Browsing by Author "Grohn, Janne"

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    Generalized Schwarzian derivatives and higher order differential equations
    (CAMBRIDGE UNIV PRESS, 2011) Chuaqui, Martin; Grohn, Janne; Rattya, Jouni
    It is shown that the well-known connection between the second order linear differential equation h '' B(z) h = 0, with a solution base {h(1), h(2)}, and the Schwarzian derivative
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    Possible Intervals for T- and M-Orders of Solutions of Linear Differential Equations in the Unit Disc
    (HINDAWI PUBLISHING CORPORATION, 2011) Chuaqui, Martin; Grohn, Janne; Heittokangas, Janne; Rattya, Jouni
    In the case of the complex plane, it is known that there exists a finite set of rational numbers containing all possible growth orders of solutions of f((k)) + a(k-1)(z)f((k-1)) + ... + a(1)(z)f' + a(0)(z)f = 0 with polynomial coefficients. In the present paper, it is shown by an example that a unit disc counterpart of such finite set does not contain all possible T- and M-orders of solutions, with respect to Nevanlinna characteristic and maximum modulus, if the coefficients are analytic functions belonging either to weighted Bergman spaces or to weighted Hardy spaces. In contrast to a finite set, possible intervals for T- and M-orders are introduced to give detailed information about the growth of solutions. Finally, these findings yield sharp lower bounds for the sums of T- and M-orders of functions in the solution bases.