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	Dear authors! Submission of all materials is carried out only electronically through Online Submission System in personal account. 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). ← Contents of issue


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