Abstract
In this paper, we study the admissible meromorphic solutions to the algebraic differential equation f^{n} f' + P_{n-1}( f ) = umathrm {e}^{v} in an angular domain instead of the whole complex plane, where P_{n-1}(f) is a differential polynomial in f of degree leq n-1 with small function coefficients, u is a non-vanishing small function of f and v is an entire function. Herein, mainly, we are able to show that the equation does not admit any meromorphic solution f under some conditions unless P_{n-1}(f)equiv0. Using this result, we are able to extend or generalize a well-known result of Hayman.
Highlights
In a recent paper, Liao and Ye considered meromorphic solutions f to f nf + Qd(z, f ) = uev, (1.1)where Qd(z, f ) denotes a differential polynomial in f of degree d with rational function coefficients
If n ≥ d + 1 and the differential equation (1.1) admits a meromorphic solution f with finitely many poles, f has the following form: f = sev/(n+1) and Qd(z, f ) ≡ 0, where s is a rational function with sn((n + 1)s + v s) = (n + 1)u
Theorem 3.1 Let f be a transcendental meromorphic function, Pn–1(f ) be a differential polynomial in f such that its coefficients are in Sf and deg Pn–1(f ) ≤ n – 1
Summary
Theorem 1.1 ([1]) Let Qd(z, f ) be a differential polynomial in f of degree d with rational function coefficients. If n ≥ d + 1 and the differential equation (1.1) admits a meromorphic solution f with finitely many poles, f has the following form: f = sev/(n+1) and Qd(z, f ) ≡ 0, where s is a rational function with sn((n + 1)s + v s) = (n + 1)u. ∞, there exists a continuously differentiable function ρ(r), which satisfies the following conditions.
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