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

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system's guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	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 + References 1. Djrbashyan, M. M. The Parametric Representation of Some General Classes of Meromorphic Functions in the Unit Circle, Doklady Akademii Nauk SSSR, 1964, vol. 157, no. 5, pp. 1024-1027 (in Russian). 2. Djrbashyan, M. M. and Zakaryan, V. S. Klassy i granichnye svojstva funkcij meromorfnyh v kruge [Classes and Boundary Properties of Meromorphic Functions in a Circle], Moscow, Nauka, 1993 (in Russian). 3. Nevanlinna, R. Einduetige Analytische Funktionen, Berlin, Springer, 1937. 4. Privalov, I. I. Granichnye svojstva analiticheskih funkcij [Boundary Properties of Analytic Functions], Moscow, Leningrad, State Publishing House of Technical-Theoretical Literature, 1950 (in Russian). 5. Duren, P. L. Theory of \(H^p\) Spaces, New York and London, Academic Press, 1970, 260 p. 6. Jerbashian, A. M. Functions of \(\alpha\)-Bounded Type in the Half-Plane, Springer Sience+Business Media, Ins, 2005. DOI: 10.1007/b102102. 7. Zakaryan, V. S. and Dallakyan, R. V. On the Boundary Values of Functions of \(A_{\omega} \) Classes, Reports of the National Academy of Sciences of Armenia, 2012, vol. 112, no. 2, pp. 135-140 (in Russian). 8. Zakaryan, V. S., Dallakyan, R. V. and Djrbashyan, A. M. About the \(\omega\)-Characteristics of Analytic Functions in the Unit Circle, Reports of the National Academy of Sciences of Armenia, 2013, vol. 113, no. 1, pp. 22-29 (in Russian). 9. Dallakyan, R. V. On \(C\)-Growth of \(\omega \)-Characteristics of Analytic Functions in the Unit Circle, Reports of the National Academy of Sciences of Armenia, 2013, vol. 113, no. 2, pp. 142-149 (in Russian). 10. Bari, N. K.Trigonometricheskie ryady [Trigonometric Series], Moscow, Fizmatgiz, 1951 (in Russian). 11. Zakaryan, V. S. About one Estimate for the Product of M. М. Djrbashyan, Izvestia AN Arm. SSR, 1988, vol. 23, no. 2, p. 189-192. (in Russian). The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. The terms of use of this work are governed by an open license (Creative Commons Attribution-NonCommercial 4.0 International). The use of metadata of the scientific article, including the title, abstract, author information, references, identifiers, and other bibliographic description elements for subsequent unrestricted use, is carried out under the terms of the CC BY or CC0 open licenses. ← Contents of issue


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