Abstract
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MATHEMATICS
Academician of the Academy of Sciences of the Armenian SSR M. M. DZHRBASHYAN
ON A PARAMETRIC REPRESENTATION OF CERTAIN GENERAL CLASSES OF MEROMORPHIC FUNCTIONS IN THE UNIT DISK
One of the fundamental results in the theory of meromorphic functions is the well-known theorem of R. Nevanlinna (¹) on the parametric representation of the class \(N\), introduced by him, of meromorphic functions in the disk \(|z|<1\) with bounded characteristic (or, what is the same thing, the class of functions of bounded type).
An attempt to extend this theorem to meromorphic functions with unbounded characteristic was undertaken by us earlier (², ³). At that time the classes \(N_\alpha^*\) \((0<\alpha<\infty)\) were introduced, consisting of functions \(F(z)\), meromorphic in the disk \(|z|<1\), for which the characteristic function \(T(r)\) satisfies the condition
\[ \int_0^1 (1-r)^{\alpha-1}T(r)\,dr<+\infty . \]
However, the canonical representation of the class \(N_\alpha^*\) obtained in that work did not have a parametric character, as is the case in R. Nevanlinna’s theorem for the class \(N\).
In the present note a certain general class \(N_\alpha\) of functions meromorphic in the disk \(|z|<1\), depending on an arbitrary parameter \(\alpha\) \((-1<\alpha<\infty)\), is defined, and a parametric representation of the class is established.
\(1^\circ\). We first recall the definitions of the fractional integral and the fractional derivative in the sense of Riemann—Liouville, which will be needed below.
Let the function \(\varphi(r)\) be defined on \((0,1)\) and belong to the class \(L(0,1)\). For any number \(\alpha\in(0,+\infty)\), the fractional integral of order \(\alpha\) of \(\varphi(r)\) is the function
\[ D^{-\alpha}\varphi(r)\equiv \frac{1}{\Gamma(\alpha)}\int_0^r (r-t)^{\alpha-1}\varphi(t)\,dt,\qquad r\in(0,1), \tag{1} \]
where the operator \(D^{-\alpha}\varphi(r)\) is defined almost everywhere on \((0,1)\) and belongs to the class \(L(0,1)\).
It can be shown that almost everywhere on \((0,1)\)
\[ \lim_{\alpha\to+0}D^{-\alpha}\varphi(r)=\varphi(r). \tag{2} \]
Further, for any \(\alpha\in(-1,0)\), the fractional derivative of order \(\gamma=-\alpha\) of \(\varphi(r)\in L(0,1)\) is customarily called the function
\[ D^{-\alpha}\varphi(r)=\frac{d}{dr}D^{-(1+\alpha)}\varphi(r),\qquad r\in(0,1), \tag{3} \]
if it exists.
Finally, in view of (2) and (3), it is natural to identify both the integral and the derivative of zero order with the function itself. Let
\[ D^0\varphi(r)\equiv \varphi(r). \tag{4} \]
Thus, the operator \(D^{-\alpha}\varphi(r)\) is defined for an arbitrary value of the parameter \(\alpha\) \((-1<\alpha<+\infty)\), provided, of course, that for values \(\alpha\in(-1,0)\) it has meaning.
Lemma 1. Let the function
\[ f\left(re^{i\varphi}\right)=\sum_{k=0}^{\infty} a_k\left(re^{i\varphi}\right)^k \tag{5} \]
be holomorphic in the unit disk. Then:
a) For any \(\alpha \in (-1,+\infty)\) the function
\[ f_\alpha\left(re^{i\varphi}\right) \equiv r^{-\alpha}D^{-\alpha}f\left(re^{i\varphi}\right) =\sum_{k=0}^{\infty} a_k\,\frac{\Gamma(1+k)}{\Gamma(1+\alpha+k)}\left(re^{i\varphi}\right)^k \tag{6} \]
is also holomorphic in the unit disk.
b) For any \(\alpha \in (-1,+\infty)\) and \(r\in(0,1)\) the integral formulas
\[ f(z)=\frac{r}{2\pi}\int_0^{2\pi} \frac{\Gamma(1+\alpha)}{\left(r-ze^{-i\vartheta}\right)^{1+\alpha}} D^{-\alpha}f\left(re^{i\vartheta}\right)\,d\vartheta \qquad (|z|<r); \tag{7} \]
\[ f(z)=-\overline{f(0)}+\frac{r}{\pi}\int_0^{2\pi} \frac{\Gamma(1+\alpha)}{\left(r-ze^{-i\vartheta}\right)^{1+\alpha}} D^{-\alpha}\operatorname{Re} f\left(re^{i\vartheta}\right)\,d\vartheta \qquad (|z|<r). \tag{8} \]
\(2^\circ\). Let the function \(F(z)\) be meromorphic in the disk \(|z|<1\). Let \(\{a_\mu\}\) and \(\{b_\nu\}\) be, respectively, the sequences of its zeros and poles distinct from \(z=0\) and numbered in the order of nondecreasing moduli,
\[ 0<|a_1|\le |a_2|\le\cdots\le |a_\mu|\le\cdots, \]
\[ 0<|b_1|\le |b_2|\le\cdots\le |b_\nu|\le\cdots, \]
where we assume that each zero or pole is written as many times as its multiplicity.
Finally, if in a neighborhood of the origin there is a Laurent expansion
\(F(z)=c_\lambda z^\lambda+c_{\lambda+1}z^{\lambda+1}+\cdots\) \((c_\lambda\ne0)\), then we shall regard the point \(z=0\) as a zero (for \(\lambda\ge1\)) or a pole (for \(\lambda\le -1\)) of multiplicity \(\lambda\). With the aid of formula (8) of Lemma 1 one establishes
Lemma 2. For any \(\alpha\in(-1,+\infty)\) and \(r\) \((0<r<1)\) the formula
\[ \begin{aligned} \log F(z) ={}&-\log \overline{c_\lambda}-2\lambda\psi_\alpha(r)+\lambda\log z \\ &+\sum_{|a_\mu|\le r} \log\left\{\left(1-\frac{z}{a_\mu}\right) e^{-V_\alpha\left(\frac{z}{r},\,\frac{a_\mu}{r}\right)}\right\} -\sum_{|b_\nu|\le r} \log\left\{\left(1-\frac{z}{b_\nu}\right) e^{-V_\alpha\left(\frac{z}{r},\,\frac{b_\nu}{r}\right)}\right\} \\ &+\frac{r}{\pi}\int_0^{2\pi} \frac{\Gamma(1+\alpha)}{\left(r-ze^{-i\vartheta}\right)^{1+\alpha}} D^{-\alpha}\log\left|F\left(re^{i\vartheta}\right)\right|\,d\vartheta \qquad (|z|<r), \end{aligned} \tag{9} \]
where
\[ \psi_\alpha(r)=\log r-\alpha\sum_{n=1}^{\infty}\frac{1}{n(n+\alpha)}, \]
\[ \begin{aligned} V_\alpha(z;\zeta) &=\frac{1}{\pi}\int_0^{2\pi} \frac{\Gamma(1+\alpha)}{\left(1-ze^{-i\vartheta}\right)^{1+\alpha}} \left\{D^{-\alpha}\log\left|1-\frac{re^{i\vartheta}}{\zeta}\right|\right\}_{r=1} \,d\vartheta -2\int_{|\zeta|}^{1}\frac{(1-x)^\alpha}{x} \\ &\quad -\sum_{k=1}^{\infty} \frac{\Gamma(1+\alpha+k)}{\Gamma(1+k)\Gamma(1+\alpha)} \left\{ \zeta^{-k}\int_0^{|\zeta|}(1-x)^\alpha x^{k-1}\,dx -\overline{\zeta}^{\,k}\int_{|\zeta|}^{1}(1-x)^\alpha x^{-k-1}\,dx \right\}z^k . \end{aligned} \tag{10} \]
Let us note that in the case \(\alpha=0\) this formula is equivalent to the well-known Jensen–Nevanlinna formula, which lies at the basis of the theory of meromorphic functions.
Taking into account the asymptotic properties of the function \(V_\alpha(z;\zeta)\), the following important theorem is proved.
Theorem 1. Let \(\{z_k\}_1^\infty\) \((0<|z_k|<1)\) be any sequence of complex numbers, numbered in the order
\[ 0<|z_1|\le |z_2|\le\cdots\le |z_k|\le\cdots . \]
and satisfying the condition
\[ \sum_{k=1}^{\infty} (1-|z_k|)^{1+\alpha}<+\infty, \tag{11} \]
where \(\alpha\in(-1,+\infty)\). Then:
a) The infinite product
\[ \pi_{\alpha}(z;z_k)=\prod_{k=1}^{\infty}\left(1-\frac{z}{z_k}\right)e^{-V_{\alpha}(z;z_k)} \tag{12} \]
converges uniformly and absolutely in every closed subdomain of the disk \(|z|<1\), representing an analytic function that vanishes only on the sequence \(\{z_k\}_1^\infty\).
b) For any \(r\) \((0<r<1)\) and \(\vartheta\) \((0\leq \vartheta\leq 2\pi)\) we also have
\[ D^{-\alpha}\log|\pi_{\alpha}(re^{i\vartheta};z_k)|\leq 0. \tag{13} \]
Let us further note that the function
\[ B_{\alpha}(z;z_k)=\frac{\pi_{\alpha}(z;z_k)}{\sqrt{\pi_{\alpha}(0;z_k)}}= \prod_{k=1}^{\infty}\left(1-\frac{z}{z_k}\right)e^{-\frac12 V_{\alpha}(z;z_k)} \tag{14} \]
is a natural analogue of the Blaschke product.
This should be understood in the sense that if
\[ \sum_{k=1}^{\infty}(1-|z_k|)<+\infty, \tag{11'} \]
then we shall have
\[ \lim_{\alpha\to+0} B_{\alpha}(z;z_k)=B(z;z_k)= \prod_{k=1}^{\infty}\frac{z_k-z}{1-\overline{z_k}z}\cdot\frac{|z_k|}{z_k}. \tag{14'} \]
\(3^\circ\). We now pass to the definition of the classes of meromorphic functions \(N_{\alpha}\). Let \(n(t,0)\) and \(n(t,\infty)\) be, respectively, the number of numbers \(\{a_\mu\}\) and \(\{b_\nu\}\) lying in the disk \(|z|\leq t\) \((0<t<1)\). We shall further agree that \(n(0,0)\) and \(n(0,\infty)\) denote, respectively, the multiplicity of the zero or pole of the function \(F(z)\) at the origin \(z=0\), so that we shall have
\[ n(0,0)-n(0,\infty)=\lambda. \]
Introduce the functions
\[ N_{\alpha}(r;0)= \frac{r^{-\alpha}}{\Gamma(1+\alpha)} \int_0^r \frac{(r-t)^\alpha}{t}\,[n(t,0)-n(0,0)]\,dt +\frac{n(0,0)}{\Gamma(1+\alpha)}[\log r-K_{\alpha}], \]
\[ N_{\alpha}(r;\infty)= \frac{r^{-\alpha}}{\Gamma(1+\alpha)} \int_0^r \frac{(r-t)^\alpha}{t}\,[n(t,\infty)-n(0,\infty)]\,dt + \tag{15} \]
\[ +\frac{n(0,\infty)}{\Gamma(1+\alpha)}[\log r-K_{\alpha}], \]
where the value of the parameter \(\alpha\in(-1,+\infty)\) is arbitrary, and
\[ K_{\alpha}=-\psi_{\alpha}(1)=\alpha\sum_{n=1}^{\infty}\frac{1}{n(n+\alpha)}. \]
Next, setting
\[ D_{(+)}^{-\alpha}\varphi(r)= \begin{cases} D^{-\alpha}\varphi(r), & \text{if } D^{-\alpha}\varphi(r)\geq 0,\\ 0, & \text{if } D^{-\alpha}\varphi(r)\leq 0, \end{cases} \tag{16} \]
and also
\[ D_{(-)}^{-\alpha}\varphi(r)=D_{(+)}^{-\alpha}\varphi(r)-D^{-\alpha}\varphi(r), \tag{16'} \]
denote
\[ m_\alpha(r,\infty)=\frac{r^{-\alpha}}{2\pi}\int_0^{2\pi}D_{(+)}^{-\alpha}\log\left|F\left(re^{i\vartheta}\right)\right|\,d\vartheta, \]
\[ m_\alpha(r,0)=\frac{r^{-\alpha}}{2\pi}\int_0^{2\pi}D_{(-)}^{-\alpha}\log\left|F\left(re^{i\vartheta}\right)\right|\,d\vartheta. \tag{17} \]
Lemma 3. a) For any \(\alpha\in(-1,+\infty)\) the identity
\[ m_\alpha(r,\infty)+N_\alpha(r,\infty)= \]
\[ =m_\alpha(r,0)+N_\alpha(r,0)+\frac{\log|C_\lambda|}{\Gamma(1+\alpha)} \quad(0<r<1) \tag{18} \]
holds.
b) The function
\[ T_\alpha(r)\equiv m_\alpha(r,\infty)+N_\alpha(r,\infty)\quad(0<r<1) \tag{19} \]
is monotonically increasing.
We shall call the function \(T_\alpha(r)\) the \(\alpha\)-characteristic of the meromorphic function \(F(z)\). Finally, let \(N_\alpha\) denote the set of all functions \(F(z)\) meromorphic in the disk \(|z|<1\) for which, for a given \(\alpha\in(-1,+\infty)\), the \(\alpha\)-characteristic is bounded, i.e., for which
\[ T_\alpha(1)=\lim_{r\to1-0}T_\alpha(r)<+\infty. \tag{20} \]
From the definition (19) of the function \(T_\alpha(r)\) and the class \(N_\alpha\), in view of (4), it is obvious that for the value of the parameter \(\alpha=0\) we shall have
\[ T_0(r)\equiv T(r),\qquad N_0\equiv N, \tag{21} \]
where \(T(r)\) is the characteristic, and \(N\) is the class of functions with bounded characteristic, introduced by R. Nevanlinna.
It has been established that the inclusion
\[ N_{\alpha_1}\subset N_{\alpha_2}\quad\text{for }-1<\alpha_1<\alpha_2<+\infty, \tag{22} \]
holds, and therefore also the inclusions
\[ N_\alpha\subset N_0\quad\text{for }-1<\alpha\leq0;\qquad N_0\subset N_\alpha\quad\text{for }0\leq\alpha<+\infty. \tag{22'} \]
Thus, we have a family \(\{N_\alpha\}\), depending on the parameter \(\alpha\) \((-1<\alpha<+\infty)\), of classes of functions meromorphic in the disk \(|z|<1\). These classes possess property (22), i.e. they expand as \(\alpha\) increases, and the known class \(N\), introduced by R. Nevanlinna, is contained in our family \(\{N_\alpha\}\) for the value of the parameter \(\alpha=0\).
Despite the considerable generality of the family of classes \(\{N_\alpha\}\), the theorem on their parametric representation has the same complete character as R. Nevanlinna’s theorem on the class \(N\equiv N_0\).
Theorem 2. The class \(N_\alpha\) \((-1<\alpha<\infty)\) coincides with the set of functions admitting in the disk \(|z|<1\) a representation of the form
\[ F(z)=A_F z^\lambda \frac{\pi_\alpha(z;a_\mu)}{\pi_\alpha(z;b_\nu)} \exp\left\{ \frac{1}{\pi}\int_0^{2\pi} \frac{d\psi(\theta)}{(1-e^{-i\theta}z)^{1+\alpha}} \right\} \quad(|z|<1), \tag{23} \]
where \(\pi_\alpha(z;a_\mu)\) and \(\pi_\alpha(z;b_\nu)\) are convergent products of the form (12), \(\psi(\theta)\) is an arbitrary function of bounded variation on \([0,2\pi]\), and, finally, \(A_F\) is an arbitrary constant.
This theorem, in particular, contains the well-known theorem of R. Nevanlinna on the parametric representation of the class \(N\equiv N_0\).
Institute of Mathematics and Mechanics
Academy of Sciences of the ArmSSR
Received
18 V 1964
References
- R. Nevanlinna, Single-Valued Analytic Functions, 1941.
- M. M. Dzhrbashyan, Dokl. AN ArmSSR, 3, No. 1, 1945.
- M. M. Dzhrbashyan, Communications of the Institute of Mathematics and Mechanics, Academy of Sciences of the ArmSSR, vol. 2, 1948.