On the absence of an exceptional set in the main relation of Wiman-Valiron theory
DOI:
https://doi.org/10.64700/mmm.94Keywords:
Analytic function, Wiman-Valiron theory, main relation, exceptional setAbstract
The object of investigation is a class of functions analytic in the right half-plane. The half-plane contains all points whose real part is greater than some fixed \(a\) and for all \(x\) from the interval \((a,+\infty)\) the corresponding supremum of modulus of the function \(F\) inside some vertical strip is finite. There is selected subclass consisting of those functions, for which the right-hand derivative of the supremum logarithm tends to infinity. For these functions for which there exists an auxiliary function with some local behavior. The following theorem is proved: If an analytic function belongs to the described class, then for each positive natural \(k\) the \(k\)-th order derivative equals \(k\)-th order power of the right-hand derivative of the supremum logarithm for every point.
References
[1] V. Baksa, A. Bandura and O. Skaskiv: Maximum modulus of slice entire regular functions of quaternionic variable with bounded index, ALTAY Conf. Proc. Math., 2 (1) (2025), 32–39.
[2] V. Baksa, A. Bandura and O. Skaskiv: Growth estimates for analytic vector-valued functions in the unit ball having bounded L-index in joint variables, Constr. Math. Anal., 3 (1) (2020), 9–19.
[3] S. I. Dubei, O. B. Skaskiv and O. M. Trusevych: On the absence of an exceptional set in the relation from Wiman’s theorem, Precarpathian Bulletin of Shevchenko Scientific Society, 21 (79) (2025), 28–34.
[4] S. I. Dubei, O. B. Skaskiv: On the main relation of the Wiman-Valiron theory and asymptotic h-density of an excepTional sets, Precarpathian Bulletin of Shevchenko Scientific Society, 19 (73) (2024), 18–23.
[5] R. Gorenflo: Über ganze transzendente funktionen von regelmäßigem wachstum, Math. Ann., 146 (3) (1962), 226–231.
[6] W. K. Hayman: The local growth of power series: a survey of the Wiman–Valiron method, Canad. Math. Bull., 17 (3) (1974), 317–358.
[7] W. K. Hayman: Subharmonic functions: Volume 2, Academic Press, London (1989).
[8] R. R. London: The behaviour of certain entire functions near points of maximum modulus, J. Lond. Math. Soc. (2), 12 (4) (1976), 485–504.
[9] M. Milgram, R. Hughes: On a generalized moment integral containing Riemann’s zeta function: Analysis and experiment, Modern Math. Methods, 3 (1) (2025), 14–41.
[10] G. Polya, G. Szegö: Problems and theorems in Analysis. I, Springer-Verlag, Heidelberg (1964).
[11] T. M. Salo, O. B. Skaskiv and Ya. Z. Stasyuk: On a central exponent of entire Dirichlet series, Mat. Stud., 19 (1) (2003), 61–72.
[12] M. N. Sheremeta: The Wiman-Valiron method for Dirichlet series, Ukrainian Math. J., 30 (4) (1978), 376–383. Translated from Ukr. Mat. Zh., 30 (6) (1978), 488–497.
[13] M. N. Sheremeta: Asymptotic properties of entire functions defined by Dirichlet series and of their derivatives, Ukrainian Math. J., 31 (6) (1979), 558–564. Translated from Ukr. Mat. Zh., 31 (6) (1979), 723–730.
[14] M. N. Sheremeta: On the derivative of an entire Dirichlet series, Math. USSR Sb., 65 (1) (1990), 133–145.
[15] M. M. Sheremeta: Entire Dirichlet series, ISDO (1993). (in Ukrainian).
[16] O. Skaskiv, A. Bandura, T. Salo and S. Dubei: Entire functions of several variables: analogs of Wiman’s Theorem, Axioms, 14 (3) (2025), Article ID: 216.
[17] O. B. Skaskiv, Ya. Z. Stasyuk: On the Wiman theorem for absolutely convergent Dirichlet series, Mat. Stud., 20 (2) (2003), 133–142.
[18] O. B. Skaskiv, Ya. Z. Stasyuk: On derivatives of Dirichlet series, Nauk. Visn. Chernivets’kogo Univ. Mat., 239 (2005), 107–111. (in Ukrainian).
[19] O. B. Skaskiv, Ya. Z. Stasyuk: On equivalence of the logarithmic derivative and the central exponent of the Dirichlet series, Mat. Stud., 33 (1) (2010), 49–55.
[20] O. B. Skaskiv: A generalization of the little Picard theorem, J. Math. Sci., 48 (5) (1990), 570–578. Translated from Teoriya Funktsii, Funktsional’nyi Analiz i Ikh Prilozheniya, 46 (1986), 90–100.
[21] O. Skaskiv, A. Bandura and A. Bodnarchuk: About Borel type relation for some positive integrals, ALTAY Conf. Proc. Math., 2 (1) (2025), 40–47.
[22] Sh. I. Strelitz: Asymptotic properties of analytical solutions of differential equations, Vilnius: Mintis (1972). (In Russian).
[23] H. Wittich: Neuere untersuchungen über eindeutige analytische functionen, Springer-Verlag, Berlin (1955).
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