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Theory of factorization of functions meromorphic in the disk
M. M. Dzhrbashyan
Abstract:
The factorization of functions of the class
$N$ of functions meromorphic in the disk has been established in the well-known theorem due to R. Nevanlinna.
In a monograph the author has constructed a theory of factorization of a family of classes
$N_\alpha$ of functions meromorphic in the disk
$|z|<1$, which classes are monotonically increasing with increasing
$\alpha$ (
$-1<\alpha<+\infty$) and in addition
$N_0=N$.
In the present work, a complete theory of factorization is established, which essentially can be applied to arbitrarily restricted or arbitrarily broad classes of meromorphic functions in the disk
$|z|<1$.
By applying the generalized operator
$L^{(\omega)}$ of Riemann–Liouville type associated with an arbitrary positive continuous function
$\omega(x)$ on
$[0,1)$,
$\omega(x)\in L(0,1)$ (
$\omega(0)=1$), a general formula of Jensen–Nevanlinna type is established which relates the values of a meromorphic function to the distribution of its zeros and its poles.
This formula leads, essentially, to a new concept of the
$\omega$-characteristic function
$T_\omega(r)$ in the class
$N\{\omega\}$ of bounded
$\omega$-characteristic, and of functions
$B_\omega(z;z_k)\in N\{\omega\}$ with zeros
$\{z_k\}_1^\infty$ which satisfy the condition $\sum_{k=1}^\infty\int_{|z_k|}^1\omega(x)\,dx<+\infty$.
Finally, in a series of theorems, parametric representations of the classes
$N\{\omega\}$, as well as of the more restricted classes
$A\{\omega\}$ of functions analytic in the disk, are established. Also their boundary properties are determined. Along with the above it is proved that every function
$F(z)\notin N$ meromorphic in the unit disk belongs to some class
$N\{\omega\}$, and hence admits a suitable factorization.
Bibliography: 17 titles.
UDC:
517.53
MSC: 30D30,
30D35,
30D50 Received: 03.01.1969
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