On an estimate of M.~M.~Djrbashyan's product $B_{\omega}$

T. V. Tavaratsyan

Vladikavkazskij matematičeskij žurnal, Tome 24 (2022) no. 3, pp. 133-143

Résumé

In the mid-60s, by M. M. Djrbashyan proposed a new method for the definition and factorization of wide classes of functions meromorphic in the unit circle. These classes, which are denoted by $N\{\omega\}$, have a complex structure and cover all meromorphic functions in the unit circle due to the fact that they depend on a functional parameter $\omega (x)$. They go to classes $N_{\alpha }$ in case $\omega (x)=(1-x)^{\alpha}$, $-1\alpha +\infty$, and in special case $\omega (x)\equiv 1$, the class $N\{ \omega\}$ is the same as Nevanlinna's class. The fundamental role in the theory of factorization of these classes is played by the products $B_{\omega}$ of M. M. Djrbashyan, which in the case $\omega (x)=(1-x)^{\alpha}$, $-1\alpha +\infty$, turn into the products $B_{\alpha}$ of M. M. Djrbashyan. In a special case $\omega (x)\equiv 1$, products $B_{\omega}$ are transformed into products by Blaschke. Using the well-known theorem on nonnegative trigonometric series, V. S. Zakaryan, obtained upper estimations for the modules of functions $B_{\alpha}$, for $-1\alpha 0$ . In this work, using a similar method, it is proved that $U_{\omega}(z;\zeta )\ge 0$, where $U_{\omega}$ is some auxiliary function. Next, using this result, upper estimations are given for the modules of products $B_{\omega}$ when $\omega (x)\in \Omega_0$.