Entire function with prescribed growth

O.Ya.Brodyak; Ya.V.Vasyl'kiv

It is established that for an entire function constructed by A. Edrei and W. H. J. Fuchs (Can. J. Math. — 1965. — V. 17, No 3. — P. 383–395) in the form of a certain infinite product, in connection with appropriate Valiron's problem (Ann. Fac. Sci. Toulouse. — 1913. — Vol. 5. — P. 117–208), the following asymptotic relations are valid:

$ \log m_p(r,f) \sim m_q(r,\log f) \sim \lambda(r) \quad (r \to +\infty), $

where $ 0 \le p < +\infty,\ 1 \le q \le \frac{s}{s-1} $, $ m_s(r,*) $ are s-th Lebesgue integral means, $ \lambda(r) $ is a positive, continuous, increasing to $ +\infty $ and convex with respect to $ \log r $ function such that

$ \lambda\left(r + \frac{1}{\lambda(r)}\right) < \exp\left(\lambda^\eta(r)\right) $

for a certain $ \eta \in (0, \tfrac{1}{2}) $, and $ r\lambda'(r)\,(\log\log(r\lambda'(r)))^s = o(\lambda^s(r)) $ as $ r \to +\infty,\ 1 < s \le 2 $.

1. Valiron~G. Sur les fonctions entières d'ordre fini et d'ordre nul, et en particulier les fonctions a correspondance régulière// Ann. Fac. Sci. Toulouse. - 1913. - V.5. - P.117-208.

2. Edrei~A., Fuchs W.H.P. Entire and meromorphic functions with asymtotically prescribed characteristic// Canad. J. Math. - 1965. - V.17, № 3. - P.383-395.

3. Clunie~J. On integral functions having prescribed asymptotic growth// Canad. J. Math. - 1965. - V.17, № 3. - P.396-404.

4. Clunie~J., Kövari T. On integral functions having prescribed asymptotic growth. II// Canad. J. Math. - 1966. - V.18. - P.7-20.

5. Miles~J.~B., Shea D.F. An extremal problem in value distribution theory// Quart. J. Math. (Oxford). - 1973. - V.24. - P.377-383.

6. Kalynets~R.Z., Kondratyuk A.A. On the growth regularity of modulus and argument of entire function in $L^p[0, 2\pi]$-metrics. // Ukr. Math. J. - 1998. - V.50, № 7. - P.889-896. (in Ukrainian)

7. Vasyl'kiv~Ya.~V. Asymptotic behavior of the logarithmic derivatives and the logarithms of meromorphic functions of completely regular growth in $L^p[0, 2\pi]$-metric.I// Mat. Studii. - 1999. - V.12, № 1. - P.37-58. (in Ukrainian)

8. Brydun~A.~M., Lyzun O.Ya., Mytsyk R.Z. Generalization of Lindelöf theorem for entire functions. // Visnyk L'viv Univ. - 2000. - V.56. - P.20-27. (in Ukrainian)

9. Kondratyuk~А.~А. Fourier series and meromorphic functions. - L'viv: Vyscha Shkola, 1988. (in Russian)

10. Hardy~G.~H., Wright E.M. An Introduction to the Theory of Numbers. - Oxford: Oxford Univ. Press, Fifth Edition, 1979.

11. Robin~G. Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann// J. Math. Pures Appl. - 1984. - V.63. - P.187-213.

12. Hayman~W.~K. Meromorphic functions. - Moscow: Mir, 1966. (in Russian)

13. Gol'dberg~A.~A., Ostrovskii I.V. Value distributions of meromorphic functions. - Moscow: Nauka, 1970. (in Russian)

14. Zygmund~А. Trigonometric series: Vols I,II. - Moscow: Mir, 1965. (in Russian)

	Entire function with prescribed growth
Author	O.Ya.Brodyak, Ya.V.Vasyl'kiv brodyakoksana@mail.ru, YaVasylkiv@gmail.com Lviv Polytechnical National University, Lviv Ivan Franko National University
Abstract	It is established that for an entire function constructed by A. Edrei and W. H. J. Fuchs (Can. J. Math. — 1965. — V. 17, No 3. — P. 383–395) in the form of a certain infinite product, in connection with appropriate Valiron's problem (Ann. Fac. Sci. Toulouse. — 1913. — Vol. 5. — P. 117–208), the following asymptotic relations are valid: \( \log m_p(r,f) \sim m_q(r,\log f) \sim \lambda(r) \quad (r \to +\infty), \) where \( 0 \le p < +\infty,\ 1 \le q \le \frac{s}{s-1} \), \( m_s(r,*) \) are s-th Lebesgue integral means, \( \lambda(r) \) is a positive, continuous, increasing to \( +\infty \) and convex with respect to \( \log r \) function such that \( \lambda\left(r + \frac{1}{\lambda(r)}\right) < \exp\left(\lambda^\eta(r)\right) \) for a certain \( \eta \in (0, \tfrac{1}{2}) \), and \( r\lambda'(r)\,(\log\log(r\lambda'(r)))^s = o(\lambda^s(r)) \) as \( r \to +\infty,\ 1 < s \le 2 \).
Keywords	entire function, prescribed growth, infinite product, Valiron's problem, Lebesgue integral mean
DOI	doi:10.30970/ms.33.1.56-64
Reference	1. Valiron~G. Sur les fonctions entières d'ordre fini et d'ordre nul, et en particulier les fonctions a correspondance régulière// Ann. Fac. Sci. Toulouse. - 1913. - V.5. - P.117-208. 2. Edrei~A., Fuchs W.H.P. Entire and meromorphic functions with asymtotically prescribed characteristic// Canad. J. Math. - 1965. - V.17, № 3. - P.383-395. 3. Clunie~J. On integral functions having prescribed asymptotic growth// Canad. J. Math. - 1965. - V.17, № 3. - P.396-404. 4. Clunie~J., Kövari T. On integral functions having prescribed asymptotic growth. II// Canad. J. Math. - 1966. - V.18. - P.7-20. 5. Miles~J.~B., Shea D.F. An extremal problem in value distribution theory// Quart. J. Math. (Oxford). - 1973. - V.24. - P.377-383. 6. Kalynets~R.Z., Kondratyuk A.A. On the growth regularity of modulus and argument of entire function in $L^p[0, 2\pi]$-metrics. // Ukr. Math. J. - 1998. - V.50, № 7. - P.889-896. (in Ukrainian) 7. Vasyl'kiv~Ya.~V. Asymptotic behavior of the logarithmic derivatives and the logarithms of meromorphic functions of completely regular growth in $L^p[0, 2\pi]$-metric.I// Mat. Studii. - 1999. - V.12, № 1. - P.37-58. (in Ukrainian) 8. Brydun~A.~M., Lyzun O.Ya., Mytsyk R.Z. Generalization of Lindelöf theorem for entire functions. // Visnyk L'viv Univ. - 2000. - V.56. - P.20-27. (in Ukrainian) 9. Kondratyuk~А.~А. Fourier series and meromorphic functions. - L'viv: Vyscha Shkola, 1988. (in Russian) 10. Hardy~G.~H., Wright E.M. An Introduction to the Theory of Numbers. - Oxford: Oxford Univ. Press, Fifth Edition, 1979. 11. Robin~G. Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann// J. Math. Pures Appl. - 1984. - V.63. - P.187-213. 12. Hayman~W.~K. Meromorphic functions. - Moscow: Mir, 1966. (in Russian) 13. Gol'dberg~A.~A., Ostrovskii I.V. Value distributions of meromorphic functions. - Moscow: Nauka, 1970. (in Russian) 14. Zygmund~А. Trigonometric series: Vols I,II. - Moscow: Mir, 1965. (in Russian)
Pages	56-64
Volume	33
Issue	1
Year	2010
Journal	Matematychni Studii
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