We obtain a fully explicit quantitative version of the Eisenstein theorem on algebraic power series which is more suitable for certain applications than the existing version due to Dwork, Robba, Schmidt and van der Poorten. We also treat ramified series and Laurent series, and we demonstrate some applications; for instance, we estimate the discriminant of the number field generated by the coefficients.

In a scale of Fock spaces with radial weights ϕ we study the existence of Riesz bases of (normalized) reproducing kernels. We prove that these spaces possess such bases if and only if ϕ(x) grows at most like (log x)2.

A proof is given to show that a positive function on the unit disk admits a harmonic majorant if and only if it has a certain explicit upper envelope that admits a superharmonic majorant. The (logarithmic) Lipschitz regularity of this superharmonic majorant is discussed.

Let f be a continuous function on an open subset Ω of ℝ2 such that for every x ∈ Ω there exists a continuous map γ : [−1, 1] → Ω with γ(0) = x and f ∘ γ increasing on [−1, 1]. Then for every γ ∈ Ω there exists a continuous map γ : [0, 1) → Ω such that γ(0) = y, f ∘ γ is increasing on [0; 1), and for every compact subset K of Ω, max{t : γ(t) ∈ K} < 1. This result gives an answer to a question posed by M. Ortel. Furthermore, an example shows that this result is not valid in higher dimensions.