×

zbMATH — the first resource for mathematics

Three results in the value-distribution theory of solutions of linear differential equations. (English) Zbl 0635.34009
The author develops, through three results, new information about the distribution of zeros of all solutions \(f\not\equiv 0\) for three different classes of equations of the form: (1) \(f''+A(z)f=0\); \(A(z)=arbitrary\) entire function, and thereby obtains a complete value- distribution theory for all solutions. The first result concerns a class of equations (1), where the order of A(z) can be any nonnegative real number, and shows that, if the equation possesses a solution \(f_ 1\not\equiv 0\) satisfying \({\bar \lambda}\)(f\({}_ 1)<\infty\), and which is of a certain form, then for any solution \(f_ 2\), which is linearly independent with \(f_ 1\), it must be valid \({\bar \lambda}\)(f\({}_ 2)=\infty\). The second result concerns the value-distribution theory for the solutions of a class of equations (1) where A(z) is a periodic entire function of the form \(B(e^{az})\) \((B(\zeta)=rational\) function; \(a=nonzero\) constant). Firstly, it is proved that, the conclusion \({\bar \lambda}\)(f)\(=\infty\) for every solution \(f\not\equiv 0\), can be replaced by the stronger conclusion \((2)\quad \log^+\bar N(r,1/f)\neq o(r)\) as \(r\to +\infty\). Furthermore it is shown that, instead of requiring both of the poles of B(\(\zeta)\) at \(\zeta =0\) and \(\zeta =\infty\) to be of odd order, the stronger conclusion (2) will hold for all solutions \(f\not\equiv 0\), when at least one of these poles is of odd order. The third result is an improvement of an existing one in the literature result, namely of the value-distribution theory given for the solution of (1) in the case when A(z) is a nonconstant polynomial of degree n. The paper is original, with strict mathematical construction and it is very clearly written.
Reviewer: D.E.Panayotounakos

MSC:
34A30 Linear ordinary differential equations and systems, general
34M99 Ordinary differential equations in the complex domain
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] S. BANK, On the value distribution theory for entire solutions of second-order linear differential equations, Proc. London Math. Soc. (3), 50 (1985), 505-534. · Zbl 0545.30022 · doi:10.1112/plms/s3-50.3.505
[2] S. BANK AND I. LAINE, On the oscillation theory of /” +Af= 0 where A is entire, Trans. Amer. Math. Soc. 273 (1982), 351-363. · Zbl 0505.34026 · doi:10.2307/1999210
[3] S. BANK AND I. LAINE, Representations of solutions of periodic second-order linear differential equations, J. reine angew. Math. 344 (1983), 1-21. · Zbl 0524.34007 · crelle:GDZPPN002200724 · eudml:152558
[4] S. BANK, I. LAINE AND J. LANGLEY, On the frequency of zeros of solutions of second-order linear differential equations, Resultate der Math. (To appear). · Zbl 0635.34007 · doi:10.1007/BF03322360
[5] M. FREI, Uber die subnormalen Losungen der Differentialgleichung wff+e zwf+ (const.) w=Q. Comm. Math. Helv. 36 (1962), 1-8. · Zbl 0115.06904 · doi:10.1007/BF02566887 · eudml:139223
[6] W. HAYMAN, Meromorphic Functions. Clarendon Press, Oxford (1964). · Zbl 0115.06203
[7] R. KAUFMAN, Riccati equations, zeros, and independence, J. reine angew. Math. 345 (1983), 63-68. · Zbl 0512.34023 · doi:10.1515/crll.1983.345.63 · crelle:GDZPPN002200880 · eudml:152574
[8] R. NEVANLINNA, Le Theoreme de Picard-Borel et la Theorie des Functions Meromorphes, Gauthier-Villars, Pans, 1929.
[9] M. OZAWA, On a solution of w?+e-zw’ + (az+b)w= 0. Kodai Math. J. 3 (1980), 295-309. · Zbl 0463.34028 · doi:10.2996/kmj/1138036197
[10] S. SAKS AND A. ZYGMUND, Analytic Functions, Monografie Mat. (Engl. Transl.), Tom 28, Warsaw, 1952. · Zbl 0048.30803 · eudml:219298
[11] G. VALIRON, Lectures on the General Theory of Integral Functions, Chelsea Publ. Co., New York, 1949.
[12] H. WITTICH, Eindeutige Losungen der Differentialgleichung w’–R(z, w), Math. Zeit. 74 (1960), 278-288. · Zbl 0091.26102 · doi:10.1007/BF01180487 · eudml:169884
[13] H. WITTICH, Subnormale Losungen der Differentialgleichung wff+p (ez) w’+q(ez) w = 0. Nagoya Math. J. 30 (1967), 29-37. · Zbl 0219.34005
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.