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arXiv · 2011.01560

Irregular finite order solutions of complex LDE's in unit disc

Igor Chyzhykov·Petro Filevych·Janne Gröhn·Janne Heittokangas·Jouni Rättyä
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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 concrete 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 highly non-trivial 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, and on a good understanding of the right-derivative of the logarithm of the maximum modulus.

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Igor Chyzhykov, Petro Filevych, Janne Gröhn, Janne Heittokangas, Jouni Rättyä. 2020-11-03. Irregular finite order solutions of complex LDE's in unit disc. https://doi.org/10.1016/j.matpur.2022.02.001

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