# On estimates of a fractional counterpart of the logarithmic derivative of a meromorphic function

Author I. E. Chyzhykov, N. S. Semochko
chyzhykov@yahoo.com, semochkons@mail.ru
Lviv Ivan Franko National University

Abstract We consider the problem of finding lower bounds for growth of solutions of a fractional differential equation in the complex plane. We estimate a fractional integral of the logarithmic derivative of a meromorphic function.
Keywords Riemann-Liouville operator; fractional derivative; fractional differential equation; logarithmic derivative; meromorphic function; Nevanlinna’s characteristic; growth
Reference 1. W. Bergweiler, Ph.J. Rippon, G.M. Stallard, Dynamics of meromorphic functions with direct or logarithmic singullarities, Proc. London Math. Soc., (3) 97 (2008), 368–400.

2. I. Chyzhykov, G.G. Gundersen, J. Heittokangas, Linear differential equations and logarithmic derivative estimates, Proc. London Math. Soc., (3) 86 (2003), 735–754.

3. I. Chyzhykov, J. Heittokangas, J. Rattya, On the finiteness of phi-order of solutions of linear differential equations in the unit disc, J. d’Analyse Math., 109 (2009), V.1, 163–196.

4. I. Chyzhykov, J. Heittokangas, J. R.atty.a, Sharp logarithmic derivative estimates with applications to ODE’s in the unit disc, J. Austr. Math. Soc., 88 (2010), 145–167.

5. S.D. Eidelman, S.D. Ivasyshen, A.N. Kochubei, Analitic Methods in the Theory of Differential and Pseudo-Differential Equations of Parabolic Type, Birkh.auser, Basel, 2004.

6. P.C. Fenton, M.M. Strumia, Wiman-Valiron theory in the disc, Journal of the London Mathematical Society, 79 (2009), 478–496.

7. A.A. Gol’dberg, I.V. Ostrovskii, Value distribution of meromorphic functions, Transl. Math. Monographs, V.236, Amer. Math. Soc., 2008.

8. A.A. Gol’dberg, N.N. Strochik, Asymptotic behavior of meromorphic functions of completely regular growth and of their logarithmic derivatives, Siberian Math. Journal, 26 (1985), ¹6, 802–809.

9. G. Gundersen, Estimates for the logarithmic derivative of a meromorphic function, plus similar estimates, J. London Math. Soc., (2) 37 (1988), 88–104.

10. H. Hedenmaln, B. Korenblum, K. Zhu, Theory of Bergman spaces. – Springer Verlag, 2001.

11. A.A. Kilbas, M. Rivero, L. Rodriguez-Germa, J.J. Trujilo, alpha-Analitic solutions of some linear fractional differential equations with variable coefficients, Appl. Math. Comput., 187 (2007), 239–249.

12. A.A. Kilbas, H.M. Srivastava, J.J. Trujilo, Theory and Applications of Fractiinal Differential Equations. – Elsevier, Amsterdam, 2006.

13. A.N. Kochubei, Fractional differential equations: alpha-entire solutions, regular and irregular singularities, Fract. Calc. Appl. Anal., 12 (2009), ¹2, 135–158.

14. I. Laine, Nevanlinna Theory and Complex Differential Equations. – Walter de Gruyter, Berlin, 1993.

15. T.M. Salo, O.B. Skaskiv, Ya.Z. Stasyuk, On a central exponent of entire Dirichlet series, Mat. Stud. 19 (2003), ¹1, 61–72.

16. S.G. Samko, A.A. Kilbas, O.I. Marichev, Integrals and derivatives of fractional order and some of their applications, Nauka i tekhnika, Minsk, 1987. (in Russian)

17. O. Skaskiv, On the logarithmic derivative and central exponent of Dirichlet series, Math. Bull. of Shevchenko Scientific Soc., 2 (2005), 213–230. (in Ukrainian)

18. O.B. Skaskiv, Ya.Z. Stasyuk, On derivatives of Dirichlet series, Visnyk of Lviv Univ., 64 (2005), 258–265. (in Ukrainian)

19. Sh. Strelitz, Asymptotic properties of analytic solutions of differential equations. – Izdat Mintis, Vilnius, 1972, 468 p. (in Russian)

20. H. Wittich, Neuere Untersuchungen uber eindeutige analytische Funktionen, 2nd edn. – Springer, Berlin, 1968.

Pages 107-112
Volume 39
Issue 1
Year 2013
Journal Matematychni Studii
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