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The Schwarzian Derivative and Disconjugacy of nth order Linear Differential Equations

Published online by Cambridge University Press:  20 November 2018

Meira Lavie
Meira Lavie*
Affiliation:
Carnegie-Mellon University, Pittsburgh, Pennsylvania
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In this paper we deal with the number of zeros of a solution of the nth order linear differential equation

1.1

where the functions pj(z) (j = 0, 1, …, n – 2) are assumed to be regular in a given domain D of the complex plane. The differential equation (1.1) is called disconjugate in D, if no (non-trivial) solution of (1.1) has more than (n – 1) zeros in D. (The zeros are counted by their multiplicity.)

The ideas of this paper are related to those of Nehari (7; 9) on second order differential equations. In (7), he pointed out the following basic relationship. The function

1.2

where y1(z) and y2(z) are two linearly independent solutions of

1.3

is univalent in D, if and only if no solution of equation(1.3) has more than one zero in D, i.e., if and only if(1.3) is disconjugate in D.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 235 - 249
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Ahronov, D., The Schwarzian derivative and univalent functions, Doctoral Thesis, Technion- Israel Institute of Technology, Haifa, Israel, 1967. (Hebrew, English synopsis)Google Scholar
2. Beesack, P. R., Integral inequalities of the Wirtinger type, Duke Math. J. 25 (1958), 477–498.Google Scholar
3. Duren, P. L., Coefficient estimates for univalent functions, Proc. Amer. Math. Soc. 13 (1962), 168–169.Google Scholar
4. Golusin, G. M., Some estimates of the coefficients of schlicht functions, Mat. Sb. 3 (1938), 321–330. (Russian)Google Scholar
5. Hadass, R., On the zeros of the solutions of the differential equation y(n)(z) +p﹛z)y﹛z) = 0, Doctoral Thesis, Technion-Israel Institute of Technology, Haifa, Israel, 1967. (Hebrew, English synopsis)Google Scholar
6. Hille, E., Remarks on a paper by Z. Nehari, Bull. Amer. Math. Soc. 55 (1949), 552–553.Google Scholar
7. Nehari, Z., The Schwarzian derivative and schlicht functions, Bull. Amer. Math. Soc. 55 (1949), 545–551.Google Scholar
8. Nehari, Z., On the zeros of solutions of n-th order linear differential equations, J. London Math. Soc. 39 (1964), 327–332.Google Scholar
9. Nehari, Z., Some function-theoretic aspects of linear second-order differential equations, J. Analyse Math. 18 (1967), 259–276.Google Scholar
10. Schwarz, B., Complex nonoscillation theorems and criteria of univalence, Tran. Amer. Math. Soc. 80 (1955), 159–186.Google Scholar
11. Wilczynski, E. J., Projective differential geometry of curves and ruled surfaces, p. 26 (B. G. Teubner, Leipzig, 1906).Google Scholar