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Irregular finite order solutions of complex LDE's in unit disc by Chyzhykov, Igor; Filevych, Petro; Gröhn, Janne; Heittokangas, Janne; Rättyä, Jouni is a scholarly article available to read on EtoBox.

What is Irregular finite order solutions of complex LDE's in unit disc about?

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.

Author

Chyzhykov, Igor; Filevych, Petro; Gröhn, Janne; Heittokangas, Janne; Rättyä, Jouni

Published

2020

Language

EN