Description of growth and oscillation of solutions of complex LDE's

It is known that, equally well in the unit disc as in the whole complex plane, the growth of the analytic coefficients $A_0,\dotsc,A_{k-2}$ of \begin{equation*} f^{(k)} + A_{k-2} f^{(k-2)} + \dotsb + A_1 f'+ A_0 f = 0, \quad k\geq 2, \end{equation*} determines, under certain growth restrictions, not only the growth but also the oscillation of its non-trivial solutions, and vice versa. A uniform treatment of this principle is given in the disc $D(0,R)$, $0<R\leq \infty$, by using several measures for growth that are more flexible than those in the existing literature, and therefore permit more detailed analysis. In particular, results obtained are not restricted to cases where solutions are of finite (iterated) order of growth in the classical sense. The new findings are based on an accurate integrated estimate for logarithmic derivatives of meromorphic functions, which preserves generality in terms of three free parameters.Comment: 24 pages. This is a revision of a previously announced preprint. There are many changes throughout the manuscrip

Oscillation of solutions of LDE's in domains conformally equivalent to unit disc

Oscillation of solutions of $f^{(k)} + a_{k-2} f^{(k-2)} + \dotsb + a_1 f' +a_0 f = 0$ is studied in domains conformally equivalent to the unit disc. The results are applied, for example, to Stolz angles, horodiscs, sectors and strips. The method relies on a new conformal transformation of higher order linear differential equations. Information on the existence of zero-free solution bases is also obtained.Comment: 14 page

Asymptotic ψ\psi-densities of subsets of natural numbers

The sizes of subsets of the natural numbers are typically quantified in terms of asymptotic (linear) and logarithmic densities. These concepts have been generalized to weighted $w$-densities, where a specific weight function $w$ plays a key role. In this paper, a parallel theory of asymptotic $\psi$-densities is introduced, where the weight is expressed slightly differently in terms of differentiable functions $\psi$, which are either concave or convex and satisfy certain asymptotic properties. Alternative new proofs for known results on analytic and Abel densities are also given.Comment: 27 page

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

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.Comment: 41 page

OPISKELUN JOUSTAVUUDEN LISÄÄMINEN MATEMATIIKAN YLIOPISTO-OPINTOJEN TAITEKOHDASSA

Matematiikan yliopisto-opintojen taitekohta sijoittuu opiskelujen vaiheeseen, jossa opiskelun pääpaino siirtyy laskennosta abstraktiin päättelyyn. Useimmissa yliopistossa tämä taitekohta kulminoituu reaalianalyysin opiskeluun, jonka haasteet ovat yleisesti tunnustetut. Itä-Suomen yliopistossa reaalianalyysin kurssin toteutukseen tehtiin muutoksia lukuvuonna 2021–2022, joilla tavoiteltiin opetusjärjestelyn joustavuuden ja kommunikoinnin lisäämistä, sekä kurssin läpimenoprosentin ja kurssiarvosanojen parantamista vaatimustasoa laskematta. Artikkelissa tarkastellaan toteutusta niin toteuman kuin siitä saatujen kokemusten suhteen. Opiskelijapalautteen perusteella kurssitoteutus oli hyvin onnistunut, joustava ja oppimista tukeva, vaikka kommunikaation lisääminen jäikin tavoitteesta. Opiskelijoiden kurssimenestys parani läpimenon ja kurssiarvosanojen osalta