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2 results

  • SHARP LOGARITHMIC DERIVATIVE ESTIMATES WITH APPLICATIONS TO ORDINARY DIFFERENTIAL EQUATIONS IN THE UNIT DISC

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  • I. CHYZHYKOV, J. HEITTOKANGAS, J. RÄTTYÄ
  • Journal:
    Journal of the Australian Mathematical Society / Volume 88 / Issue 2 / April 2010
    Published online by Cambridge University Press:
    07 April 2010, pp. 145-167
    Print publication:
    April 2010
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  • New estimates are obtained for the maximum modulus of the generalized logarithmic derivatives f(k)/f(j), where f is analytic and of finite order of growth in the unit disc, and k and j are integers satisfying k>j≥0. These estimates are stated in terms of a fixed (Lindelöf) proximate order of f and are valid outside a possible exceptional set of arbitrarily small upper density. The results obtained are then used to study the growth of solutions of linear differential equations in the unit disc. Examples are given to show that all of the results are sharp.

  • Growth and oscillation theory of non-homogeneous linear differential equations

  • Gary G. Gundersen, Enid M. Steinbart, Shupei Wang
  • Journal:
    Proceedings of the Edinburgh Mathematical Society / Volume 43 / Issue 2 / June 2000
    Published online by Cambridge University Press:
    20 January 2009, pp. 343-359
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  • We investigate the growth and the frequency of zeros of the solutions of the differential equation f(n) + Pn–1 (z) f(n–1) + … + P0 (z) f = H (z), where P0 (z), P1(z), …, Pn–1(z) are polynomials with P0 (z) ≢ 0, and H (z) ≢ 0 is an entire function of finite order.