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Generalized Schwarzian derivatives and higher order differential equations

Published online by Cambridge University Press:  20 June 2011

MARTIN CHUAQUI
Affiliation:
P. Universidad Católica de Chile, Av. Vicuña Mackenna 4860, Santiago, Chile. e-mail: mchuaqui@mat.puc.cl
JANNE GRÖHN
Affiliation:
University of Eastern Finland, Campus of Joensuu, P.O. Box 111, 80101 Joensuu, Finland. e-mail: janne.grohn@uef.fi and jouni.rattya@uef.fi
JOUNI RÄTTYÄ
Affiliation:
University of Eastern Finland, Campus of Joensuu, P.O. Box 111, 80101 Joensuu, Finland. e-mail: janne.grohn@uef.fi and jouni.rattya@uef.fi

Abstract

It is shown that the well-known connection between the second order linear differential equation h″ + B(z)h = 0, with a solution base {h1, h2}, and the Schwarzian derivative of f = h1/h2, can be extended to the equation h(k) + B(z) h = 0 where k ≥ 2. This generalization depends upon an appropriate definition of the generalized Schwarzian derivative Sk(f) of a function f which is induced by k−1 ratios of linearly independent solutions of h(k) + B(z) h = 0. The class k(Ω) of meromorphic functions f such that Sk(f) is analytic in a given domain Ω is also completely described. It is shown that if Ω is the unit disc or the complex plane , then the order of growth of fk(Ω) is precisely determined by the growth of Sk(f), and vice versa. Also the oscillation of solutions of h(k) + B(z) h = 0, with the analytic coefficient B in or , in terms of the exponent of convergence of solutions is briefly discussed.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2011

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References

REFERENCES

[1]Astala, K. and Zinsmeister, M.Teichmüller spaces and BMOA. Math. Ann. 289 (1991), no. 4, 613625.CrossRefGoogle Scholar
[2]Bank, S.A general theorem concerning the growth of solutions of first-order algebraic differential equations. Compositio Math. 25 (1972), no. 1, 6170.Google Scholar
[3]Bertilsson, D.Coefficient estimates for negative powers of the derivative of univalent functions. Ark. Mat. 36 (1998), no. 2, 255273.CrossRefGoogle Scholar
[4]Bishop, C. J. and Jones, P. W.Harmonic measure, L 2 estimates and the Schwarzian derivative. J. Anal. Math. 62 (1994), 77113.CrossRefGoogle Scholar
[5]Cartwright, M. L.On analytic functions regular in the unit circle (II). Quart. J. Math. 4 (1933), 246257.Google Scholar
[6]Chyzhykov, I., Heittokangas, J. and Rättyä, J.Finiteness of ϕ-order of solutions of linear differential equations in the unit disc. J. Anal. Math. 109 (2009), 163198.CrossRefGoogle Scholar
[7]Chyzhykov, I., Heittokangas, J. and Rättyä, J.Sharp logarithmic derivative estimates with applications to ordinary differential equations in the unit disc. J. Aust. Math. Soc. 88 (2010), no. 2, 145167.CrossRefGoogle Scholar
[8]Gundersen, G. G.Finite order solutions of second order linear differential equations. Trans. Amer. Math. Soc. 305 (1988), no. 1, 415429.CrossRefGoogle Scholar
[9]Gundersen, G. G., Steinbart, E. M. and Wang, S.The possible orders of solutions of linear differential equations with polynomial coefficients. Trans. Amer. Math. Soc. 350 (1998), no. 3, 12251247.CrossRefGoogle Scholar
[10]Heittokangas, J. and Rättyä, J.Zero distribution of solutions of complex linear differential equations determines growth of coefficients. Math. Nachr. 284 (2011), no. 4, 412420.CrossRefGoogle Scholar
[11]Heittokangas, J., Korhonen, R. and Rättyä, J.Growth estimates for solutions of linear complex differential equations. Ann. Acad. Sci. Fenn. Math. 29 (2004), no. 1, 233246.Google Scholar
[12]Heittokangas, J., Korhonen, R. and Rättyä, J.Linear differential equations with coefficients in the weighted Bergman and Hardy spaces. Trans. Amer. Math. Soc. 360 (2008), 10351055.CrossRefGoogle Scholar
[13]Kim, W. J.The Schwarzian derivative and multivalence. Pacific J. Math. 31 (1969), 717724.CrossRefGoogle Scholar
[14]Korhonen, R. and Rättyä, J.Finite order solutions of linear differential equations in the unit disc. J. Math. Anal. Appl. 349 (2009), no. 1, 4354.CrossRefGoogle Scholar
[15]Laine, I.Nevanlinna Theory and Complex Differential Equations (Walter de Gruyter, Berlin, 1993).CrossRefGoogle Scholar
[16]Pau, J. and Peláez, J. A.Logarithms of the derivative of univalent functions in Q p spaces. J. Math. Anal. Appl. 350 (2009), 184194.CrossRefGoogle Scholar
[17]Pérez-González, F. and Rättyä, J.Dirichlet and VMOA domains via Schwarzian derivative. J. Math. Anal. Appl. 359 (2009), 543546.CrossRefGoogle Scholar
[18]Pommerenke, Ch.Boundary Behaviour of Conformal Maps. (Springer-Verlag, Berlin, 1992).CrossRefGoogle Scholar
[19]Schippers, E.Distortion theorems for higher order Schwarzian derivatives of univalent functions. Proc. Amer. Math. Soc. 128 (2000), no. 11, 32413249.CrossRefGoogle Scholar
[20]Wittich, H.Zur theorie linearer differentialgleichungen im komplexen. Ann. Acad. Sci. Fenn. Ser. A I Math. 379 (1966), 118.CrossRefGoogle Scholar