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On The Zeros Of Solutions Of Second-Order Linear Differential Equations

Published online by Cambridge University Press:  20 November 2018

P. R. Beesack
Affiliation:
Hamilton College, McMaster University, Hamilton, Canada
Binyamin Schwarz
Affiliation:
Technion (Israel Institute of Technology), Haifa, Israel
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Introduction. In §1 of this paper we consider the complex differential equation

1, |z| < 1,

where q(z) is a regular function in the open unit circle. We shall give a lower bound for the non-Euclidean distance of any pair of zeros of any non-trivial (i.e., not identically zero) solution u(z) of (1).

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1956

References

1. Beesack, P. R., Non-oscillation and disconjugacy in the complex domain, Trans. Amer. Math. Soc. 81 (1956), 211242.Google Scholar
2. Bôcher, M., Proc. Fifth International Cong, of Math., Cambridge (1912), 1, 163195.Google Scholar
3. Hardy, G. H., Littlewood, J. E., and Pólya, G., Inequalities (2nd edition, Cambridge, 1952).Google Scholar
4. Ince, E. L., Ordinary Differential Equations (Dover, New York, 1944).Google Scholar
5. Kamke, E., Differentialgleichungen Lösungsmethoden und Lösungen (Chelsea, New York, 1948).Google Scholar
6. Nehari, Z., The Schwarzian derivative and schlicht functions. Bull. Amer. Math. Soc. 55 (1949), 545551.Google Scholar
7. Nehari, Z., Some criteria of univalence, Proc. Amer. Math. Soc. 5 (1954), 700704.Google Scholar
8. Pokornyi, V. V., On some sufficient conditions for univalance, Doklady Akademii Nauk SSSR (N.S.) 79 (1951), 743746.Google Scholar
9. Pólya, G. and Szegö, G., Isoperimetric Inequalities in Mathematical Physics (Princeton 1951).Google Scholar
10. Schwarz, B., Complex non-oscillation theorems and criteria of univalence, Trans. Amer. Math. Soc. 80 (1955), 159186.Google Scholar
11. Taam, C. T., Oscillation theorems, Amer. J. of Math. 74 (1952), 317324.Google Scholar