Irregular solutions of complex linear differential equations in the unit disc

Abstract: It is shown that the order and the lower order of growth are equal for all non-trivial solutions of $f^{(k)}+A f=0$ if and only if the coefficient $A$ is analytic in the unit disc and $\log^+ M(r,A)/\log(1-r)$ tends to a~finite limit as $r\to 1^-$. A~family of examples is constructed, where the order of solutions remain the same while the lower order may vary on a~certain interval depending on the irregular growth of the coefficient. These coefficients emerge as the logarithm of their modulus approximates smooth radial subharmonic functions of prescribed irregular growth on a~sufficiently large subset of the unit disc. A~result describing the phenomenon behind these examples is also established. En route to results of general nature, a~new sharp logarithmic derivative estimate involving the lower order of growth is discovered. In addition to these estimates, arguments used are based, in particular, on the Wiman-Valiron theory adapted for the lower order.

complex variablesdynamical systems

Audience: researchers in the topic

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