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 \(z=0\). [\(\ldots\)]”

\[ M(f,R)=\max_{| z|\leq R}| f(z)|. \]

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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\textit{F. S. Myshakov}, Math. Notes 96, No. 6, 831--835 (2014; Zbl 1317.30037); translation from Mat. Zametki 96, No. 5, 794--798 (2014)

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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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