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An analog of the Valiron-Goldberg theorem under a restriction condition on the averaged counting function of zeros. (English. Russian original) Zbl 1317.30037
Math. Notes 96, No. 6, 831-835 (2014); translation from Mat. Zametki 96, No. 5, 794-798 (2014).
From the introduction: “A century ago, G. Valiron [Toulouse Ann. (3) 5, 117–257 (1914; JFM 46.1462.03)] studied the following problem. There is an entire function \(f\) of finite nonintegral order. The number \(n_f(R)\) of roots of \(f\) in the disk \(| z|\leq R\) is given, and it is a “sufficiently regular” function. It is required to find the best possible asymptotic upper bound of the quantity \(\ln M(f,R)\), where
\[ M(f,R)=\max_{| z|\leq R}| f(z)|. \]
This problem was solved by Valiron in the “first approximation” in [loc.cit.], and later A. A. Goldberg [Mat. Sb., N. Ser. 58 (100), 289–334 (1962; Zbl 0121.29101)] rigorously proved this result and showed that it cannot be improved. Further results in this area were recently obtained by A. Yu. Popov [Sb. Math. 204, No. 5, 683–725 (2013); translation from Mat. Sb. 204, No. 5, 67–108 (2013; Zbl 1283.30061)]. In the present paper, we solve a similar problem with a prescribed majorant of the averaged counting function of the set of nonzero roots of \(f(z)\):
\[ N_f(R)=\int_0^R\bigg(\frac{n_f^0(x)}{x}\bigg) dx, \]
where \(n^0_f(x)=n_f^{}(x)-m\) and \(m\) is the multiplicity of the root of \(f(z)\) at the pooint \(z=0\). [\(\ldots\)]”

MSC:
30D20 Entire functions of one complex variable, general theory
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
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References:
[1] Valiron, G, No article title, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys. (3), 5, 117-257, (1913)
[2] Goldberg, A A, No article title, Mat. Sb., 58, 289, (1962)
[3] Popov, A Yu, No article title, Mat. Sb., 204, 67, (2013)
[4] B. Ya. Levin, Distribution of Zeros of Entire Functions (Gostekhizdat, Moscow, 1956) [in Russian]. · Zbl 0111.07401
[5] Denjoy, A, No article title, J. Math., 6, 1, (1910)
[6] A. A. Goldberg and I. V. Ostrovskii, The Distribution of Values of Meromorphic Functions (Nauka, Moscow, 1970) [in Russian].
[7] Khabibullin, B N, No article title, Mat. Sb., 200, 129, (2009)
[8] E. Seneta, Regularly Varying Functions (Springer-Verlag, Berlin, 1976; Nauka, Moscow, 1985). · Zbl 0324.26002
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