Uniqueness Distributions for Entire Functions with Uniform Constraints on Their Growth

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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/v3523-1431-1350-j Uniqueness Distributions for Entire Functions with Uniform Constraints on Their Growth Khabibullin, B. N. Vladikavkaz Mathematical Journal 2025. Vol. 27. Issue 1. Abstract: Let \(M=M_{\mathsf{up}}-M_{\mathsf{low}}\) be the difference of subharmonic functions on the complex plane \(\mathbb C\). First, we discuse the following general problem: What are the conditions for the distribution of points \(Z\) on \({\mathbb{C}}\), under which there is an entire nonzero function \(f\) that vanishes on \(Z\) and satisfies the inequality \(\|f\|\leq e^M\) on \(\mathbb{C}\)? We formulate some known results for the general problem from one of our papers with co-authors. The next step is to discuss a specific problem of when \(M_{\mathsf{up}}=b\|\mathrm{Im}\|\) is the module of the imaginary part with a numerical multiplier \(b\geq 0\), and \(M_{\mathsf{low}}\) is the Poisson transformation of a positive even function \(w\) on the real axis \({\mathbb{R}}\), increasing on the positive semi-axis \({\mathbb{R}}^+\), and with a finite logarithmic integral. A very significant contribution to this theory is contained in a number of fundamental works by A. V. Abanin, including his known monograph. It is precisely such classes of entire functions that arise after the Fourier--Laplace transform of test functions on compacts. In this direction, the article discusses the limits of applicability of the Beurling--Malliavin theory, and also provides our criterion with co-authors, but only for the zero function \(w=0\). The final main result of the article extends the last criterion to the cases of a nonzero function \(w\neq 0\). Keywords: entire function, point distribution, zero distribution, subharmonic function, mass distribution, Cartwright class, logarithmic integral, Poisson integral, ultradifferentiable function, ultradistribution Language: Russian Download the full text For citation: Khabibullin, B. N. Uniqueness Distributions for Entire Functions with Uniform Constraints on Their Growth, Vladikavkaz Math. J., 2025, vol. 27, no. 1, pp. 112-126 (in Russian). DOI 10.46698/v3523-1431-1350-j + References 1. Hayman, W. and Kennedy, P. Subharmonic Functions, Vol. 1, London Mathematical Society Monographs, no. 9, London, Academic Press, 1976, xvii+284 p. 2. Ransford, Th. Potential Theory in the Complex Plane, Cambridge, Cambridge University Press, 1995, 232 p. 3. Khabibullin, B. N. and Khabibullin, F. B. Necessary and Sufficient Conditions for Zero Subsets of Holomorphic Functions with Upper Constraints in Planar Domains, Lobachevskii Journal of Mathematics, 2021, vol. 42, no. 4, pp. 800-810. 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