Theory of factorization of functions meromorphic in the disk

M. M. Dzhrbashyan

Sbornik. Mathematics, Tome 8 (1969) no. 4, pp. 493-592

Résumé

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.