Approximation of subharmonic functions in the unit disk

Let u be a subharmonic function in D={|z|<1}. There exist an absolute constant C and an analytic function f in D such that \int_D |u(z)-log|f(z)|| dm(z)<C where m denotes the plane Lebesgue measure. We also consider uniform approximation.Comment: 25 page

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