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DOI: 10.46698/n0335-8321-3720-b

On an Estimate of M. M. Djrbashyan's Product \(B_{\omega}\)

Tavaratsyan, T. V.
Vladikavkaz Mathematical Journal 2022. Vol. 24. Issue 3.
Abstract:
† 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 \(\omega (x)\in \Omega_0\).
Keywords: Djrbashyan products, Blaschke products, convex sequences, class of functions \(\Omega_0\), Fourier series
Language: Russian Download the full text  
For citation: Tavaratsyan, T. V. On an Estimate of M. M. Djrbashyan's Product \(B_{\omega}\),† Vladikavkaz Math. J., †2022, vol. 24, no. 3, pp. 133-143 (in Russian). DOI 10.46698/n0335-8321-3720-b
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