Abstract
Full Text
MATHEMATICS
I. V. VITENKO, A. N. KOSTOVSKII
DETERMINATION OF THE PRINCIPAL INDICES OF LAURENT SERIES
(Presented by Academician A. A. Dorodnitsyn, 20 XII 1963)
Let a Laurent series be given
\[ f(z)=\sum_{m=-\infty}^{\infty} a_m z^m , \tag{1} \]
converging in the annulus \(r<|z|<R\).
If \(f(xe^{i\varphi})\ne 0\) for \(0\leq \varphi<2\pi,\ r<x<R\), then the value
\[ I(x)=\frac{1}{2\pi i}\int_{|z|=x}\frac{f'(z)}{f(z)}\,dz \tag{2} \]
is called the principal index of the series (1).
In \((^1)\) sufficient conditions are given for the existence of the principal index for values of \(x\) from the interval \((r,R)\). In \((^2)\) necessary and sufficient conditions for the principal index are given when \(f(z)\) is a polynomial. In the present note the results of \((^2)\) are generalized to Laurent series, and also a number of other results are given.
Introduce the notation
\[ u_m^{(k)}=\min_{\alpha>0}\left|a_m^{(k)}:a_{m+\alpha}^{(k)}\right|^{1/\alpha}, \qquad v_m^{(k)}=\max_{\beta>0}\left|a_{m-\beta}^{(k)}:a_m^{(k)}\right|^{1/\beta}, \]
\[ D_m^{(k)}=u_m^{(k)}:v_m^{(k)}, \tag{3} \]
where \(a_{m-\beta}^{(k)},\ a_m^{(k)},\ a_{m+\alpha}^{(k)}\) are the coefficients of the transformed Laurent series
\[ f_k(z)=\sum_{m=-\infty}^{\infty} a_m^{(k)}z^m =\prod_{j=0}^{k-1} f\left(\omega_j^{(k)} z^{1/k}\right), \qquad \omega_j^{(k)}=\exp\left(\frac{2\pi i}{k}j\right), \]
\[ r^k<|z|<R^k \qquad (k=1,2,\ldots). \]
Choose \(x_1\) and \(x_2\) so that \(f(z)\ne 0\) for \(x_1<|z|<x\) and \(x<|z|<x_2\), and \(r\leq x_1<x<x_2\leq R\).
Using the relations between the quantities \(a_m^{(k)}\) and \(x\) given in (3), we can write:
\[ \lim_{k\to\infty}\frac{u_p^{(k)}}{x^k}>0, \qquad \frac{v_p^{(k)}}{x^k}=O(\rho^k) \qquad \left(\frac{x_1}{x}<\rho<1\right), \]
\[ \lim_{k\to\infty}\frac{u_{p+j}^{(k)}}{x^k}>0, \qquad \lim_{k\to\infty}\frac{v_{p+j}^{(k)}}{x^k}<\infty \qquad (j=1,2,\ldots,q-1), \tag{4} \]
\[ \frac{x^k}{u_{p+q}^{(k)}}=O(\tau^k), \qquad \lim_{k\to\infty}\frac{v_{p+q}^{(k)}}{x^k}<\infty \qquad \left(\frac{x}{x_2}<\tau<1\right), \]
\[ p=I(x-0),\qquad q=I(x+0). \]
From (3) and (4) it follows
Theorem 1. In order that the number \(p\) from (2), which satisfies the inequalities \(I(r+0)\le p\le I(R-0)\), be a principal index, it is necessary and sufficient that
\[ \lim_{k\to\infty} D_p^{(k)}=\infty . \tag{5} \]
Without the requirement \(I(r+0)\le p\le I(R-0)\), condition (5) of the theorem will be necessary but, generally speaking, will not be sufficient. This is easy to see from the following example:
\[ f(z)=\sum_{m=0}^{\infty} a_m z^m=(z-2)\exp\left(\sum_{m=0}^{\infty}\frac{1}{r_m^2}z^{r_m}\right), \]
where \(r_0,r_1,\ldots\) is an increasing sequence of prime numbers of the natural series. This series has a unique principal index, equal to zero; nevertheless \(\lim_{k\to\infty}D_1^{(k)}=\infty,\ k=h,h^2,\ldots,h^\nu,\ldots;\ h\) is an integer \(\ge 2\).
Theorem 2. If the series (1) converges in the domain \(0<|z|<\infty\), then condition (5) is necessary and sufficient for the number \(p\) from (2) to be a principal index.
In (¹) it is shown that if \(D_p^{(1)}>9\) for the function (1), then for this function the number \(p\) will be a principal index. Therefore condition (5) in Theorems 1 and 2 may be replaced by the relation
\[ \max_{k\ge 1} D_p^{(k)}>9, \]
where \(D_p^{(k)}\) in (3) is taken for \(f_k(z)\).
In particular, if (1) is a Taylor series (\(a_m=0\) for \(m<0\)), having in its disk of convergence of radius \(R\) the zeros \(0\le |z_1|\le |z_2|\le \cdots \le |z_n|\le \cdots <R\), then Theorem 2 may be formulated as follows:
Theorem 2′. For the inequality \(|z_n|<|z_{n+1}|\) (or \(|z_n|<R\), if the power series has only \(n\) zeros in the disk of convergence) to hold between the moduli of the zeros of a Taylor series, it is necessary and sufficient that
\[ \lim_{k\to\infty}\left( \left|\frac{a_{n-\beta}^{(k)}}{a_n^{(k)}}\right|^{1/\beta} : \min_{\alpha>0}\left|\frac{a_n^{(k)}}{a_{n+\alpha}^{(k)}}\right|^{1/\alpha} \right)=0 \qquad (\beta=1,2,\ldots,n). \]
Let an arbitrary function \(v(t)\), holomorphic in the annulus \(r<|z|<R\), be given. Knowing the principal indices of the series (1), we compute the value of the function
\[ I_v(x)=\frac{1}{2\pi i}\int_{|z|=x}\frac{f'(z)}{f(z)}v(z)\,dz,\qquad r<x<R. \]
Consider the auxiliary Laurent series
\[ Q_{k,v}(z)=\frac{1}{k}\sum_{l=0}^{k-1}\prod_{\substack{j=0\\ j\ne l}}^{k-1} f\!\left(\omega_j^{(k)}z^{1/k}\right) F_v\!\left(\omega_l^{(k)}z^{1/k}\right) =\sum_{m=-\infty}^{\infty} b_{m,v}^{(k)}z^m, \tag{6} \]
where
\[ F_v(u)=u f'(u)v(u),\qquad r<|u|<R,\qquad r^k<|z|<R^k. \]
Suppose that the function \(f(z)\) has no zeros in the annulus \(r\le x_1<|z|<x_2\le R\); then in the annulus \(x_1^k<|z|<x_2^k\) the function \(Q_{k,v}(z)\) can be represented in the form
\[ Q_{k,v}(z)=f_k(z)\frac{1}{k}\sum_{l=0}^{k-1}T\!\left(\omega_l^{(k)}z^{1/k}\right) =f_k(z)\sum_{m=-\infty}^{\infty} t_m z^m, \tag{7} \]
where
\[ T(z)=z\frac{f'(z)}{f(z)}\,v(z)=\sum_{m=-\infty}^{\infty} t_m z^m,\qquad t_0=I_v(x) \]
\[ (x_1<|z|<x_2;\quad x_1<x<x_2). \]
Using the relations for the quantities \(a_m^{(k)}\) and \(t_m\) given in \((3,4)\), from (6) and (7) we can write
\[ I_v(x)=-\frac{b_{p,v}^{(k)}}{a_p^{(k)}}+O(\rho^k)\qquad \left(\frac{x_1}{x_2}<\rho<1\right), \tag{8} \]
\[ p(x)=I(x)\qquad (x_1<x<x_2). \]
Suppose now that in the annulus \(r\le r'<|z|<R'\le R\) the function \(f(z)\) has in all \(q\) zeros \(z_1,z_2,\ldots,z_q\); then
\[ I_v(R'-0)-I_v(r'+0)=\sum_{m=1}^{q} v(z_m). \]
Hence, from (8), we obtain the theorem.
Theorem 3. If the Laurent series (1) in the annulus \(r'<|z|<R'\) has in all \(q\) zeros \(z_1,z_2,\ldots,z_q\), then
\[ \lim_{k\to\infty}\left(\frac{b_{p+q,v}^{(k)}}{a_{p+q}^{(k)}}-\frac{b_{p,v}^{(k)}}{a_p^{(k)}}\right) =\sum_{m=1}^{q} v(z_m),\qquad p=I(r'+0), \tag{9} \]
where \(v(z)\) is an arbitrary holomorphic function in this annulus.
The coefficients \(a_m^{(k)}, b_m^{(k)}\) are determined by the formulas proposed in (4). In particular, if \(k\) takes the discrete values \(1,2,4,8,16,\ldots\), then
\[ a_m^{(2k)}=(-1)^m\left[(a_m^{(k)})^2+2\sum_{j=1}^{\infty}(-1)^j a_{m-j}^{(k)}a_{m+j}^{(k)}\right], \tag{10} \]
\[ b_m^{(2k)}=(-1)^m\left[a_m^{(k)}b_{m,v}^{(k)}+\sum_{j=1}^{\infty}(-1)^j\left(a_{m-j}^{(k)}b_{m+j,v}^{(k)}+a_{m+j}^{(k)}b_{m-j,v}^{(k)}\right)\right], \]
\[ m=0,\pm1,\pm2,\ldots \]
Here
\[ a_m^{(1)}=a_m,\qquad \sum_{m=-\infty}^{\infty} b_{m,v}^{(1)}z^m=zf'(z)v(z),\qquad r<|z|<R. \]
Taking \(v(z)\) successively equal to \(z,z^2,\ldots,z^q\), we find the coefficients of the functions \(f_k(z), Q_{k,z}(z), Q_{k,z^2}(z),\ldots,Q_{k,z^q}(z)\). The transformation of the coefficients of these functions is carried out by formulas (10) or by the formulas given in (4). By formulas (9) we find \(\sum_{m=1}^{q} z_m,\sum_{m=1}^{q} z_m^2,\ldots,\sum_{m=1}^{q} z_m^q\), and hence we can compose a polynomial whose roots will be the roots \(z_1,z_2,\ldots,z_q\) of the function (1) from the annulus \(r'<|z|<R'\).
Lviv State University
named after Ivan Franko
Received
30 XI 1963
REFERENCES
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