In this paper, we study the value distribution of zeros of certain nonlinear difference polynomials of entire functions of finite order.

On Picard Value Problem of Some Difference Polynomials

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On Picard value problem of some difference polynomials

Let f be a meromorphic function of finite order ρ with few poles in the sense Sλ(r, f) := O(r λ+ε) + S(r, f), where λ < ρ and ε ∈ (0, ρ − λ), and let g(f) := ∑k j=1 bj(z)f j(z + cj) be a linear delaydifferential polynomial of f with small meromorphic coefficients bj in the sense Sλ(r, f). The zero distribution of f n(g(f))s − b0 is considered in this paper, where b0 is a small function in the sense Sλ(r, f).

Zero distribution of some delay-differential polynomials

In this paper, we consider the zeros distribution of $f(z)P(z,f) -q(z)$
 , where $P(z,f)$
 is a linear differential-difference polynomial of a finite-order transcendental entire function $f(z)$
 , and $q(z)$
 is a nonzero polynomial. To a certain extent, Theorem 1.1 generalizes the recent results (Latreuch and Belaidi in Arab. J. Math. 7(1):27–37, 2018; Lu et al. in Kodai Math. J. 39(3):500–509, 2016) related to Hayman conjecture (Hayamn in Ann. Math. 70:9–42, 1959).

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The zeros on complex differential-difference polynomials of certain types

In this note, we study the admissible meromorphic solutions for algebraic differential equation fnf' + Pn−1(f) = R(z)eα(z), where Pn−1(f) is a differential polynomial in f of degree ≤ n − 1 with small function coefficients, R is a non-vanishing small function of f, and α is an entire function. We show that this equation does not possess any meromorphic solution f(z) satisfying N(r, f) = S(r, f) unless Pn−1(f) ≡ 0. Using this result, we generalize a well-known result by Hayman.

Meromorphic Solutions for a Class of Differential Equations and Their Applications

Let f denote a transcendental meromorphic function with N(r, f) = S(r, f) and k be an integer. By using methods different from others, we have been able to derive several new results and pose some new conjectures that relate to the yet to be resolved conjecture concerning the quantitative estimates on the zeros of ff(k)-b, for a non-vanishing small function b.

Caiping Zhuo

Papers

Notes on the value distribution of ff(k)-b