Let u be a subharmonic function of order zero in ℝm, m ≥ 2, with Riesz measure μ on the negative semiaxis Ox1, n(r, u) = μ ({x ∈ ℝm: |x| ≤ r}), dm = m − 2 for m ≥ 3, d2 = 1, and \(N\left(r,u\right)={d}_{m}{\int }_{1}^{r}\frac{n\left(t,u\right)}{{t}^{m-1}}dt.\) Under the condition of slow growth of N(r, u), we determine the asymptotics of u(x) as |x| = r → +∞. We also study the inverse relationship between the regular growth of u and the behavior of N(r, u) as r → +∞.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
G. Valiron, “Sur les fonctions entières d’ordre nul et d’ordre fini, et en particulier sur les fonctions à correspondance régulière,” Ann. Fac. Sci. Univ. Toulouse, 5, No. 3, 117–257 (1913).
E. C. Titchmarsh, “On integral functions with real negative zero,” Proc. London Math. Soc., 26, No. 2, 185–200 (1927).
M. V. Zabolots’kyi, “Valiron-type and Valiron–Titchmarsh-type theorems for entire function of order zero,” Ukr. Mat. Zh., 48, No. 3, 315–325 (1996); English translation: Ukr. Math. J., 48, No. 3, 354–366 (1996).
A. A. Gol’dberg and I. V. Ostrovskii, Distribution of Values of Meromorphic Functions [in Russian], Nauka, Moscow (1970).
P. Z. Agranovich, “Polynomial asymptotic representation of subharmonic functions with masses on one ray in the space,” Mat. Studii, 23, No. 2, 169–178 (2005).
A. I. Kheyfits, “Valiron–Titchmarsh theorem for subharmonic functions in ℝn with masses on a half-line,” Ann. Fac. Sci. Univ. Toulouse, 23, No. 1, 159–173 (2014).
E. Seneta, Regularly Varying Functions [Russian translation], Nauka, Moscow (1985).
M. V. Keldysh, “On one Tauberian theorem,” Tr. Mat. Inst. Akad. Nauk SSSR, 38, 77–86 (1951).
W. K. Hayman and P. B. Kennedy, Subharmonic Functions [Russian translation], Mir, Moscow (1980).
M. V. Zabolots’kyi, “Asymptotics of Blaschke products the counting function of zeros of which is slowly increasing,” Ukr. Mat. Zh., 52, No. 12, 1650–1660 (2000); English translation: Ukr. Math. J., 52, No. 12, 1882–1895 (2000).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, No. 11, pp. 1523–1532, November, 2022. Ukrainian DOI: https://doi.org/10.37863/umzh.v74i11.7251.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Zabolotskyy, M.V., Zabolotskyy, T.M. & Tarasyuk, S.I. Valiron-Type and Valiron–Titchmarsh-Type Theorems for Subharmonic Functions of Slow Growth. Ukr Math J 74, 1739–1751 (2023). https://doi.org/10.1007/s11253-023-02167-w
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-023-02167-w