Growth of solutions of second order linear differential equations with extremal functions for Denjoy's conjecture as coefficients

Main Article Content

Jianren Long

Abstract

The classical problem of finding conditions on the entire coefficients $A(z)$ and $B(z)$ guaranteeing that all nontrivial solutions of $f''+A(z)f'+B(z)f=0$ are of infinite order is discussed. Some such conditions which involve deficient value, Borel exceptional value and extremal functions for Denjoy's conjecture are obtained.

Article Details

How to Cite
Long, J. (2016). Growth of solutions of second order linear differential equations with extremal functions for Denjoy’s conjecture as coefficients. Tamkang Journal of Mathematics, 47(2), 237–247. https://doi.org/10.5556/j.tkjm.47.2016.1914
Section
Papers
Author Biography

Jianren Long

School ofMathematical Science, Guizhou Normal University, 550001, Guiyang, P.R. China. Department of Physics andMathematics, University of Eastern Finland, P.O. Box 111, 80101, Joensuu, Finland.

References

L. Ahlfors, Untersuchungen zur Theorie der konformen abbildung und der Theorie der ganzen Funktionen, Acta Soc. Sci. Fenn.,1(1930), 1--40.

T. B. Cao, Growth of solutions of a class of complex differential equations, Ann. Polon. Math. 95(2009), 141--152.

Z. X. Chen, The growth of solutions of $f''+e^{-z}f'+Q(z)f=0$ where the order (Q)=1, Sci. China Ser. A, 45(2002), 290--300.

Z. X. Chen and C. C. Yang, Some further results on the zeros and growths of entire solutions of second order linear differential equations, Kodai Math. J., 22(1999), 273--285.

C. J. Dai and S. Y. Ji, Radial line of order $rho$ and its relation to the distribution of Borel directions, J. Shanghai Normal Uni. Nat. Sci., 22(1980), 16--24.

A. Denjoy, A. Sur les fonctions entieres de genre fini, C. R. Acad. Sci. Paris, 45(1907), 106--109.

D. Drasin, On asymptotic curves of functions extremal for Denjoy's conjecture, Proc. London Math. Soc., 26(1973), 142--166.

D. Drasin and A. Weitsman, On the Julia Directions and Borel Directions of Entire Functions, Proc. London Math. Soc., 32(1976), 199--212.

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

G. G. Gundersen, Finite order solution of second order linear differential equations, Trans. Amer. Math. Soc., 305(1988), 415--429.

S. Hellerstein, J. Miles and J. Rossi, On the growth of solutions of $f''+gf'+hf=0$, Trans, Amer. Math. Soc., 324(1991), 693--705.

W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1964.

P. B. Kennedy, A class of integral functions bounded on certain curves, Proc. London Math. Soc., 6(1956), 518--547.

K. Kwon, On the growth of entire functions satisfying second order linear differential equations, Bull. Korean Math. Soc., 33(1996),

--496.

K. Kwon, K. and J. Kim, Maximum modulus, characteristic, deficiency and growth of solutions of second order linear differential equation, Kodai Math. J., 24(2001), 344-351.

I. Laine, Nevanlinna Theory and Complex Differential Equations, Walter de Gruyter Berlin, New York, 1993.

I. Laine and P. C. Wu, Growth of solutions of second order linear differential equations, Proc. Amer. Math. Soc., 128(2000), 2693--2703.

J. R. Long, P. C. Wu and Z. Zhang, On the growth of solutions of second order linear differential equations with extremal coefficients, Acta Math. Sinica (English Series), (2013), 365--372.

A. I. Markushevich, Theory of Functions of a Complex Variable, Vol. $Pi$, Revised English Edition Translated and Edited by Richard A. Silverman. Prentice-Hall, Inc., Englewood Cliffs, N. J., 1965.

M. Tsuji, On Borel's directions of meromorphic functions of finite order, Tohoku Math. J., 2(1950), 97--112.

M. Ozawa, On a solution of $w''+e^{-z}w'+(az+b)w=0$, Kodai Math. J., 3(1980), 295--309.

J. Wang and I. Laine, Growth of solutions of second order linear differential equations, J. Math. Anal. Appl.,342(2008),39--51.

S. P. Wang, On the sectorial oscillation theory of $f''+A(z)f=0$, Ann. Acad. Sci. Fenn. Ser. A I Math. Diss., 92(1994), 1--60.

H. Wittich, Neuere untersuchungen uber eindeutige analytische Funktionen, Springer-Verlag, 1955.

P. C. Wu and J. Zhu, On the growth of solutions of the complex differential equation $f''+Af'+Bf=0$}, Sci. China Ser. A 54(2011), 939--947.

S. J. Wu, On the distribution of Borel directions of entire function, Chinese Ann. Math., 14A(1993), 400--406.

C. C. Yang and H. X. Yi, Uniqueness Theory of Meromorphic Functions, Kluwer Academic Publishers, New York, 2003.

L. Yang, Value Distribution Theory, Springer-Verlag, Berlin, 1993.

L. Yang, Deficient values and angular distibution of entire functions, Trans. Amer. Math. Soc., 308(1988), 583--601.

L. Yang and C. C. Yang, Angular distribution of $ff'$, Sci. China Ser. A, 37(1994), 284--294.

L. Yang and G. H. Zhang, Distribution of Borel directions of entire functions, Acta Math. Sinica, 19(1976), 157--168.

G. H. Zhang, On entire function extremal for Denjoy's conjecture, Sci. Sinica (I), 24(1981), 885-898.

G. H. Zhang, Theory of Entire and Meromorphic Functions-Deficient and Asymptotic values and Singular Directions, Springer-Verlag, Berlin, 1993.