Reports of the Academy of Sciences of the USSR
1960. Volume 134, No. 2

MATHEMATICS

M. A. EVGRAFOV and I. A. CHEGIS

A GENERALIZATION OF THE PHRAGMÉN–LINDELÖF THEOREM FROM ANALYTIC FUNCTIONS TO HARMONIC FUNCTIONS IN SPACE

(Presented by Academician M. V. Keldysh on 3 V 1960)

Theorem 1. Let \(u(r,\varphi,x)\) be a harmonic function in the cylinder
\(r\leq a,\ 0\leq \varphi<2\pi,\ -\infty<x<\infty\). If the conditions

\[ u(a,\varphi,x)=0,\qquad \left|\frac{\partial u}{\partial r}(a,\varphi,x)\right|<C; \tag{1} \]

\[ \max_{(r,\varphi)} |u(r,\varphi,x)|<C\exp e^{\pi |x|/(2+\varepsilon)a},\qquad \varepsilon>0, \tag{2} \]

are satisfied, then \(u(r,\varphi,x)\equiv 0\).

Theorem 2. Let \(u(r,\theta,\varphi)\) be a harmonic function in the cone
\(0<r<\infty,\ 0\leq\varphi<2\pi,\ 0\leq\theta\leq\theta_0<\pi\). If the conditions

\[ u(r,\theta_0,\varphi)=0,\qquad \left|\frac{\partial u}{\partial \theta}(r,\theta_0,\varphi)\right|<C; \tag{1'} \]

\[ \max_{(\theta,\varphi)} |u(r,\theta,\varphi)|<C\exp\left(r+\frac{1}{r}\right)^{\pi/2\theta_0-\varepsilon},\qquad \varepsilon>0, \tag{2'} \]

are satisfied, then \(u(r,\theta,\varphi)\equiv 0\).

The proof of these two results is based on the following uniqueness theorem for Dirichlet series:

Theorem 3. Let

\[ F(z)=\sum_{n=1}^{\infty} a_n e^{\lambda_n z} \]

be an entire function, and

\[ |a_n|^{1/n}<\frac{C}{n^{2+\varepsilon}},\qquad \varepsilon>0; \tag{3} \]

\[ \lim_{n\to\infty}\frac{n}{\lambda_n}=\alpha,\qquad 0<\alpha<\infty,\qquad \lambda_n>0. \tag{4} \]

If, moreover, \(|F(x)|<C,\ -\infty<x<\infty\), then \(F(z)\equiv 0\).

We begin with the proof of Theorem 3. Put

\[ F_\rho(z)=\rho\int_{-\infty}^{\infty} F(t)\exp\left(-e^{\rho(t-z)}+\rho(t-z)\right)\,dt. \tag{5} \]

Lemma 1. If

\[ |F(t)|<Ce^{-\delta |t|},\qquad -\infty<t<\infty,\qquad 0<\delta<\rho, \tag{6} \]

then \(F_\rho(x+iy)\) is regular in the strip

\[ -\infty<x<\infty,\qquad |y|<\pi/2\rho-\eta,\qquad \eta>0, \tag{7} \]

and satisfies in this strip the inequality

\[ |F_\rho(x+iy)|<Ce^{-\delta |x|}. \tag{8} \]

Proof. The uniform convergence of (5) in the strip (7) follows from the estimate

\[ |F(t)\exp(-e^{\rho(t-x-iy)}+\rho(t-x-iy))|< \]

\[ <C\exp(-\delta |t|-e^{\rho(t-x)}\cos \rho y-\rho(t-x)), \]

since there \(\cos \rho y>0\). Let us estimate \(F_\rho(x+iy)\). For \(x>0\) we have

\[ |F_\rho(x+iy)|<C_\rho \int_{-\infty}^{\infty} \exp(-\delta |u+x|-e^{\rho u}\cos \rho y+\rho u)\,du<Ce^{-\delta x}, \]

since \(|u+x|\ge u+x\). The estimate for \(x<0\) is analogous.

Lemma 2. If

\[ F(z)=\sum_{n=1}^{\infty} a_n e^{\lambda_n z} \]

is an entire function and conditions (3), (4) are satisfied, and \(\rho>1/(2+\varepsilon)\alpha\), then

\[ F_\rho(z)=\sum_{n=1}^{\infty} a_n \Gamma\left(\frac{\lambda_n}{\rho}+1\right)e^{\lambda_n z}. \tag{9} \]

Proof. Put \(z=x\) in formula (5) and make the change of variable \(u=e^{\rho(t-x)}\), \(t=\frac{1}{\rho}\ln u+x\). Then we obtain

\[ F_\rho(x)=\int_0^\infty \left(\sum_{n=1}^{\infty} a_n e^{\lambda_n x} u^{\lambda_n/\rho}\right)e^{-u}\,du. \]

Applying the theorem on termwise integration (see (1), Ch. I, § 7.9), we obtain that \(F_\rho(z)\) is representable by the Dirichlet series (9), provided this series converges. The condition for convergence of the series (9) has the form
\(\rho\ge 1/(2+\varepsilon_1)\alpha,\ 0<\varepsilon_1<\varepsilon\), since, when it is fulfilled, the coefficients of the series (9) satisfy the inequality
\[ |a_n|\Gamma(\lambda_n/\rho+1)<C\cdot n^{-\varepsilon_2 n},\qquad \varepsilon_2>0. \]

Lemma 3. Let the function \(f(t+i\lambda)\) be regular in the strip \(|\lambda|\le \gamma\), \(-\infty<t<\infty\), and satisfy there the inequality
\[ |f(t+i\lambda)|<Ce^{-\delta |t|}. \]
Then for the function

\[ \varphi(z)=\int_{-\infty}^{\infty} f(t)e^{-tz}\,dt, \]

regular in the domain \(|\operatorname{Re} z|<\delta\), the estimate
\[ |\varphi(iy)|<Ce^{-\gamma |y|} \]
holds.

Proof. Since \(f(t+i\lambda)\to 0\) as \(t\to \pm\infty\), \(|\lambda|\le \gamma\), we have

\[ \varphi(z)=\int_{i\beta-\infty}^{i\beta+\infty} f(\zeta)e^{-z\zeta}\,d\zeta,\qquad |\beta|\le \gamma, \]

therefore, for \(y>0\),

\[ \varphi(iy)=\int_{i\gamma-\infty}^{i\gamma+\infty} f(\zeta)e^{-iy\zeta}\,d\zeta =e^{-\gamma y}\int_{-\infty}^{\infty} f(t-i\gamma)e^{-ity}\,dt, \]

whence \(|\varphi(iy)|<Ce^{-\gamma y}\), \(y>0\), and analogously for \(y<0\).

Lemma 4. Denote

\[ G_\rho(z)=\prod_{n=1}^{\infty}\left(1-\frac{z^2}{\lambda_n^2}\right) \int_{-\infty}^{\infty} F_\rho(t)e^{-tz}\,dt,\qquad \rho>\frac{1}{(2+\varepsilon)\alpha}. \tag{10} \]

The function \(G_\rho(z)\) can be analytically continued into the half-plane \(\operatorname{Re} z \geqslant 0\) and satisfies there the inequalities

\[ |G_\rho(iy)| < C e^{\pi |y|\left(\alpha-\frac{1}{2\rho}+\varepsilon_3\right)} \qquad (\varepsilon_3>0\text{ arbitrary}); \tag{11} \]

\[ |G_\rho(z)| < C e^{b|z|}. \tag{12} \]

Proof. Consider the functions

\[ \Phi_1(z)=\int_0^\infty F_\rho(t)e^{-tz}\,dt, \qquad \Phi_2(z)=\int_{-\infty}^{0} F_\rho(t)e^{-tz}\,dt. \]

Since, by Lemma 1, \(|F_\rho(t)|<Ce^{-\delta |t|}\), the function \(\Phi_1(z)\) is regular for \(\operatorname{Re} z \geqslant 0\), and the integral for \(\Phi_2(z)\) when \(\operatorname{Re} z<\delta\) can be evaluated by termwise integration. Namely, since \(\rho>1/(2+\varepsilon)\alpha\),

\[ \Phi_2(z)= \int_{-\infty}^{0} \left(\sum_{n=1}^{\infty} a_n \Gamma\left(\frac{\lambda_n}{\rho}+1\right)e^{\lambda_n t}\right)e^{-zt}\,dt = \sum_{n=1}^{\infty} \frac{a_n \Gamma\left(\lambda_n/\rho+1\right)}{\lambda_n-z}. \tag{13} \]

The formula obtained gives an analytic continuation of \(\Phi_2(z)\) to the whole \(z\)-plane. Thus \(G_\rho(z)\) is continued into the half-plane \(\operatorname{Re} z \geqslant 0\), and from representations (10) and (13) it is clear that \(G_\rho(z)\) is regular for \(\operatorname{Re} z \geqslant 0\).

We now prove inequality (11). Apply Lemma 3 to the function \(F_\rho(t+i\lambda)\). By Lemma 1 we obtain for the parameter \(\gamma\) in Lemma 3 the value \(\gamma=\pi/2\rho-\eta\), where \(\eta>0\) is arbitrary. Therefore

\[ \left| \int_{-\infty}^{\infty} F_\rho(t)e^{-iyt}\,dt \right| < C e^{-(\pi/2\rho-\eta)|y|}. \tag{14} \]

For the canonical product

\[ \psi(z)=\prod_{n=1}^{\infty}\left(1-\frac{z^2}{\lambda_n^2}\right), \qquad \lim_{n\to\infty}\frac{n}{\lambda_n}=\alpha, \]

the inequality (see, for example, (2), p. 87)

\[ \max_{|z|=y}|\psi(z)|=\psi(\pm iy)\leqslant C e^{(\pi\alpha+\varepsilon)|y|} \tag{15} \]

holds, where \(\varepsilon>0\) is arbitrary. From (14) and (15), (11) follows.

To prove (12), note that the function \(\Phi_2(z)\) is bounded on a sequence of circles \(|z|=r_k\), \(r_k\to\infty\), \(r_{k+1}-r_k<C\), since outside the circles \(|z-\lambda_n|<n^{-2}\) we have

\[ |\Phi_2(z)|<C\sum_{n=1}^{\infty} n^{-\varepsilon n}n^2<C, \]

and the sum of the diameters of the circles inside which this inequality is not valid is less than a constant. This observation and inequality (15) prove (12).

Proof of Theorem 3. Note that without loss of generality we may assume that \(F(x)\) satisfies inequality (6), since otherwise we could take the function \(F(z)e^{-\lambda_0 z/2}\), for which all the conditions would be fulfilled.

Now apply Lemmas 1–4 successively, taking \(\rho=1/(2+\varepsilon_1)\alpha\). The constructed function \(G_\rho(z)\) is regular in the half-plane \(\operatorname{Re} z \geqslant 0\) and satisfies inequalities (11), (12). But from the Phragmén—Lindelöf principle it follows (see (3), Sec. III, Ch. 6, No. 327) that such a \(G_\rho(z)\equiv 0\). This, in turn, means that \(F_\rho(z)\equiv 0\), i.e. \(a_n=0\), \(n=1,2,\ldots\).

Proof of Theorem 1. Since \(u(r,\varphi,x)\) has in the cylinder \(r\leqslant a,\ 0\leqslant \varphi<2\pi,\ -\infty<x<\infty\) two continuous derivatives and satisfies the boundary condition \(u(a,\varphi,x)=0\), it can be expanded in the series

\[ u(r,\varphi,x)= \sum_{n=-\infty}^{\infty}\sum_{m=1}^{\infty} e^{in\varphi}J_n\left(\lambda_{n,m}\frac{r}{a}\right) \frac{1}{J_n'(\lambda_{n,m})} \left(a_{n,m}e^{\lambda_{n,m}x/a}+b_{n,m}e^{-\lambda_{n,m}x/a}\right), \]

where \(J_n(x)\) are Bessel functions; \(\lambda_{n,m}\) is the \(m\)-th positive root, in order of magnitude, of \(J_n(x)=0\). Put

\[ v(r,\varphi,x)=\sum_{n=-\infty}^{\infty}\sum_{m=1}^{\infty} \frac{a_{n,m}}{J'_n(\lambda_{n,m})}\,e^{in\varphi}J_n\!\left(\lambda_{n,m}\frac{r}{a}\right)e^{\lambda_{n,m}x/a}. \]

It is clear that conditions (1), (2) hold also for \(v(r,\varphi,x)\), if only they hold for \(u(r,\varphi,x)\). Therefore it suffices to consider \(v(r,\varphi,x)\).

By virtue of the boundedness of \(\dfrac{\partial v}{\partial r}(a,\varphi,x)\) we have

\[ \left|\int_0^{2\pi}\frac{\partial v}{\partial r}(a,\varphi,x)e^{-in\varphi}\,d\varphi\right| <C\left|\sum_{m=1}^{\infty} a_{n,m}\lambda_{n,m}e^{\lambda_{n,m}x/a}\right|<C. \]

Consider the entire function

\[ f_n(z)=\sum_{m=1}^{\infty}a_{n,m}\lambda_{n,m}e^{\lambda_{n,m}z} \]

and show that it satisfies all the conditions of Theorem 3.

First, it is well known that

\[ \lim_{m\to\infty}\frac{m}{\lambda_{n,m}}=\frac{1}{\pi}, \]

i.e. condition (4) is fulfilled and \(\alpha=1/\pi\). Secondly, from condition (2) of Theorem 1 we have

\[ \int_0^{2\pi}\int_0^a |v(r,\varphi,x)|^2\,r\,dr\,d\varphi = \sum_{n=-\infty}^{\infty}\sum_{m=1}^{\infty}|a_{n,m}|^2 e^{2\lambda_{n,m}x/a} <C\exp 2\rho x, \]

where \(\rho=\pi/(2+\varepsilon)a\). Consequently,

\[ |a_{n,m}|<C\min_x \exp\left\{\rho x-\lambda_{n,m}\frac{x}{a}\right\} = C\exp\left[-\frac{\lambda_{n,m}}{a\rho}\left(\ln\frac{\lambda_{n,m}}{a\rho}-1\right)\right], \]

which gives us (for fixed \(n\))

\[ (\lambda_{n,m}|a_{n,m}|)^{1/m}<Cm^{-2-\varepsilon}. \]

Thus all the conditions of Theorem 3 are satisfied, \(f_n(z)\equiv0\), and hence also \(v(r,\varphi,x)\equiv0\). The theorem is proved.

The proof of Theorem 2 is carried out in exactly the same way; only, for \(u(r,\theta,\varphi)\), one uses the expansion into the series

\[ u(r,\theta,\varphi)=\sum_{n=-\infty}^{\infty}\sum_{m=1}^{\infty} e^{in\varphi}H_{n,m}(\theta)\left(a_{n,m}r^{\lambda_{n,m}}+b_{n,m}r^{-\lambda_{n,m}-1}\right). \]

Here \(\lambda_{n,m}\) are eigenvalues, and \(H_{n,m}(\theta)\) are eigenfunctions of the problem

\[ H''(\theta)+H'(\theta)\operatorname{ctg}\theta+ \left(\lambda(\lambda+1)+\frac{m^2}{\sin^2\theta}\right)H(\theta)=0, \qquad |H(0)|<\infty,\quad H(\theta_0)=0. \]

By refining Theorem 3 one could obtain, in Theorems 1 and 2, the replacement of conditions (2) and \((2')\) respectively by the conditions

\[ \ln\max_{(r,\varphi)}|u(r,\varphi,x)|=o\!\left(e^{\pi|x|/2a}\right),\qquad x\to\pm\infty, \]

\[ \ln\max_{(\theta,\varphi)}|u(r,\theta,\varphi)| =o\!\left(\left(r+\frac{1}{r}\right)^{\pi/2\theta_0}\right),\qquad r\to0,\infty. \]

A further strengthening of Theorems 1 and 2 is already impossible. In the case of Theorem 2 and \(\theta_0=\pi/2\), this is seen from the example

\[ u(r,\theta,\varphi)=\cos(\mu r\sin\theta\cos\varphi)\operatorname{sh}(\mu r\cos\theta) \qquad (\mu>0\text{ arbitrary}). \]

Theorems 1 and 2 contain the answer to certain questions posed by S. N. Mergelyan in \((^4)\).

Received
28 IV 1960

REFERENCES

\(^1\) E. C. Titchmarsh, Theory of Functions, Moscow, 1951.
\(^2\) M. A. Evgrafov, Asymptotic Estimates and Entire Functions, Moscow, 1957.
\(^3\) G. Pólya, G. Szegő, Problems and Theorems in Analysis, Moscow, 1956.
\(^4\) S. N. Mergelyan, UMN, 11, 5 (71), 3 (1956).