MATHEMATICS

A. A. Bonami

On Mean Moduli of Analytic Functions

(Presented by Academician A. N. Kolmogorov on 17 XI 1956)

Denote by \(S(\alpha,\beta)\) the strip \(\alpha<\operatorname{Re} z<\beta\) of finite width, and by \(S[\alpha,\beta]\) the strip \(\alpha\leqslant \operatorname{Re} z\leqslant \beta\). Hardy, Ingham, and Pólya \((^{1})\) considered means of the form

\[ \varphi_p(x,y)=\frac{1}{2y}\int_{-y}^{y}|f(x+i\eta)|^p\,d\eta \quad (p>0), \tag{1} \]

where the functions \(f(z)=f(x+iy)\) satisfy the following condition E:

a) \(f(z)\) is analytic in \(S(\alpha,\beta)\) and continuous in every finite part of \(S[\alpha,\beta]\);

b) \(f(z)=O(e^{k|y|})\) uniformly in \(S(\alpha,\beta)\), \(0<k<\dfrac{\pi}{\beta-\alpha}\).

The present paper contains analogous theorems in which the functions are free of the restriction of continuity on the boundary of the domain.

In what follows, as a rule, we shall take \(\alpha=0\) and \(\beta=\pi\), although the results are valid in the general case. We shall always assume that the functions under consideration are analytic inside the strip.

Definition 1. \(f(z)\in A\) (respectively \(H_p\)) if the subharmonic function \(\ln^{+}|f(z)|\) (respectively \(|f(z)|^p\)) has a harmonic majorant in \(S\).

Definition 2. \(f(z)\in \mathfrak{M}_p\) if \(\varphi_p(x,y)\leqslant M(f)=\mathrm{const}\) in \(S\).

Lemma 1. Let \(u(x,y)\) be a nonnegative subharmonic function in \(S(0,\pi)\); let \(\lambda(t)\geq 0\) be a continuous nondecreasing function \((t>0)\), and

\[ \int_{-\infty}^{\infty} u(x,y)\lambda(|y|)\,dy<M=\mathrm{const}. \]

Then for every \(\delta\) \((0<\delta<\pi/2)\) in the strip \(S(\delta,\pi-\delta)\) the following hold:

1) \(\lambda(|y|+\delta)u(x,y)\to 0\) uniformly as \(|y|\to\infty\);

2) \[ u(x,y)\leqslant \frac{M}{\delta\lambda(|y|+\delta)}. \]

Lemma 2. If a nonnegative subharmonic function in \(S(0,\pi)\), \(u(x+iy)\), satisfies the condition

\[ \int_{-\infty}^{\infty} u(x+iy)e^{-|y|}\,dy<M(u)=\mathrm{const} \quad (0<x<\pi), \tag{2} \]

then it has a harmonic majorant in \(S(0,\pi)\).

Indeed, by means of the substitution

\[ \frac{w-1}{w+1}=ie^{iz} \tag{3} \]

\(S(0,\pi)\) is mapped onto the disk \(|w|<1\), and condition (2) is transformed into

\[ \int_\gamma u(w)\,|dw| \leqslant \mathrm{const}, \]

where \(\gamma\) is the image of the line \(\operatorname{Re} z=\mathrm{const}\).

With the aid of the images of the lines \(x=\delta,\ x=\pi-\delta\) and of line segments perpendicular to the \(x\)-axis, one can form a convex contour \(\Gamma\) approximating the circle \(|w|=1\). On the basis of Lemma 1 we have

\[ \int_\Gamma u(w)|dw| \leqslant 4\left(M+\frac{\varepsilon}{\operatorname{tg}\delta}\right) \qquad (\varepsilon>0). \]

On the basis of Gabriel’s theorem \(\left({}^{2}\right)\)

\[ \int_{|w|=\rho<1} u(w)|dw| \leqslant 8\left(M+\frac{\varepsilon}{\operatorname{tg}\delta}\right). \]

Since \(\varepsilon\) is arbitrary, we conclude (3), that \(u(w)\) has a harmonic majorant in \(|w|<1\), and \(u(z)\) in \(S(0,\pi)\).

Corollary. \(\mathfrak{M}_p \subset H_p\).

Indeed, let

\[ \Phi(x,t)=\int_0^t \{|f(x+i\sigma)|^p+|f(x-i\sigma)|^p\}\,d\sigma . \]

Then

\[ \int_{-\infty}^{\infty} e^{-|y|}|f(x+iy)|^p\,dy = \int_0^\infty e^{-y}\Phi(x,y)\,dy \leqslant 2M, \]

and on the basis of Lemma 2 we obtain the assertion.

Theorem 1. If \(f(z)\in\mathfrak{M}_p\), then:

a) \(f(z)\) has finite angular boundary values almost everywhere on the lines \(x=0\) and \(x=\pi\);

b) these boundary values are summable to the power \(p\) on every finite interval of variation of \(y\); moreover

\[ \varphi_p(\xi,y)\leqslant M(f) \qquad \text{for } \xi=0 \text{ and } \xi=\pi; \]

c) \(f(z)=B(z)g(z)\), where \(B(z)\) is the Blaschke product for the strip and the zeros of \(f(z)\):

\[ B(z)=\operatorname{ctg}^{\lambda}\left(\frac{z}{2}+\frac{\pi}{4}\right) \prod_{k=1}^{\infty} \frac{\sin \dfrac{z-z_k}{2}}{\sin \dfrac{z+z_k}{2}} \left(\frac{\cos \overline{z}_k}{\cos z_k}\right)^{1/2}; \]

\(\lambda\) is the multiplicity of the point \(z=\pi/2\) as a zero of \(f(z)\), and the convergence condition is

\[ \sum_{(z_k)} e^{-|y_k|}\sin x_k < +\infty; \]

\(g(z)\ne 0\) in \(S(0,\pi)\); \(g(z)\in\mathfrak{M}_p\), and \(g(z)\) has almost everywhere on the lines \(x=0\) and \(x=\pi\) angular boundary values whose moduli are equal to \(|f(z)|\);

d) if \(E\) is any bounded measurable set in \(-\infty<y<\infty\), then

\[ \int_E |f(x+iy)|^pdy \to \int_E |f(\xi+iy)|^pdy \qquad \text{as } x\to \xi \;(\xi=0 \text{ and } \xi=\pi); \]

d) for every bounded measurable set \(E\)
\[ \int_E |f(x+iy)-f(\xi+iy)|^p\,dy \to 0 \quad \text{as } x\to \xi\;(\xi=0 \text{ and } \xi=\pi). \]

Let us consider propositions that may be called theorems of the Phragmén–Lindelöf type for the means \(\varphi_p(x,y)\). The proofs are based on the following integral representations of the classes \(A\) and \(H_p\).

Lemma 3. Every positive harmonic function in \(S(0,\pi)\) is representable in the form
\[ u(x,y)=\frac1{2\pi}\int_{-\infty}^{\infty} \frac{\sin x\,\operatorname{ch}\eta}{\Delta_1}\,d\psi_0(\eta) +\frac1{2\pi}\int_{-\infty}^{\infty} \frac{\sin x\,\operatorname{ch}\eta}{\Delta_2}\,d\psi_\pi(\eta) \]
\[ +(Me^y+Ne^{-y})\sin x, \]
where
\[ \Delta_1=\operatorname{ch}(y-\eta)-\cos x,\qquad \Delta_2=\operatorname{ch}(y-\eta)+\cos x; \]
\(\psi_0(\eta)\) and \(\psi_\pi(\eta)\) are nondecreasing functions bounded for \(-\infty<\eta<\infty\); \(M\) and \(N\) are nonnegative constants determined by the relations
\[ M=\lim_{y\to\infty} e^{-y}u\!\left(\frac{\pi}{2},y\right), \qquad N=\lim_{y\to-\infty} e^{y}u\!\left(\frac{\pi}{2},y\right). \]

Theorem 2. In order that \(f(z)\in A\) (respectively \(H_p\)), it is necessary and sufficient that \(f(z)\) can be represented in the form
\[ f(z)=e^{i\lambda}\exp i\{Me^{iz}-Ne^{-iz}\}B(z)D(z)g(z), \tag{4} \]
where \(\lambda, M, N\) are real constants; \(B(z)\) is a Blaschke product;
\[ D(z)=\exp\frac1{2\pi i}\left\{ \int_{-\infty}^{\infty}\ln p_0(\eta)\, \frac{e^{iz}+e^\eta}{(1-e^{iz}e^\eta)\operatorname{ch}\eta}\,d\eta + \int_{-\infty}^{\infty}\ln p_\pi(\eta)\, \frac{e^{iz}-e^\eta}{(1+e^{iz}e^\eta)\operatorname{ch}\eta}\,d\eta \right\}, \]
\[ p_0(\eta)\ge 0;\qquad p_\pi(\eta)\ge 0;\qquad \int_{-\infty}^{\infty} e^{-|\eta|}\ln p_0(\eta)\,d\eta<+\infty; \]
\[ \int_{-\infty}^{\infty} e^{-|\eta|}\ln p_\pi(\eta)\,d\eta<+\infty; \]
\[ g(z)=\exp\frac1{2\pi i}\left\{ \int_{-\infty}^{\infty}\frac{e^{iz}+e^\eta}{1-e^{iz}e^\eta}\,d\psi_0(\eta) + \int_{-\infty}^{\infty}\frac{e^{iz}-e^\eta}{1-e^{iz}e^\eta}\,d\psi_\pi(\eta) \right\}; \]
\(\psi_0(\eta)\) and \(\psi_\pi(\eta)\) have bounded variation in \((-\infty,\infty)\) and derivative almost everywhere equal to zero.

For the classes \(H_p\), in addition,
\[ M\ge 0;\qquad N\ge 0;\qquad \int_{-\infty}^{\infty} e^{-|\eta|}[p_0(\eta)]^p\,d\eta<+\infty;\qquad \int_{-\infty}^{\infty} e^{-|\eta|}[p_\pi(\eta)]^p\,d\eta<+\infty; \]
\(\psi_0(\eta)\) and \(\psi_\pi(\eta)\) are nondecreasing functions.

Theorem A. In order that \(f(z)\in H_p\) have bounded means \(\varphi_p(x,y)\) in \(S(\alpha,\beta)\), it is necessary and sufficient that
\[ \varphi_p(\alpha,y)\le K,\qquad \varphi_p(\beta,y)\le K \quad (K=\mathrm{const}). \tag{5} \]
Moreover, from (5) it follows that \(\varphi_p(x,y)\le K\) for \(\alpha\le x\le\beta\).

Let us show the sufficiency of the conditions. From (4) it follows that
\[ |f(x+iy)|\le \exp\frac1{2\pi}\left\{\int_{-\infty}^{\infty}\ln p_0(\eta)\frac{\sin x}{\Delta_1}\,d\eta\right\} \exp\frac1{2\pi}\left\{\int_{-\infty}^{\infty}\ln p_\pi(\eta)\frac{\sin x}{\Delta_2}\,d\eta\right\}. \]
Applying Hölder’s inequality and the inequality between the arithmetic mean and the geometric mean, we obtain

\[ \varphi_p(x,y)\leq \left\{ \frac1{2y}\int_{-y}^{y}dt\int_{-\infty}^{\infty} [p_0(\eta)]^p\,\frac{\sin x}{2(\pi-x)\Delta_1}\,d\eta \right\}^{(\pi-x)/\pi} \times \left\{ \frac1{2y}\int_{-y}^{y}dt\int_{-\infty}^{\infty} [p_\pi(\eta)]^p\,\frac{\sin x}{2x\Delta_2}\,d\eta \right\}^{x/\pi}. \]

If we introduce the functions
\[ F_\xi(\eta)=\int_0^\eta\{[p_\xi(\sigma)]^p+[p_\xi(-\sigma)]^p\}\,d\sigma \qquad(\xi=0\ \text{and}\ \xi=\pi) \]
and integrate by parts, then we find that
\[ \varphi_p(x,y)\leq K\quad \text{in } S(0,\pi). \]

Theorem B. In order that a function \(f(z)\) of class \(A\) also belong to the class \(\mathfrak M_p\), it is necessary and sufficient that the following conditions be satisfied:

a) the least harmonic majorant of the function \(\ln^+|f(z)|\) is the limit of a uniformly convergent, inside \(S(\alpha,\beta)\), minorizing sequence of harmonic functions bounded above;

b) \(\varphi_p(\alpha,y)\leq K,\quad \varphi_p(\beta,y)\leq K\) \((K=\mathrm{const})\).

Moreover, from a) and b) it follows that \(\varphi_p(x,y)\leq K\) for \(\alpha\leq x\leq \beta\).

The proof follows from the theorem of P. Ya. Polubarinova-Kochina \((^3)\) and from \((^4)\).

Let us note that functions \(f(z)\in A\) satisfying condition a) of Theorem B in \(S(0,\pi)\) are characterized also by the fact that in their integral representation (4) one has \(M\geq0,\; N\geq0\); \(\psi_0(\eta)\) and \(\psi_\pi(\eta)\) are nonincreasing functions.

Theorem C. If \(f(z)\in\mathfrak M_p\) in \(S(\alpha,\beta)\), then
\[ \varphi_p(x,y)\leq A^{\frac{\beta-x}{\beta-\alpha}} B^{\frac{x-\alpha}{\beta-\alpha}}, \]
where \(\varphi_p(\alpha,y)\leq A\) and \(\varphi_p(\beta,y)\leq B\), i.e. the logarithm of the upper bound of the means \(\varphi_p(x,y)\) is a convex function of \(x\) for \(\alpha\leq x\leq\beta\).

The proof follows directly from Theorem A.

Theorem D. Suppose \(f(z)\) satisfies condition a) of Theorem B and, in addition:
\[ \text{b′)}\quad \varlimsup_{y\to\infty} y^{-a}\int_{-y}^{y}|f(\alpha+i\eta)|^p\,d\eta\leq A,\qquad \varlimsup_{y\to\infty} y^{-b}\int_{-y}^{y}|f(\beta+i\eta)|^p\,d\eta\leq B,\quad a\geq0, \]
\[ b\geq0,\quad A>0,\quad B>0. \]

Then
\[ \varlimsup_{y\to\infty} y^{-\mu}\int_{-y}^{y}|f(x+i\eta)|^p\,d\eta\leq M \qquad(\alpha\leq x\leq\beta), \]
where
\[ \mu=a\frac{\beta-x}{\beta-\alpha}+b\frac{x-\alpha}{\beta-\alpha}, \qquad M=A^{\frac{\beta-x}{\beta-\alpha}}B^{\frac{x-\alpha}{\beta-\alpha}}. \]

In particular, if \(\varphi_p(\alpha,y)\leq M,\; \varphi_p(\beta,y)\leq M\), then \(\varphi_p(x,y)\leq M\) for \(\alpha\leq x\leq\beta\), and
\[ \ln\left[\varlimsup_{y\to\infty}\varphi_p(x,y)\right] \]
is a convex function of \(x\).

Vladimir State
Pedagogical Institute
named after P. I. Lebedev-Polyanskii

Received
17 XI 1956

CITED LITERATURE

1. G. H. Hardy, A. E. Ingham, G. Pólya, Proc. Roy. Soc., ser. A, 113, A 765, 541 (1927).
2. R. M. Gabriel, J. Lond. Math. Soc., 21, 8, 87 (1946).
3. I. I. Privalov, Boundary Properties of Analytic Functions, 1950.
4. I. I. Privalov, P. I. Kuznetsov, Matem. sborn., 6 (48), 3, 345 (1939).