Ping Li’s research while affiliated with University of Science and Technology of China and other places

Let h be a meromorphic function with few poles and zeros. By Nevanlinna's value distribution theory we prove some new properties on the polynomials in h with the coefficients being small functions of h. We prove that if f is a meromorphic function and if is identically a polynomial in h with the constant term not vanish identically, then f is a polynomial in h. As an application, we are able to find the entire solutions of the differential equation of the type , where P(f) is a differential polynomial in f of degree at most n-1, and Q() is a polynomial in of degree k max {n-1, s(n-1)/n} with small functions of as its coefficients.

We discuss possible relations between two meromorphic functions f and g when they share some pairs of small functions. By utilizing the generalized Nevanlinna's second main theorem for small functions obtained recently, we have been able to show that two meromorphic functions f and g must be linked by a quasi-Möbius transformation if they share three pairs of small functions CM* and share another pair of small function IM*. Moreover, we also improves a known result due to T. Czubiak and G. Gundersen on two meromorphic functions sharing five pairs of values and the results on the unicity of meromorphic functions that share five small functions obtained by Li Bao-Qin and Li Yu-Hua as well.

By utilizing Nevanlinna's value distribution theory of meromorphic functions, we solve the transcendental entire solutions of the following type of nonlinear differential equations in the complex plane: f(n)(z) + P(f) = p(2)e(alpha 2z), where p(1) and p(2) are two small functions of e(z), and alpha(1), alpha(2) are two nonzero constants with some additional conditions, and P(f) denotes a differential polynomial in f and its derivatives (with small functions of f as the coefficients) of degree no greater than n - 1.

By Nevanlinna's value distribution theory, we give a more precise form of the meromorphic function f provided that f m can be expressed as a summation of n + 1 (n!≤! m) functions in family The results in this article generalize some known results obtained by Green, M., 1974, On the functional equation a new Picard theorem. Transactions of the American Mathematical Society, 195, 223–230; Noda, Y., 1993, On the fuctional equation and rigidity theorems for holomorphic curves. Kodai Mathematical Journal, 16, 90–117.

In this paper, by studying the properties of meromorphic functions which have few zeros and poles, we find all the entire functions f(z) which share a small and finite order meromorphic function a(z) with its derivative, and f(n)(z)−a(z)=0 whenever f(z)−a(z)=0 (n⩾2). This result is a generalization of several previous results.

We have continued, by utilizing Nevanlinna's value distribution theory, our previous studies on the existence or solvability of meromorphic solutions of functional equations with constant coefficients to that of similar types of functional equations with meromorphic (small functions) coefficients. The results obtained are relating to value sharing or unicity of meromorphic functions.

In this paper, we find all the forms of meromorphic functions f(z) that share the value 0 CM∗, and share b(z)IM∗ with g(z)=a1(z)f(z)+a2(z)f′(z). And a1(z), a2(z) and b(z) (a2(z),b(z)≢0) be small functions with respect to f(z). As an application, we show that some of nonlinear differential equations have no transcendental meromorphic solution.

By utilizing Nevanlinna's value distribution theory of meromorphic functions, it is shown that the following type of nonlinear differential equations:fn(z)+Pn−3(f)=p1eα1z+p2eα2z has no nonconstant entire solutions, where n is an integer ⩾4, p1 and p2 are two polynomials (≢0), α1, α2 are two nonzero constants with α1/α2≠ rational number, and Pn−3(f) denotes a differential polynomial in f and its derivatives (with polynomials in z as the coefficients) of degree no greater than n−3. It is conjectured that the conclusion remains to be valid when Pn−3(f) is replaced by Pn−1(f) or Pn−2(f).

Suppose that f is a nonconstant entire function and L[f] a linear differential polynomial in f with constant coefficients. In this paper, by considering the existence of the solutions of some differential equations, we find all the forms of entire functions f in most cases when f and L[f] share two values counting multiplicities jointly. This result generalize some known results due to Rubel–Yang and Li–Yang.