MATHEMATICS

V. G. RYABYKH

ON SOME PROPERTIES OF ANALYTIC FUNCTIONS OF THE CLASS \(H'_p\)

(Presented by Academician P. Ya. Kochina on 16 IV 1964)

By the class \(H'_p\) we shall mean the set of analytic functions \(f(z)\) in the disk \(|z|<1\) such that

\[ H'_p(f)=\int_0^1 \int_0^{2\pi} |f(re^{i\theta})|^p r\,dr\,d\theta<\infty,\qquad p>0. \]

This class is a natural generalization of the class \(H_p\) of F. Riesz, which is defined as the set of functions analytic in the disk such that

\[ H_p(f)=\lim_{r\to 1}\int_0^{2\pi} |f(re^{i\theta})|^p\,d\theta<\infty. \]

The question naturally arises of the relation between the classes \(H'_p\) and \(H_p\).

\(1^\circ.\) Theorem 1. If \(f'(z)\in H'_1\), then \(f(z)\in H_1\).

An example constructed by S. N. Mergelyan \((^1)\) shows that there exists a function \(f_0(z)\in H_1\) for which \(f'_0(z)\notin H'_1\).

To a certain extent the following assertion is the converse of Theorem 1.

Theorem 2. If \(f(z)\in \bigcap_{p<1} H_p\), then \(f'(z)\in \bigcap_{p<1} H'_p\).

Proof. Since \(f(z)\) is analytic in the disk \(|z|<1\), we have

\[ f'(z)=\frac{1}{2\pi i}\int_{|t|=R}\frac{f(t)}{(t-z)^2}\,dt. \]

Hence, for \(0<r<R<1\), one obtains the inequality

\[ \int_0^{2\pi} |f'(re^{i\theta})|\,d\theta \le \frac{1}{2\pi}\int_0^{2\pi} |f(Re^{i\varphi})| \int_0^{2\pi}\frac{d\theta}{R^2-2Rr\cos(\theta-\varphi)+r^2}\,d\varphi, \]

from which it follows that

\[ \int_0^{2\pi} |f'(re^{i\theta})|\,d\theta \le \frac{1}{R-r}\int_0^{2\pi}|f(Re^{i\varphi})|\,d\varphi. \tag{1} \]

Using the known estimate \((^2)\), for \(0<\delta<1\),

\[ \int_0^{2\pi}|f(Re^{i\theta})| \le C(1-R)^{1-1/\delta}\{H_\delta(f)\}^{1/\delta}, \]

where the constant \(C\) does not depend on \(R\), we obtain from (1)

\[ \int_0^{2\pi}|f'(Re^{i\theta})|\,d\theta \le \frac{C_1}{(1-R)^{1/\delta-1}(R-r)}\{H_\delta(f)\}^{1/\delta}. \]

Applying to the left-hand side of (2) Hölder’s inequality for \(p<1\) \(({}^3)\) and putting \(r=R^2\), we shall have

\[ \left\{\int_0^{2\pi} |f'(R^2 e^{i\theta})|^p\,d\theta\right\}^{1/p} \le \frac{C_3}{R(1-R)^{1/\delta}}\{H_\delta(f)\}^{1/\delta}. \]

Raising both sides to the power \(p\) and integrating with respect to \(R\) from zero to one, we obtain

\[ \int_0^1\int_0^{2\pi} |f'(R^2 e^{i\theta})|^p R^2\,dR^2\,d\theta \le A(p,f)\int_0^1 \frac{R^2\,dR^2}{R^p(1-R)^{p/\delta}}. \]

Thus, \(f'(z)\in H'_p\) for every \(p<\delta\) and, consequently,
\(f'(z)\in \bigcap_{p<1} H'_p\). The theorem converse to Theorem 2 does not hold. This follows from the following assertion.

Theorem 3. For every \(0<\delta<1\) there exists a function \(f_0(z)\in H_\delta\) for which \(f_0(z)\notin H_{\delta+\varepsilon}\) for every \(\varepsilon>0\), but \(f'_0(z)\in \bigcap_{p<1} H'_p\).

The proof is based on a theorem obtained by V. P. Khavin in \(({}^4)\). It is interesting to note that V. P. Khavin’s theorem does not extend to the class \(H'_p\). Here the following theorem holds:

Theorem 4. Let \(f(z)\in H'_p\), \(p>0\), but \(f(z)\notin H'_{p+\varepsilon}\) for every \(\varepsilon>0\); if \(\max |f(z)|=O(\varphi(r))\), where \(\varphi(r)\) is a continuous nondecreasing function, \(z=re^{i\theta}\), then \(\int_0^1 \varphi^q(r)\,dr=\infty\) for all \(q>p\).

For functions belonging to \(H_p\), \(p<1\), the following is true.

Theorem 5. If \(f(r)\in H_p\), \(p<1\), then \(f'(z)\in H'_q\), where \(q<2p/(p+1)\).

We first prove the theorem for functions having no zeros in the disk \(|z|<1\). Let \(F(z)\ne 0\) in the disk \(|z|<1\); then for \(\varepsilon>0\) and \(|z|<R<1\)

\[ [F(z)]^\varepsilon = \frac{1}{2\pi i} \int_{|t|=R} \frac{[F(t)]^\varepsilon}{(t-z)}\,dt, \]

therefore,

\[ \varepsilon [F(z)]^{\varepsilon-1}F'(z) = \frac{1}{2\pi i} \int_{|t|=R} \frac{[F(t)]^\varepsilon}{(t-z)^2}\,dt. \]

Hence, for \(z=re^{i\varphi}\), \(t=Re^{i\theta}\), we have the estimate

\[ |F'(z)| \le \frac{1}{2\pi\varepsilon}|F(z)|^{1-\varepsilon} \int_0^{2\pi} \frac{|F(Re^{i\theta})|^\varepsilon R} {R^2-2Rr\cos(\theta-\varphi)+r^2}\,d\theta . \tag{3} \]

Using the known inequality \(({}^2)\) for \(F(z)\in H_p\),

\[ |F(z)| \le \frac{C}{(1-r)^{1/p}},\qquad |z|=r, \tag{4} \]

we derive from (3)

\[ |F'(z)| \le \frac{C_1 R\,|F(z)|^{1-\varepsilon}} {(1-R)^{\varepsilon/p}(R^2-r^2)} . \]

Raise both sides of the inequality to the power \(q\), \(p<q<1\), and suppose that \(r=R^2\); then

\[ \int_0^{2\pi} |F'(R^2 e^{i\varphi})|^q\,d\varphi \le \frac{C_1^q} {R^q(1-R)^{(1+\varepsilon/p)q}} \int_0^{2\pi} |F(Re^{i\theta})|^{(1-\varepsilon)q}\,d\theta . \tag{5} \]

Using (4) we easily obtain an estimate, valid for the whole class \(H_p\) and for any \(p>0,\ \alpha>p\):

\[ \int_0^{2\pi}\left|F\left(re^{i\varphi}\right)\right|^\alpha d\varphi \leq \frac{C_2}{(1-r)^{(\alpha-p)/p}}\,H_p(F). \tag{6} \]

We shall now assume that \(\varepsilon<1-p/q\); applying inequality (6) to the right-hand side of (5), we obtain

\[ \int_0^{2\pi}\left|F'\left(R^2e^{i\theta}\right)\right|^p d\theta \leq \frac{C_3}{(1-R)^{q(1+1/p)-1}R^q}. \]

If the condition \(q<\dfrac{2}{1/p+1}\) is satisfied, then \(F'(z)\in H'_q\).

If \(f(z)\) is an arbitrary function in \(H_p,\ p<1\), then it is representable in the form \(f(z)=F_1(z)-F_2(z)\) ((5), p. 540), where \(F_j(z)\) \((j=1,2)\) belong to \(H_p\) and have no zeros in the disk \(|z|<1\); this case reduces to the preceding one.

The number \(2p/(p+1)\) cannot be replaced by a larger number, as is shown by the example of a function of the form
\(z^2(1-z)^{-1/p}\left(\ln\dfrac{1}{1-z}\right)^2\).

\(2^\circ\). The question of the zeros of functions of the class \(H'_p\) was considered by M. M. Dzhrbashyan \((^6)\). The following result is due to him.

Let \(f(z)\in A(\alpha)\), i.e., let \(f(z)\) be analytic in the disk \(|z|<1\), \(f(z)\not\equiv0\), \(\alpha>-1\), and

\[ \frac{\alpha+1}{\pi}\int_0^1\int_0^{2\pi} \left(1-\rho^2\right)^\alpha \ln^+\left|f\left(\rho e^{i\theta}\right)\right| \,\rho\,d\rho\,d\theta<\infty; \]

if \(a_1,a_2,\ldots\) are the zeros of \(f(z)\), counted with multiplicity, and

\[ a_1\leq a_2\leq a_3\leq\cdots, \]

then

\[ \sum_{k=1}^{\infty}\left(1-|a_k|\right)^{2+\alpha}<\infty. \]

It turns out that if a function belongs to the class \(H'_p,\ p>0\), then the following is true.

Theorem 6. If \(f(z)\in H'_p,\ p>0\), and \(a_1,a_2,\ldots\) are the zeros of \(f(z)\), counted with multiplicity, with

\[ |a_1|\leq |a_2|\leq |a_3|\leq\cdots, \]

then for every \(\varepsilon>0\)

\[ \sum_{k=1}^{\infty}\left(1-|a_k|\right)^{1+\varepsilon}<\infty. \]

With the aid of one result \((^7)\) it follows that there exists a function
\(f(z)\in H'_p,\ 0<p<1\), such that

\[ \sum_{k=1}^{\infty}\left(1-|a_k|\right)=\infty, \]

where \(a_1,a_2,\ldots\) are the zeros of the function \(f(z)\).

\(3^\circ\). The following theorem of G. M. Goluzin is known \((^8)\).

If

\[ f(z)=\sum_{k=0}^{\infty}a_k z^k \]

is an analytic function in the disk \(|z|<1\) and \(H_1(f)\leq 2\pi\), then for all \(n=0,1,2,\ldots\) the estimate \(|a_n|\leq1\) holds.

In this case the equality sign for a certain \(n\) is attained if and only if

\[ f(z)=\varepsilon \sum_{k=0}^{n} a_k z^k \sum_{k=0}^{n} \bar a_k z^{\,n-k}, \]

where for \(a_k\) and \(\varepsilon\) the condition

\[ |\varepsilon|\sum_{k=0}^{\infty} |a_k|^2 \]

is satisfied.

With respect to the coefficients of functions belonging to the class \(H'_1\), one may assert:

Theorem 7. If \(f(z)=\sum_{k=0}^{\infty} a_k z^k\) is an analytic function in the disk \(|z|<1\) and \(H'_1(f)=2\pi\), then for all \(n=0,1,2,\ldots\) the estimate \(|a_n|\le n+2\) is valid. In this case the equality sign for a certain \(n\) is attained if and only if

\[ f(z)=e^{i\alpha}(n+2)z^n, \]

where \(\alpha\) is an arbitrary real number.

The author expresses deep gratitude to S. Ya. Al’per for supervising the work.

Received
14 IV 1964

CITED LITERATURE

1. S. N. Mergelyan, Izv. Akad. Nauk SSSR, Ser. Mat., 15, 395 (1951).
2. I. I. Privalov, Boundary Properties of Analytic Functions, 2nd ed., Moscow–Leningrad, 1950.
3. G. G. Khardi, J. E. Littlewood, G. Polya, Inequalities, IL, 1948.
4. V. P. Khavin, Vestn. Leningrad Univ., 1, 102 (1962).
5. N. K. Bari, Trigonometric Series, Moscow, 1961.
6. M. M. Dzhrbashyan, Reports of the Institute of Mathematics and Mechanics, Academy of Sciences of the Armenian SSR, 2 (1948).
7. P. Porcelli, Rend. mat. e appl., 20, 385 (1961).
8. G. M. Goluzin, Trudy Mat. Inst. im. V. A. Steklova Akad. Nauk SSSR, 18 (1946).