We find all nonrational meromorphic solutions of the equation ww′′ − (w′)2 = α(z)w + β(z)w′ + γ (z), where α, β, and γ are rational functions of z. In so doing, we answer a question of Hayman by showing that all such solutions have finite order. Apart from special choices of the coefficient functions, the general solution is not meromorphic and contains movable branch points. For some choices f...

The numbers of a-points of different classes of meromorphic functions have been widely studied for more than 120 years. As a culmination of similar studies, there arose classical Nevanlinna theory (1920s) and Ahlfors theory (1935). Clearly, further development should touch not only the numbers of a-points but also on their locations. Similar regularities were established nearly 30 years ago: th...

In the paper the ideas of relative Nevanlinna L∗-order and relative Nevanlinna L∗-lower order of an analytic function with respect to an entire function in the unit disc U = {z : |z| < 1} are introduced. Hence, we study some comparative growth properties of composition of two analytic functions in the unit disc U on the basis of relative Nevanlinna L∗-order and relative Nevanlinna L∗-lower order.

The aim of this paper is to give a presentation of several subjects of Schur analysis with some historical information. The class of J-inner functions plays a key role in this new mathematical eld which is situated at the seam of various mathematical disciplines (operator theory, scattering theory, complex function theory, prediction theory for stochastic processes, spectral theory for diierent...

The Borel lemma says that any positive non-decreasing continuous function T satisfies T (r+1/T (r)) ≤ 2T (r) outside of a possible exceptional set of finite linear measure. This lemma plays an important role in the theory of entire and meromorphic functions, where the increasing function T is either the logarithm of the maximum modulus function, or the Nevanlinna characteristic. As a result, ex...

A tropical version of Nevanlinna theory is described in which the role of meromorphic functions is played by continuous piecewise linear functions of a real variable whose one-sided derivatives are integers at every point. These functions are naturally defined on the max-plus (or tropical) semi-ring. Analogues of the Nevanlinna characteristic, proximity and counting functions are defined and ve...