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UDC 517.535.6
MATHEMATICS
Academician of the Academy of Sciences of the Armenian SSR M. M. DZHRBASHYAN, V. S. ZAKHARYAN
ON BOUNDARY PROPERTIES OF MEROMORPHIC FUNCTIONS OF THE CLASS \(N_\alpha\)
\(1^\circ\). In the papers \((^{1,2})\), for each function \(F(z)\) meromorphic in the disk \(|z|<1\), and for any value of the parameter \(\alpha\) \((-1<\alpha<+\infty)\), three functions \(m_\alpha(r;F)\), \(N_\alpha(r;F)\), and \(T_\alpha(r;F)\equiv m_\alpha(r;F)+N_\alpha(r;F)\) were defined, which are peculiar analogues of the well-known Nevanlinna functions \(m(r;F)\), \(N(r;F)\), and \(T(r;F)\), and coincide with them when \(\alpha=0\). The class \(N_\alpha\) was defined by means of the function \(T_\alpha(r;F)\) (also called the \(\alpha\)-characteristic) as the set of those \(F(z)\) for which
\[ T^{(\alpha)}(F)=\sup_{0<r<1}\{T_\alpha(r;F)\}<+\infty . \tag{1} \]
Since \(T_0(r;F)=T(r;F)\), it follows that \(N_0=N\), where \(N\) is the class of functions of bounded type of R. Nevanlinna \((^3)\).
As is known \((^3)\), if \(F(z)\in N=N_0\), then the limit
\[ F(e^{i\vartheta})=\lim_{r\to 1-0} F(re^{i\vartheta}) \tag{2} \]
exists everywhere on \([0,2\pi]\), except, perhaps, for a certain exceptional set \(E\subset [0,2\pi]\) of linear measure zero.
In view of the fact that the classes \(N_\alpha\) \((-1<\alpha<+\infty)\) decrease monotonically as \(\alpha\) decreases \((^{1,2})\), and, in particular, \(N^\alpha\subset N_0\) \((-1<\alpha<0)\), the problem naturally arises of the possibility of a finer characterization of the exceptional set \(E\subset [0,2\pi]\) where the limit (2) for \(F(z)\in N_\alpha\) \((-1<\alpha<0)\) may fail to exist. In the present note a solution of this problem is given by using the notion of \(\gamma\)-capacity of sets (see, for example, \((^4)\)).
The following basic theorem is proved.
Theorem 1. If \(F(z)\in N_\alpha\) \((-1<\alpha<0)\), then the limit (2) exists everywhere on \([0,2\pi]\), except, perhaps, for a certain set \(E\), whose \(\gamma\)-capacity (where \(1+\alpha<\gamma<1\) is any number) is equal to zero.
Below we shall give a proof of this theorem, as well as three auxiliary theorems which are also of independent interest.
An important example of a function of the class \(N_\alpha\) with zeros at the points of a given sequence \(\{z_k\}_1^\infty\) \((0<|z_k|\le |z_{k+1}|<1)\), subject only to the condition
\[ \sum_{k=1}^{\infty} (1-|z_k|)^{1+\alpha}<+\infty , \tag{3} \]
is the product, convergent in the disk \(|z|<1\), \((^{1,2})\)
\[ B_\alpha(z;z_k)=\prod_{k=1}^{\infty}\left(1-\frac{z}{z_k}\right)e^{-W_\alpha(z;z_k)}, \tag{4} \]
where, for \(|z|<1\) and \(|\zeta|<1\),
\[ W_\alpha(z;\zeta)=\int_{|\zeta|}^{1}\frac{(1-x)^\alpha}{x}\,dx- \]
\[ -\sum_{k=1}^{\infty}\frac{\Gamma(1+\alpha+k)}{\Gamma(1+\alpha)\Gamma(1+k)} \left\{\xi^{-k}\int_{0}^{|\xi|}(1-x)^\alpha x^{k-1}\,dx-\bar{\xi}^{\,k}\int_{|\xi|}^{1}(1-x)^\alpha x^{-k-1}\,dx\right\}z^k . \tag{5} \]
The proof of Theorem 1 rests essentially on the parametric representation of the class \(N_\alpha\) \(({}^1,{}^2)\) and on the well-known theorem of Salem and Zygmund \(({}^4)\).
Theorem A. The class \(N_\alpha\) \((-1<\alpha<\infty)\) coincides with the set of functions admitting the representation
\[ F(z)=cz^\lambda \frac{B_\alpha(z;a_\mu)}{B_\alpha(z;b_\nu)} \exp\left\{\frac{1}{2\pi}\int_{0}^{2\pi} S_\alpha(e^{-i\vartheta}z)\,d\psi(\vartheta)\right\}, \tag{6} \]
where \(c\) is a constant, \(\lambda\ge 0\) is an integer,
\[ S_\alpha(z)=\Gamma(1+\alpha)\left\{\frac{2}{(1-z)^{1+\alpha}}-1\right\},\qquad \operatorname{Re} S_\alpha(z)\ge 0, \tag{7} \]
\(\psi(\vartheta)\) is an arbitrary real function of finite total variation on \([0,2\pi]\).
Theorem B \(({}^4)\). If
\[ \sum_{k=1}^{\infty}(a_k^2+b_k^2)k^\beta<+\infty\qquad (0<\beta<1), \]
then the trigonometric series
\[ \frac{a_0}{2}+\sum_{k=1}^{\infty} a_k\cos kx+b_k\sin kx \]
can diverge only on a set whose \((1-\beta)\)-capacity is zero.
We note that before the appearance of \(({}^1,{}^2)\), other classes \(T_\beta\) \((0<\beta<1)\) included in the class \(N\) were known \(({}^5)\): \(w(z)\in T_\beta\) if
\[ \int_{0}^{1}\frac{A(r;w)}{(1-r)^\beta}\,dr<+\infty,\qquad \text{where }\quad A(r;w)=\iint_{|z|<r}\frac{|w'(z)|^2}{(1+|w(z)|^2)^2}\,dx\,dy . \tag{8} \]
From this definition it follows directly that
Lemma C. If the function \(w(z)=\sum_{k=0}^{\infty} a_k z^k\) \((|z|<1)\) is bounded and belongs to the class \(T_\beta\) \((0<\beta<1)\), then
\[ \sum_{k=1}^{\infty}|a_k|^2 k<+\infty . \]
In his study devoted to the classes \(T_\beta\), L. Carleson first proved that for \(F(z)\in T_\beta\) \((0<\beta<1)\) the assertion of Theorem 1 is valid, with the value \(\gamma=1-\beta\).*
\(2^\circ\). Let us observe that for \(\alpha=0\) the function \(B_\alpha(z;z_k)\) coincides with the ordinary Blaschke product, i.e.
\[ B_0(z;z_k)=B(z)=\prod_{k=1}^{\infty}\frac{z_k-z}{1-\bar z_k z}\left|\frac{z_k}{z}\right|. \tag{9} \]
We first give a theorem that plays an important role in the theorems set forth below.
Theorem 2. If \(-1<\alpha<0\) and condition (4) is satisfied, then
\[ |B_\alpha(z;z_k)|\le |B_0(z;z_k)|<1\qquad (|z|<1). \tag{10} \]
* If the function \(F(z)\in T_\beta\) is bounded, then this theorem follows directly from Theorem B and Lemma C. In the general case, since there is no representation for the classes \(T_\beta\), L. Carleson established this theorem by a more complicated method. This explains also the fact that not every function \(F\in T_\beta\) can be represented in the form of a quotient \(f/g\) of bounded functions from the same class \(({}^5)\)—a “defect” of which the class \(N_\alpha\) is free.
Let us first note that, by virtue of the definition (6) of the function \(B_\alpha(z;z_k)\), it is enough to establish the inequality
\[ U_\alpha(z;\zeta)\equiv \operatorname{Re}\{W_\alpha(z;\zeta)-W_0(z;\zeta)\}\geq 0 \qquad (|z|<1,\ |\zeta|<1). \tag{11} \]
But from the expansion (5) of the function \(W_\alpha(z;\zeta)\) we have
\[ U_\alpha(z;\zeta)=\frac{a_0(\rho)}{2} +\sum_{k=1}^{\infty} a_k(\rho)|\omega|^k \cos(k\arg \omega), \tag{12} \]
where \(\rho=|\zeta|\), \(\omega=\bar\zeta z\) \((|\omega|=\rho|z|<1)\),
\[ a_0(\rho)=2\int_{\rho}^{1}\bigl[(1-x)^\alpha-1\bigr]x^{-1}\,dx, \tag{13} \]
\[ a_k(\rho)=\frac{1}{k} -\frac{\Gamma(1+\alpha+k)}{\Gamma(1+\alpha)\Gamma(1+k)} \left\{ \rho^{-2k}\int_{0}^{\rho}(1-x)^\alpha x^{k-1}\,dx -\int_{\rho}^{1}(1-x)^\alpha x^{-k-1}\,dx \right\} \]
\[ (k=1,2,\ldots). \]
Let us note further that it is enough to establish the convexity of the sequence \(\{a_k(\rho)\}_1^\infty\), i.e., the validity of the inequalities
\[ \varphi_k(\rho)\equiv \Delta^2 a_k(\rho)\equiv a_k(\rho)-2a_{k+1}(\rho)+a_{k+2}(\rho)\geq 0 \]
\[ (0<\rho\leq 1;\ k=0,1,2,\ldots), \tag{14} \]
so that (11) would follow from this by the known theorem \((^4)\) on nonnegative trigonometric series. The proof of the inequalities (14) is by no means simple, but we shall only outline it for lack of space. First, we consider the function
\[
\rho^5\varphi_0'(\rho)=2(1-\rho^2)^2[1-(1-\rho)^\alpha]
+\alpha\rho(1-\rho)^\alpha[4\rho^2-(1+\alpha)\rho-2]
\]
\((0<\rho\leq 1)\), and by means of appropriate calculations and estimates we verify that all coefficients of its expansion in powers of \(\rho\) are negative. Since \(\varphi_0(1)=0\), it follows from this that \(\varphi_0(\rho)\geq 0\) \((0<\rho\leq 1)\).
To prove the inequalities (14) for \(k\geq 1\), one must first write the functions \(a_k(\rho)\) \((k\geq 1)\) in the form \(a_k(\rho)=b_k(\rho)+d_k(\rho)\), where
\[ b_k(\rho)=\frac{1}{k} -\frac{\Gamma(1+\alpha+k)}{\Gamma(1+\alpha)\Gamma(1+k)} \rho^{-2k}\int_{0}^{\rho^2}(1-x)^\alpha x^{k-1}\,dx, \]
\[ d_k(\rho)= \frac{\Gamma(1+\alpha+k)}{\Gamma(1+\alpha)\Gamma(1+k)} \left\{ \int_{\rho}^{1}(1-x)^\alpha x^{-k-1}\,dx -\rho^{-2k}\int_{\rho^2}^{\rho}(1-x)^\alpha x^{k-1}\,dx \right\}, \]
noting that \(b_k(\rho)\geq b_k(1)=0\), \(d_k(\rho)\geq d_k(1)=0\) \((k=1,2,\ldots)\). Then, by direct estimates, the convexity of both sequences \(\{b_k(\rho)\}_1^\infty\) and \(\{d_k(\rho)\}_1^\infty\) is established. Finally, since
\[
\varphi_k(\rho)=\Delta^2 b_k(\rho)+\Delta^2 d_k(\rho),\qquad k=1,2,\ldots,
\]
it follows that \(\varphi_k(\rho)\geq 0\) \((0<\rho\leq 1;\ k=1,2,\ldots)\).
It is known \((^5)\) that, under condition (3), \(B(z)\in T_{-\alpha}\). Relying on Theorem 2 and on this fact, one proves
Theorem 3. If the sequence \(\{z_k\}_1^\infty\) satisfies condition (3) for \(-1<\alpha<0\), then \(B_\alpha(z;z_k)\in T_{-\alpha}\).
\(3^\circ\). Consider the generalized Cauchy–Stieltjes integral
\[ K_\alpha(z)=\frac{1}{2\pi}\int_{0}^{2\pi} \frac{d\psi(\vartheta)}{(1-e^{-i\vartheta}z)^{1+\alpha}} \qquad (-1<\alpha<0), \tag{15} \]
where \(\psi(\vartheta)\) is an arbitrary real function of bounded variation on \([0,2\pi]\).
Theorem 4. If \(\psi(\vartheta)\) is nondecreasing on \([0,2\pi]\), then the function
\[ w(z)=\exp\{-K_\alpha(z)\} \tag{16} \]
is bounded in the disk \(|z|<1\) and, for any \(\alpha'\) \((-1<\alpha<\alpha'<0)\), belongs to the class \(T_{-\alpha'}\).
Let us outline the proof of this theorem. Since, for \(|z|<1\),
\[ \operatorname{Re}\left[\frac{1}{(1-ze^{-i\vartheta})^{1+\alpha}}\right] \ge \frac{(1-r)^{1+\alpha}}{|1-re^{i(\varphi-\vartheta)}|^{2+2\alpha}} \quad (z=re^{i\varphi}) \]
and \(d\psi(\vartheta)\ge 0\), from (17) and (18) we shall have
\[ |w(z)|^2\le \exp\{-\omega(re^{i\varphi})\};\quad \omega(re^{i\varphi})= \frac{(1-r)^{1+\alpha}}{\pi} \int_0^{2\pi} \frac{d\psi(\vartheta)} {|1-re^{i(\varphi-\vartheta)}|^{2+2\alpha}}, \tag{17} \]
and hence \(|w(z)|\le 1\).
Further, since by virtue of (16)
\[ |w'(z)|\le |w(z)|\,\frac{1+\alpha}{2\pi} \int_0^{2\pi} \frac{d\psi(\vartheta)} {|1-re^{i(\varphi-\vartheta)}|^{2+\alpha}}, \]
then, using Schwarz’s inequality, we obtain
\[ |w'(z)|^2\le \omega(re^{i\varphi})e^{-\omega(re^{i\varphi})} (1-r)^{-1-\alpha} \int_0^{2\pi} \frac{d\psi(\vartheta)} {|1-re^{i(\varphi-\vartheta)}|^2}. \]
Finally, observing that \(\omega e^{-\omega}\le e^{-1}\) \((0\le \omega<\infty)\), by virtue of (17) we arrive at the inequality
\[ |w'(z)|^2(1+|w(z)|^2)^{-2} \le e^{-1}(1-r)^{-1-\alpha} \int_0^{2\pi} \frac{d\psi(\vartheta)} {|1-re^{i(\varphi-\vartheta)}|^2}. \]
From this it already follows easily that
\[ A(r;w)\le c_1 r^2+c_2(1-r)^{-1-\alpha}\quad (0\le r<1), \tag{18} \]
where \(c_1>0\) and \(c_2>0\) do not depend on \(r\). Therefore, for any \(\alpha'\) \((-1<\alpha<\alpha'<0)\),
\[ \int_0^1 A(r;w)(1-r)^{\alpha'}\,dr<+\infty, \]
i.e. \(w(z)\in T_{-\alpha'}\).
Proof of Theorem 1 follows immediately from the main theorem A, if one takes into account Theorems 2, 3, 4, Lemma C, and Theorem B.
From Theorem 1 there follows immediately
Theorem 5. The generalized Cauchy–Stieltjes integral \(K_\alpha(re^{i\varphi})\) \((-1<\alpha<0)\) has radial limits (2) everywhere except, possibly, for some set \(E\subset[0,2\pi]\), whose \(\gamma\)-capacity (where \(1+\alpha<\gamma<1\)—arbitrary) is equal to zero.
Institute of Mathematics and Mechanics
Academy of Sciences of the ArmSSR
Received
3 II 1967
REFERENCES
- M. M. Dzhrbashyan, DAN, 157, No. 5, 1024 (1964).
- M. M. Dzhrbashyan, Integral transforms and representations of functions in the complex domain, vol. IX, “Nauka,” 1966.
- R. Nevanlinna, Single-valued analytic functions, vol. VI, Moscow–Leningrad, 1941.
- N. K. Bari, Trigonometric series, vols. I and V, Moscow, 1961.
- L. Carleson, On a Class of Meromorphic Functions and its Associated Exceptional Sets, Uppsala, 1950.