Uniqueness distributions for entire functions with uniform constraints on their growth

B. N. Khabibullin

Vladikavkazskij matematičeskij žurnal, Tome 27 (2025) no. 1, pp. 112-126

Résumé

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 $w\neq 0$.