Growth of solutions of second order linear differential equations.

*(English)*Zbl 0952.34070Consider the linear differential equation
\[
f''+A(z)f'+B(z)f=0, \tag{1}
\]
where \(A(z), B(z)\not\equiv 0\) are entire functions satisfying \(\rho(B)<\rho(A).\) The following question is natural: If \(A(z)\) has no finite deficient values, does every nonconstant solution to (1) have infinite order? The authors study the growth of solutions to (1) under a condition related to this question. The main result is stated as follows: If \(\rho(B)< \rho(A)<\infty\) and
\[
T(r,A)/\log M(r,A)\to 1 \tag{2}
\]
as \(r\to\infty\) outside a set of finite logarithmic measure, then every nonconstant solution to (1) has infinite order. By T. Murai [Ann. Inst. Fourier 33, No. 3, 39-58 (1983; Zbl 0519.30029)], if \(A(z)\) has Fejér gaps, then (2) is valid for some exceptional set of finite logarithmic measure. Hence the same conclusion holds, if \(\rho(B)<\rho(A)<\infty\) and \(A(z)\) has Fejér gaps.

Reviewer: Shun Shimomura (Yokohama)

##### MSC:

34M10 | Oscillation, growth of solutions to ordinary differential equations in the complex domain |

34M05 | Entire and meromorphic solutions to ordinary differential equations in the complex domain |

30D20 | Entire functions of one complex variable, general theory |

30D35 | Value distribution of meromorphic functions of one complex variable, Nevanlinna theory |

30D05 | Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable |

34M20 | Nonanalytic aspects differential equations in the complex domain (MSC2000) |

PDF
BibTeX
XML
Cite

\textit{I. Laine} and \textit{P. Wu}, Proc. Am. Math. Soc. 128, No. 9, 2693--2703 (2000; Zbl 0952.34070)

Full Text:
DOI

##### References:

[1] | Albert Edrei and Wolfgang H. J. Fuchs, The deficiencies of meromorphic functions of order less than one, Duke Math. J. 27 (1960), 233 – 249. · Zbl 0094.04901 |

[2] | Gary G. Gundersen, Estimates for the logarithmic derivative of a meromorphic function, plus similar estimates, J. London Math. Soc. (2) 37 (1988), no. 1, 88 – 104. · Zbl 0638.30030 · doi:10.1112/jlms/s2-37.121.88 · doi.org |

[3] | Wei Xin Gao, Linear differential equations with regular singularities, Acta Math. Sinica 30 (1987), no. 4, 566 – 576 (Chinese). · Zbl 0643.34009 |

[4] | W. K. Hayman, On Iversen’s theorem for meromorphic functions with few poles, Acta Math. 141 (1978), no. 1-2, 115 – 145. · Zbl 0382.30020 · doi:10.1007/BF02545745 · doi.org |

[5] | W. K. Hayman, Meromorphic functions, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964. · Zbl 0115.06203 |

[6] | W.K. Hayman and J.F. Rossi, Characteristic, maximum modulus and value distribution, Trans. Amer. Math. Soc. 284 (1984), 651-664. · Zbl 0547.30023 |

[7] | W.K. Hayman and F.M. Stewart, Real inequalities with application to function theory, Proc. Cambridge Philos. Soc. 50 (1953), 250-260. · Zbl 0056.07202 |

[8] | Simon Hellerstein, Joseph Miles, and John Rossi, On the growth of solutions of \?”+\?\?’+\?\?=0, Trans. Amer. Math. Soc. 324 (1991), no. 2, 693 – 706. · Zbl 0719.34011 |

[9] | Takafumi Murai, The deficiency of entire functions with Fejér gaps, Ann. Inst. Fourier (Grenoble) 33 (1983), no. 3, 39 – 58 (English, with French summary). · Zbl 0489.30028 |

[10] | H. Wittich, Zur Theorie linearer Differentialgleichungen im Komplexen, Ann. Acad. Sci. Fenn. Ser. A I No. 379 (1966), 19 (German). · Zbl 0139.03602 |

[11] | Guan Hou Zhang, Theory of entire and meromorphic functions, Translations of Mathematical Monographs, vol. 122, American Mathematical Society, Providence, RI, 1993. Deficient and asymptotic values and singular directions; Translated from the Chinese by Chung-chun Yang. · Zbl 0790.30019 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.