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On complex oscillation and a problem of Ozawa. (English) Zbl 0609.34041
The author proves that if Q(z) is a non-constant polynomial and \(\alpha\in C\), then every nontrivial solution of \(y''+(e^{z+\alpha}+Q(z))y=0\) has zeros with infinite exponent of convergence. Similar methods are used to settle a problem of Ozawa: If P(z) is a non-constant polynomial, then all nontrivial solutions of \(y''+e^{-z}y'+P(z)y=0\) have infinite order.
Reviewer: P.Marušiak

34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
Full Text: DOI
[1] I. AMEMIYA AND M. OZAWA, Non-existence of finite order solutions of wff+ e zw’+Q(z)w =0, Hokkaido Math. J., 10 (1981), Special Issue, 1-17. · Zbl 0554.34003
[2] S. BANK AND I. LAINE, On the oscillation theory of /// +^4/=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, Representation 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 AND I. LAINE, On the zeros of meromorphic solutions of second order linear differential equations, Comment. Math. Helv., 58 (1983), 656-677. · Zbl 0532.34008 · doi:10.1007/BF02564659 · eudml:139960
[5] S. BANK, I. LAINE AND J. K. LANGLEY, On the frequency of zeros of solutions of second order linear differential equations, to appear, Resultate der Mathematik. · Zbl 0635.34007 · doi:10.1007/BF03322360
[6] R. BELLMAN, Stability theory of differential equations, McGraw-Hill, NewYork, 1953. · Zbl 0053.24705
[7] M. FREI, Uber die subnormalen Lsungen der Differentialgleichung wf’ +e zw’+ (const.) u>=0, Comment. Math. Helv., 36 (1962), 1-8. · Zbl 0115.06904 · doi:10.1007/BF02566887 · eudml:139223
[8] W. H. J. FUCHS, Topics in the theory of functions of one complex variable, Van Nostrand Math. Studies, 12, 1967. · Zbl 0155.11502
[9] G. GUNDERSEN, On the question of whether f” +e-2f’ +B(z)f=Qcan admit a solution / ^ 0 of finite order, Proc. R. S. Edinburgh, 102A (1986), 9-17. · Zbl 0598.34002 · doi:10.1017/S0308210500014451
[10] W. K. HAYMAN, Slowly growing integral and subharmonic functions, Comment. Math. Helv., 34 (1960), 75-84. · Zbl 0123.26702 · doi:10.1007/BF02565929 · eudml:139185
[11] W. K. HAYMAN, Meromorphic functions, Oxford at the Clarendon Press, 1964. · Zbl 0115.06203
[12] H. HEROLD, Ein Vergleichssatz fur komplexe linearer Differentialgleichungen, Math. Zeit., 126 (1972), 91-94. · Zbl 0226.34005 · doi:10.1007/BF01580359 · eudml:171724
[13] M. OZAWA, On a solution of w” +e-*w’ + (az+b)w =0, Kodai Math. J., 3 (1980), 295-309. · Zbl 0463.34028 · doi:10.2996/kmj/1138036197
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