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Oscillation results on y″ + Ay = 0 in the complex domain with transcendental entire coefficients which have extremal deficiencies

Published online by Cambridge University Press:  20 January 2009

Y. M. Chiang
Y. M. Chiang
Affiliation:
Division of MathematicsBolton Institute of Higher EducationDeane Road Bolton BL3 5ABEngland
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Abstract

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Let A(z) be a transcendental entire function and f1, f2 be linearly independent solutions of

We prove that if A(z) has Nevanlinna deficiency δ(0, A) = 1, then the exponent of convergence of E: = flf2 is infinite. The theorems that we prove here are similar to those in Bank, Laine and Langley [3].

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 38 , Issue 1 , February 1995 , pp. 13 - 34
Copyright
Copyright © Edinburgh Mathematical Society 1995

References

REFERENCES

1.Bank, S. and Laine, I., On the oscillation theory of f″ + Af =0 where A is entire, Trans. Amer. Math. Soc. 273 (1982), 351–363.Google Scholar
2.Bank, S. and Laine, I., On the zeros of meromorphic solutions of second order linear differential equations, Comment. Math. Helv. 58 (1983), 656–677.CrossRefGoogle Scholar
3.Bank, S., Laine, I. and Langley, J. K., On the frequency of zeros of solutions of second order linear differential equations, Resultate Math. 10 (1986), 8–24.CrossRefGoogle Scholar
4.Bank, S., Laine, I. and Langley, J. K., Oscillation results for solutions of linear differential equations in the complex domain, Resultate Math. 16 (1989), 3–13.CrossRefGoogle Scholar
5.Bank, S. and Langley, J. K., On the oscillation of solutions of certain linear differential equations in the complex domain, Proc. Edinburgh Math. Soc. 30 (1987), 455–469.CrossRefGoogle Scholar
6.Bank, S. and Langley, J. K., Oscillation theory for higher order linear differential equations with entire coefficients, Complex Variables 16 (1991), 163–175.Google Scholar
7.Chiang, Y. M., Schwarzian derivative and second order differential equations (Ph.D. thesis, Univ. of London, 1991).Google Scholar
8.Edrei, A. and Fuchs, W. H. J., Valeurs deficientes et valeurs asymptotiques des fonctions méromorphes, Comment. Math. Helv. 33 (1959), 258–295.CrossRefGoogle Scholar
9.Fuchs, W. H. J.Proof of a conjecture of G. Polya concerning gap series, Illinois J. Math. 7 (1963), 661–667.CrossRefGoogle Scholar
10.Hayman, W. K., Meromorphic functions (Clarendon Press, Oxford, 1964).Google Scholar
11.Herold, H., Ein vergleichssatz für komplexe lineare Differentialgleichungen, Math. Z. 126 (1972), 91–94.CrossRefGoogle Scholar
12.Hille, E., Ordinary differential equations in the complex plane (Wiley-Interscience, New York, 1976).Google Scholar
13.Langley, J. K., On complex oscillation and a problem of Ozawa, Kodai Math. J. 9 (1986), 430–439.CrossRefGoogle Scholar
14.Rossi, J., Second order differential equations with transcendental coefficients, Proc. Amer. Math. Soc. 97 (1986), 61–66.CrossRefGoogle Scholar
15.Shen, L. C., Solution to a problem of S. Bank regarding the exponent of convergence of the solutions of a differential equation f″ + Af = 0, Kexue Tongbao 30 (1985), 1581–1585.Google Scholar
16.Valiron, G., Lectures on the general theory of integral functions, Chelsea, New York, 1949.Google Scholar