Geodesic
Lower bounds for entire functions
Ufa mathematical journal, Tome 11 (2019) no. 3, pp. 44-60 Cet article a éte moissonné depuis la source Math-Net.Ru

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We study lower bounds for entire functions of proximate order and of completely regular growth. We introduce the notion of the index condensation for sequences of complex numbers of proximate order. This notion generalizes that of the index of condensation for sequences of order one. We also introduce a properly balanced set, which is a properly distributed set with a zero condensation index. We show that a regular set is properly balanced and we prove that the properly balanced property of the zero set of an entire function is a necessary and sufficient condition for the existence of family of pairwise disjoint circles with the centers at its zeros and with relatively small radii. Outside these circles, the absolute value of the function admits lower bounds asymptotically coinciding with its upper bounds in the entire plane. Thus, we show that the notion of a properly balanced set naturally generalizes the notion of a regular set in the case of arbitrary sequences including multiples sequences. A method for constructing of an exceptional set consisting of circles with centers at zeroes of entire function is also provided. In some cases, we can make the sum of the radii of these circles arbitrarily small.
Keywords: entire function, proximate order, completely regular growth, properly balanced set, regular set. 
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O. A. Krivosheeva; A. S. Krivosheev; A. I. Rafikov. Lower bounds for entire functions. Ufa mathematical journal, Tome 11 (2019) no. 3, pp. 44-60. http://geodesic.mathdoc.fr/item/UFA_2019_11_3_a3/

[1] B. Ya. Levin, Distribution of zeros of entire functions, Amer. Math. Soc., Providence, RI, 1980 | MR

[2] A. S. Krivosheev, “A fundamental principle for invariant subspaces in convex domains”, Izv. Math., 68:2 (2004), 291–353 | DOI | DOI | MR | Zbl

[3] O. A. Krivosheyeva, “Singular points of the sum of a series of exponential monomials on the boundary of the convergence domain”, St. Petersburg Math. J., 23:2 (2012), 321–350 | DOI | MR | Zbl

[4] A. S. Krivosheev, O. A. Krivosheeva, “A basis in invariant subspace of entire functions”, St. Petersburg Math. J., 27:2 (2016), 273–316 | DOI | MR | MR | Zbl

[5] O. A. Krivosheeva, A. S. Krivosheev, “Singular points of the sum of a Dirichlet series on the convergence line”, Funct. Anal. Appl., 49:2 (2015), 122–134 | DOI | DOI | MR | Zbl

[6] A. F. Leont'ev, Entire functions. Exponential series, Nauka, M., 1983 (in Russian) | MR