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Uniqueness theorems for entire functions whose difference polynomials share a meromorphic function of a smaller order

Xiao-Min Li, Wen-Li Li, Hong-Xun Yi, Zhi-Tao Wen (2011)

Annales Polonici Mathematici

We deal with uniqueness of entire functions whose difference polynomials share a nonzero polynomial CM, which corresponds to Theorem 2 of I. Laine and C. C. Yang [Proc. Japan Acad. Ser. A 83 (2007), 148-151] and Theorem 1.2 of K. Liu and L. Z. Yang [Arch. Math. 92 (2009), 270-278]. We also deal with uniqueness of entire functions whose difference polynomials share a meromorphic function of a smaller order, improving Theorem 5 of J. L. Zhang [J. Math. Anal. Appl. 367 (2010), 401-408], where the entire...

Uniqueness theorems for meromorphic functions concerning fixed points

Xiu-Qing Lin, Wei-Chuan Lin (2011)

Annales Polonici Mathematici

This paper is devoted to the study of uniqueness of meromorphic functions sharing only one value or fixed points. We improve some related results due to J. L. Zhang [Comput. Math. Appl. 56 (2008), 3079-3087] and M. L. Fang [Comput. Math. Appl. 44 (2002), 823-831], and we supplement some results given by M. L. Fang and X. H. Hua [J. Nanjing Univ. Math. Biquart. 13 (1996), 44-48] and by C. C. Yang and X. H. Hua [Ann. Acad. Sci. Fenn. Math. 22 (1997), 395-406].

Universal sequences for Zalcman’s Lemma and Q m -normality

Shahar Nevo (2005)

Annales Polonici Mathematici

We prove the existence of sequences ϱ ₙ n = 1 ∞ , ϱₙ → 0⁺, and z ₙ n = 1 ∞ , |zₙ| = 1/2, such that for every α ∈ ℝ and for every meromorphic function G(z) on ℂ, there exists a meromorphic function F ( z ) = F G , α ( z ) on ℂ such that ϱ ₙ α F ( n z ₙ + n ϱ ₙ ζ ) converges to G(ζ) uniformly on compact subsets of ℂ in the spherical metric. As a result, we construct a family of functions meromorphic on the unit disk that is Q m -normal for no m ≥ 1 and on which an extension of Zalcman’s Lemma holds.

Value distribution and uniqueness of difference polynomials and entire solutions of difference equations

Xiaoguang Qi (2011)

Annales Polonici Mathematici

This paper is devoted to value distribution and uniqueness problems for difference polynomials of entire functions such as fⁿ(f-1)f(z+c). We also consider sharing value problems for f(z) and its shifts f(z+c), and improve some recent results of Heittokangas et al. [J. Math. Anal. Appl. 355 (2009), 352-363]. Finally, we obtain some results on the existence of entire solutions of a difference equation of the form f ⁿ + P ( z ) ( Δ c f ) m = Q ( z ) .

Value distribution of meromorphic solutions of homogeneous and non-homogeneous complex linear differential-difference equations

Li-Qin Luo, Xiu-Min Zheng (2016)

Open Mathematics

In this paper, we investigate the value distribution of meromorphic solutions of homogeneous and non-homogeneous complex linear differential-difference equations, and obtain the results on the relations between the order of the solutions and the convergence exponents of the zeros, poles, a-points and small function value points of the solutions, which show the relations in the case of non-homogeneous equations are sharper than the ones in the case of homogeneous equations.

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