UDC 517.531 + 517.51

MATHEMATICS

B. I. Korenblum

NONTRIVIALITY CONDITIONS FOR CERTAIN CLASSES OF FUNCTIONS ANALYTIC IN AN ANGLE, AND QUASIANALYTICITY PROBLEMS

(Presented by Academician A. Yu. Ishlinskii, June 2, 1965)

1°. Let \(G\) be a connected set of points of the complex plane or, more generally, of some Riemann surface, and let \(\{A_0=1, A_1, A_2,\ldots\}\) be a given sequence of nonnegative numbers (some of which may be equal to \(+\infty\)). We shall denote by \(D\{G; A_n\}\) the class of functions \(f(z)\) \((z\in G)\), infinitely differentiable on \(G\), for which

\[ \sup_{z\in G} |f^{(n)}(z)| \leq C A_n \quad (n=0,1,2,\ldots) \tag{1} \]

(the constant \(C\) depends on \(f\)). Obviously, the functions \(f(z)\equiv \mathrm{const}\) belong to any class \(D\{G; A_n\}\); if, apart from them, the given class contains not a single function, we shall call it trivial. We pose the following problem:

A. Given a set \(G\); find necessary and sufficient conditions which the sequence \(\{A_n\}_0^\infty\) must satisfy in order that the class \(D\{G; A_n\}\) be nontrivial.

As an example, we point out that if \(G\) is the entire complex plane \(K\) (without the point at infinity), then any class \(D\{K; A_n\}\) is trivial (Liouville’s theorem); an analogous result holds when \(G\) is the Riemann surface of the function \(\log z\). At the other extreme, when \(G\) is a bounded set, the necessary and sufficient condition for nontriviality of the class \(D\{G; A_n\}\) is \(A_1>0\).

In the present paper, problem A is solved for the case in which \(G\) is an angle in the complex plane or, more generally, on the Riemann surface of the function \(\log z\). In doing so, a connection is revealed between problem A and the classical problem of quasianalyticity, and with an analogous problem in the complex domain.

2°. Definitions.

1) \(I_\alpha\) \((0\leq \alpha<\infty)\) denotes the angle on the Riemann surface of the function \(\log z\) defined by the inequalities \(0\leq |z|<\infty\), \(-\pi\alpha/2\leq \arg z\leq \pi\alpha/2\) (for \(\alpha<2\), instead of the Riemann surface one may consider the ordinary complex plane); \(\gamma_\alpha\) denotes the boundary of \(I_\alpha\). Thus \(I_0=\gamma_0=[0,\infty]\), and \(\gamma_1\) is the imaginary axis.

2) The classes \(D\{I_\alpha; A_n\}\) are denoted by \(D_\alpha\{A_n\}\) \((0\leq \alpha<\infty)\); the classes \(D\{\gamma_\alpha; A_n\}\) are denoted by \(\widetilde D_\alpha\{A_n\}\) \((0<\alpha\leq 1)\).

3) The class \(D_\alpha\{A_n\}\) \((0\leq \alpha<\infty)\) is called quasianalytic at the point \(0\) if there does not exist a function \(f(z)\not\equiv 0\) belonging to this class and such that \(f^{(n)}(0)=0\) \((n=0,1,\ldots)\).

Theorem 1. Let \(0\leq \alpha\leq 1\). For the nontriviality of the class \(D_\alpha\{A_n\}\) it is necessary and sufficient that

\[ \inf_{n>0}\left(n^{1-\alpha} A_n^{1/n}\right)>0. \tag{2} \]

This same condition is necessary and sufficient also for the nontriviality of the class \(\widetilde D_\alpha\{A_n\}\).

Theorem 2. Let \(\alpha>1\). For the nontriviality of the class \(D_\alpha\{A_n\}\) it is necessary and sufficient that

\[ \int^\infty r^{-\alpha/(\alpha-1)}\log T(r)\,dr<\infty, \tag{3} \]

where \(T(r)=\sup_{n>0}(r^n/A_n)\).

Theorem 3. In order that the class \(D_\alpha\{A_n\}\) \((0\leq \alpha<\infty)\) be quasi-analytic at the point \(0\), it is necessary and sufficient that the class \(D_{\alpha+2}\{A_n\}\) be trivial, i.e. (by virtue of Theorem 2) that

\[ \int^\infty r^{-(\alpha+2)/(\alpha+1)}\log T(r)\,dr=\infty. \tag{4} \]

Remarks.

1) As is known \(((1),\) p. 55), condition (3) is equivalent to the inequality

\[ \sum^\infty \beta_n^{-1/(\alpha-1)}<\infty, \]

and condition (4) is equivalent to the equality

\[ \sum^\infty \beta_n^{-1/(\alpha+1)}=\infty,\qquad \text{where }\ \beta_n=\inf_{k\geq n} A_k^{1/k}. \]

2) For \(\alpha>1\) condition (2) remains necessary for the nontriviality of the class \(D_\alpha\{A_n\}\), but ceases to be sufficient.

3) For \(\alpha=0\) Theorem 3 gives the known classical condition of quasi-analyticity on a line. For \(\alpha=1\) we obtain the condition of quasi-analyticity in a half-plane, established earlier \((^2)\)*.

4) For \(\alpha=1\) and \(\alpha=0\) Theorem 1 is essentially known, since it follows easily from the more precise results of A. N. Kolmogorov \((^3)\) and A. Gorny \((^4)\), who established inequalities for upper bounds of successive derivatives of a function on an axis and a half-axis.

\(3^\circ\). We proceed to the proof of Theorem 1. Let \(0<\alpha\leq 1\) and

\[ \inf_{n>0}\left(n^{1-\alpha}A_n^{1/n}\right)=0. \tag{5} \]

We shall show that the class \(D_\alpha\{A_n\}\) is trivial. Obviously, we may assume that the infimum in (5) is not attained, i.e. that \(A_n>0\), \((n>0)\).

Lemma. Under condition (5) there exists a function \(F(z)\), analytic inside the angle \(K\setminus I_\alpha\) (the complement of \(I_\alpha\) to \(K\)) and continuous on its boundary, as well as a sequence of positive numbers \(\{B_n\}_0^\infty\), such that

\[ F(0)=1,\qquad |F(z)|\leq B_n|z|^{-n}\quad (z\in K\setminus I_\alpha;\ n=0,1,\ldots), \tag{6} \]

\[ \inf_{n>0}\left[\frac1n(A_nB_n)^{1/n}\right]=0. \tag{7} \]

Proof of the lemma. From the classical Watson problem it follows \(((^1),\) pp. 55—56) that, for the possibility of constructing a function \(F(z)\) satisfying in the angle \(K\setminus I_\alpha\) the inequalities (6), it is sufficient that the numbers \(B_n\) satisfy the conditions

\[ B_1\leq B_2^{1/2}\leq B_3^{1/3}\leq\cdots;\qquad \sum_{n=1}^\infty B_n^{-1/n(2-\alpha)}<\infty. \]

We shall show that it is possible to satisfy simultaneously these conditions and condition (7). By virtue of (5), one can indicate a sequence of indices
\(n_0=0<n_1<n_2<\cdots\) such that
\(A_{n_k}\leq [k^3(2-\alpha)n_k^{1-\alpha}]^{-n_k}\) \((k=1,2,\ldots)\).
Put \(B_n=(k^2n_k)^{n(2-\alpha)}\) \((n_{k-1}<n\leq n_k;\ k=1,2,\ldots)\).

* In paper \((^2)\) the problem of quasi-analyticity for a disk is solved; it is not difficult, however, to show that for a half-plane the conditions of quasi-analyticity are the same as for a disk.

Obviously,

\[ \sum_{n=1}^{\infty} B_n^{-1/n(2-\alpha)} = \sum_{k=1}^{\infty}\sum_{n=n_{k-1}+1}^{n_k}\frac{1}{k^2 n_k} < \sum_{k=1}^{\infty}\frac{1}{k^2}<\infty . \]

On the other hand,

\[ \frac{1}{n_k}(A_{n_k}B_{n_k})^{1/n_k}\le k^{-(2-\alpha)}\to 0. \]

The lemma is proved.

Now let \(f(z)\in D_\alpha\{A_n\}\) \((0<\alpha\le 1)\). Consider the integral

\[ \int_{\gamma_\alpha}\frac{F(\varepsilon \zeta)f(\zeta)\,d\zeta}{(\zeta-z)^2} \qquad (\varepsilon>0), \]

where \(z\) is an interior point of \(I_\alpha\), and the boundary \(\gamma_\alpha\) is traversed in the positive direction. Since

\[ f(\zeta)=f(0)+f'(0)\zeta+\cdots+ \frac{f^{(n-1)}(0)}{(n-1)!}\zeta^{n-1}+R_n(\zeta), \]

\[ |R_n(\zeta)|\le \frac{C A_n}{n!}|\zeta|^n \qquad (\zeta\in I_\alpha), \]

it follows that

\[ \left| \int_{\gamma_\alpha}\frac{F(\varepsilon\zeta)f(\zeta)\,d\zeta}{(\zeta-z)^2} \right| = \left| \int_{\gamma_\alpha}\frac{F(\varepsilon\zeta)R_n(\zeta)\,d\zeta}{(\zeta-z)^2} \right| \le \frac{C A_nB_n}{\varepsilon^n n!} \int_{\gamma_\alpha}\frac{|d\zeta|}{|\zeta-z|^2}, \]

because integrals of the form

\[ \int_{\gamma_\alpha}\frac{F(\varepsilon\zeta)p(\zeta)\,d\zeta}{(\zeta-z)^2}, \]

where \(p(\zeta)\) is an arbitrary polynomial, are equal to zero. Taking the infimum over \(n\) and using (7), we obtain

\[ \left| \int_{\gamma_\alpha}\frac{F(\varepsilon\zeta)f(\zeta)\,d\zeta}{(\zeta-z)^2} \right| \le C\int_{\gamma_\alpha}\frac{|d\zeta|}{|\zeta-z|^2} \inf_{n>0}\left( \frac{1}{\varepsilon}\sqrt[n]{\frac{A_nB_n}{n!}} \right)^n =0. \]

Passing to the limit as \(\varepsilon\to 0\), we find

\[ \int_{\gamma_\alpha}\frac{f(\zeta)\,d\zeta}{(\zeta-z)^2}=0 \quad \text{or} \quad f'(z)=0 \qquad (z\in I_\alpha). \]

With a slight modification of this argument, one can also cover the case \(\alpha=0\).

Thus, the necessity of condition (3) is proved. Its sufficiency follows from the fact that for \(0\le \alpha\le 1\) there exist functions \(g(z)\) such that

\[ \sup_{z\in I_\alpha}|g^{(n)}(z)|\le Ca^n/n^{n(1-\alpha)} \qquad (n=0,1,2,\ldots), \tag{8} \]

where \(C,a\) are certain constants. Obviously, \(g(\varepsilon z)\), for sufficiently small \(\varepsilon>0\), will belong to the class \(D_\alpha\{A_n\}\), if inequality (2) is satisfied. An example of such a function may be the Mittag-Leffler function ([5], p. 265)

\[ E_{2-\alpha}(-z)= \sum_{n=0}^{\infty}\frac{(-1)^n z^n}{\Gamma[1+(2-\alpha)n]}. \]

This is an entire function of order \((2-\alpha)^{-1}\), bounded in the angle \(I_\alpha\). In an additional angle \(K\setminus I_\alpha\) it satisfies the estimate

\[ \left| E_{2-\alpha}(-z)- \frac{1}{2-\alpha}e^{(-z)^{1/(2-\alpha)}} \right| < \frac{M}{|z|}. \]

Using these properties, it is easy to prove, by applying Cauchy’s integral over a circle of suitable radius to estimate derivatives, that \(E_{2-\alpha}(-z)\) satisfies inequalities (8).

The second assertion of Theorem 1 is easily reduced to the first. For this purpose one considers the Cauchy-type integral

\[ \int_{\gamma_\alpha}\frac{f(\zeta)\,d\zeta}{(\zeta-z)^2} \]

separately in the domain \(I_\alpha-\gamma_\alpha\) and in the domain \(K\setminus I_\alpha\).

4°. To prove Theorems 2 and 3, suppose that \(f(z)\in D_{\alpha}\{A_n\}\), \(0\leq \alpha<\infty\), and consider the Laplace transform of the function \(f(z)=g(re^{i\varphi})\), \((l_{\varphi}\) is the ray \(\varphi=\mathrm{const},\ 0\leq r<\infty)\):

\[ F(\zeta)=F(\rho e^{i\theta}) =\int_{l_\varphi} f(z)e^{-z\zeta}\,dz =e^{i\varphi}\int_{0}^{\infty} f(re^{i\varphi})e^{-r\rho e^{i(\varphi+\theta)}}\,dr . \tag{9} \]

Obviously, for each fixed \(\varphi\) \((-\alpha\pi/2\leq \varphi\leq \alpha\pi/2)\), this integral converges in the angle
\(-\pi/2-\varphi<\theta<\pi/2-\varphi,\ 0<\rho<\infty\), of the Riemann surface of the function \(\log \zeta\). Integrating (9) by parts, we obtain

\[ |F(\zeta)|\leq \frac{CA_0}{\rho\cos(\varphi+\theta)}; \quad \left|F(\zeta)-\frac{f(0)}{\zeta}\right| \leq \frac{CA_1}{\rho^2\cos(\varphi+\theta)}; \ldots \]

\[ \ldots;\quad \left|F(\zeta)-\frac{f(0)}{\zeta}-\ldots-\frac{f^{(n-1)}(0)}{\zeta^n}\right| \leq \frac{CA_n}{\rho^{n+1}\cos(\varphi+\theta)} . \]

Denoting by \(P_\varphi\) the half-plane on the Riemann surface of the function \(\log\zeta\), defined by the inequalities
\(-\pi/2-\varphi<\theta<\pi/2-\varphi,\ \rho\cos(\varphi+\theta)\geq 1\), we obtain

\[ |F(\zeta)|\leq CA_0;\quad \left|F(\zeta)-\frac{f(0)}{\zeta}\right| \leq \frac{CA_1}{\rho};\ldots \]

\[ \ldots;\quad \left|F(\zeta)-\frac{f(0)}{\zeta}-\ldots-\frac{f^{(n-1)}(0)}{\zeta^n}\right| \leq \frac{CA_n}{\rho^n}\quad (\zeta\in P_\varphi). \tag{10} \]

Rotating the line \(l_\varphi\) in (9), we find, by virtue of the analyticity of \(f(z)\), that \(F(\zeta)\) can be analytically continued to the angle
\(-(\alpha+1)\pi/2<\theta<(\alpha+1)\pi/2\), and in the domain
\[ S_\alpha=\bigcup_{|\varphi|\leq \alpha\pi/2} P_\varphi \]
the estimates (10) are preserved.

It is now easy to prove the necessity in Theorem 2 and the sufficiency in Theorem 3. Indeed, if \(f(0)=f'(0)=\ldots=0\), then \(F(\zeta)\), by (10), satisfies the conditions of Watson’s problem in the domain \(S_\alpha\). It is easy to show that the conditions for the solvability of Watson’s problem for the domain \(S_\alpha\) and for the angle \(I_{\alpha+1}\) are the same; consequently, when condition (4) is fulfilled, one must have \(F(\zeta)\equiv 0,\ f(z)\equiv 0\).

Under the conditions of Theorem 2 one must consider the function
\[ F_1(\zeta)=F_1(\rho e^{i\theta}) =F(\rho e^{i\theta})-F(\rho e^{i(\theta+2\pi)}) \]
(\(\theta+2\pi\) denotes passage to the next sheet of the Riemann surface). Denoting by \(\widetilde S_\alpha\) the domain consisting of the points \(\zeta=\rho e^{i\theta}\) for which \(\rho e^{i\theta}\in S_\alpha,\ \rho e^{i(\theta+2\pi)}\in S_\alpha\), we obtain, by virtue of (10),

\[ |F_1(\zeta)|\leq 2CA_n/|\zeta|^n \quad (n=0,1,2,\ldots;\ \zeta\in \widetilde S_\alpha). \]

Using Watson’s problem once again, we obtain (3).

The sufficiency in Theorem 2 follows from the fact that, when inequality (3) is fulfilled, one can construct in the angle
\(-(\alpha-1)\pi/2\leq \theta\leq (\alpha-1)\pi/2\)
an analytic function \(F(\zeta)=F(\rho e^{i\theta})\) satisfying the conditions

\[ \int_{0}^{\infty} |F(\rho e^{i\theta})|\rho^n\,d\rho \leq A_n \quad (|\theta|\leq (\alpha-1)\pi/2;\ n=0,1,2,\ldots). \]

Then the Laplace transform

\[ f(z)=f(re^{i\varphi}) =e^{i\theta}\int_{0}^{\infty} F(\rho e^{i\theta})e^{-r\rho e^{i(\theta+\varphi)}}\,d\rho \]

will satisfy the estimates (1) in a certain angle of opening \(\alpha\pi\). If \(\alpha\geq 2\), then the function
\[ f_1(re^{i\varphi})=f(re^{i\varphi})-f(re^{i(\varphi+2\pi)}) \]
will satisfy the estimates (1) in an angle of opening \((\alpha-2)\pi\), but for it, in addition,
\[ f_1(0)=f_1'(0)=f_1''(0)=\ldots=0. \]
This completes the proof of Theorems 2 and 3.

Kiev Civil Engineering Institute

Received
29 V 1965

REFERENCES

\(^{1}\) S. Mandelbrojt, Adjacent Series. Regularization of Sequences. Applications, Moscow, 1955.
\(^{2}\) B. I. Korenblum, DAN, 164, No. 1 (1965).
\(^{3}\) A. N. Kolmogorov, Uch. zap. Mosk. univ., 30 (mathematics), 3 (1939).
\(^{4}\) A. Gorny, Acta Math., 71, 317 (1939).
\(^{5}\) L. Bieberbach, Lehrbuch d. Funktionentheorie, 2, Leipzig—Berlin, 1927.