), \(|z|=r_2(\tau)\) under a continuous change of the parameter \(\tau\) in the segment \([0,1]\), and since the point \((0,0)\) is an interior point of the bicylinder \(|w|<r_1(\tau)\), \(|z|<r_2(\tau)\) for any \(\tau\), \(0<\tau<1\), it follows, by Theorem 2, that

\[ |a_{mn}|\leqslant (1-|a_{00}|^2)r_1^{-m}(\tau)r_2^{-n}(\tau), \]

\(m>0,\ n>0\), where \(\tau\) is an arbitrary number in \((0,1)\). Using the functions of one complex variable

\[ F(0,z)=\sum_{n=0}^{\infty} a_{0n}z^n, \]

\[ F(w,0)=\sum_{m=0}^{\infty} a_{m0}w^m, \]

we have

\[ |a_{0n}|\leqslant (1-|a_{00}|^2)r_2^{-n}(0),\quad n>0;\qquad |a_{m0}|\leqslant (1-|a_{00}|^2)r_1^{-m}(1),\quad m>0. \]

It is not difficult to see that

\[ \max_{0\leqslant \tau\leqslant 1} r_1^m(\tau)r_2^n(\tau)= \begin{cases} r_1^m\left(\dfrac{m}{m+n}\right)r_2^n\left(\dfrac{m}{m+n}\right), & m>0,\quad n>0;\\ r_1^m(1), & m>0,\quad n=0;\\ r_2^n(0), & m=0,\quad n>0; \end{cases} \]

therefore we have the estimate (13). Further, in the case of the bicylinder

\[ \left\{|w|<r_1\left(\frac{m_0}{m_0+n_0}\right),\quad |z|<r_2\left(\frac{m_0}{m_0+n_0}\right)\right\} \]

(\(m_0,n_0\) are any fixed ones from \(m=1,2,\ldots;\ n=1,2,\ldots\)), the equality sign in the estimate

\[ |a_{m_0,n_0}|\leqslant (1-|a_{00}|^2)M^{-1}(m_0,n_0) \]

according to Theorem 2 occurs only for the function

\[ F(w,z)=\bigl(a_{00}M(m_0,n_0)+\eta_{m_0,n_0}w^{m_0}z^{n_0}\bigr) \bigl(M(m_0,n_0)+\overline{a}_{00}\eta_{m_0,n_0}w^{m_0}z^{n_0}\bigr)^{-1}, \]

\[ |\eta_{m_0,n_0}|=1, \]

and since \(m_0,n_0\) are any fixed numbers from \(m=1,2,\ldots;\ n=1,2,\ldots\), equality in (13) for \(m>0,\ n>0\) occurs only for the function (14).

Remark 2. From Theorems 2 and 3, as corollaries, Cauchy’s inequalities for Taylor coefficients and the uniqueness of extremal functions follow easily.

Remark 3. In the case of \(n\) complex variables all arguments are carried out in exactly the same way (for a polycylinder—as for a bicylinder; for the special class of domains \((6)\)—as for the domain \(D\)).

Moscow Regional Pedagogical Institute
named after N. K. Krupskaya

Received
22 III 1962

REFERENCES

¹ I. I. Bavrin, DAN, 126, No. 5 (1959).
² I. I. Bavrin, DAN, 137, No. 3 (1961).
³ G. M. Goluzin, Geometric Theory of Functions of a Complex Variable, Moscow–Leningrad, 1952.
⁴ G. M. Goluzin, Matem. sborn., 26 (68), 7 (1950).
⁵ A. A. Temlyakov, DAN, 120, No. 5 (1958).
⁶ I. I. Bavrin, DAN, 131, No. 6 (1960).