Zur Theorie der binomischen Differentialgleichungen

The Yosida Class is Universal

“…, where m is any integer > β, s ∈ Z, and S is a polynomial with deg S ≤ m. Each zero q and pole p in the products occurs according to its multiplicity. For β an integer, the term in brackets, the sums in (15), and also the sequence of functions I k (z) in (13) remain uniformly bounded if we choose µ = m = β, which means that in (12) and ( 16) we may replace m by β if we simultaneously replace r → ∞ by r k → ∞ for some suitably chosen sequence (r k ). We finish this section by proving a technical lemma as follows:…”

“…Example. Transcendental meromorphic solutions to algebraic differential equations( 3 ) w ′n = R(z, w) have order of growth ̺ ≥ 1/3 or else ̺ = 0 (Bank and Kaufman [2], the author [13]). An example for the latter case is due to Bank and Kaufman [1] (slightly modified):…”

Meromorphic solutions of algebraic differential equations

“…The distribution of the values of the solutions of these equations is studied in detail in [55]. It is profitable to consider the following example [57]:…”

“…If the order is 0, then T(r, y) = (c + o(l))logV [24], [25], [26]. The distribution of the values of meromorphic functions determined by the equations A, B, and С is studied in detail in [54], [55], and [57]. In [24] it is conjectured that any transcendental solution of an arbitrary first-order differential equation with rational coefficients has the property that T(r, у) Ф o(log 2 r), r -* °°.…”