Abstract
Let f be an entire function of finite order, let ngeq 1, mgeq 1, L(z,f)not equiv 0 be a linear difference polynomial of f with small meromorphic coefficients, and P_{d}(z,f)not equiv 0 be a difference polynomial in f of degree dleq n-1 with small meromorphic coefficients. We consider the growth and zeros of f^{n}(z)L^{m}(z,f)+P_{d}(z,f). And some counterexamples are given to show that Theorem 3.1 proved by I. Laine (J. Math. Anal. Appl. 469:808–826, 2019) is not valid. In addition, we study meromorphic solutions to the difference equation of type f^{n}(z)+P_{d}(z,f)=p_{1}e^{alpha _{1}z}+p_{2}e^{alpha _{2}z}, where ngeq 2, P_{d}(z,f)not equiv 0 is a difference polynomial in f of degree dleq n-2 with small mromorphic coefficients, p_{i}, alpha _{i} (i=1,2) are nonzero constants such that alpha _{1}neq alpha _{2}. Our results are improvements and complements of Laine 2019, Latreuch 2017, Liu and Mao 2018.
Highlights
Introduction and main resultsIn this paper, we assume familiarity with the basic results and standard notations of Nevanlinna theory [7, 9, 21]
We denote by S(r, f ) any quantify satisfying S(r, f ) = o(T(r, f )), as r → ∞, outside of a possible exceptional set of finite logarithmic measure, we use S(f ) to denote the family of all small functions with respect to f
By applying Lemma B, we see that m r, Lm(z, f ) = S(r, f ), m r, fLm(z, f ) = S(r, f )
Summary
Halburd and Korhonen [5] established the difference analogue of Clunie lemma. Lemma B (See [5, Corollary 3.3]) Let f be a nonconstant finite-order meromorphic solution of f n(z)P(z, f ) = Q(z, f ), where P(z, f ) and Q(z, f ) are difference polynomials in f with small meromorphic coefficients, and let c ∈ C, δ < 1. In 2019, Laine [10, Theorem 2.1] presented an extension to Theorem D as follows: Theorem E (See [10]) Let f be a transcendental entire function of finite order ρ, b0 be a nonvanishing small meromorphic function of f , L(z, f ) be nonvanishing and n ≥ 2, m ≥ 1. Our purpose is to improve and extend the results in [10] for an entire function f by considering the zero distribution of f nLm(z, f ) + Pd(z, f ), where n ≥ 2, m ≥ 1, Pd(z, f ) is a difference polynomial in f of degree d ≤ n – 2, with small meromorphic coefficients. Lemma 2.2 Let f be a transcendental entire function of finite order ρ, Pd(z, f ) be difference polynomial in f of degree d ≤ n – 1, L(z, f ) :=.
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