Fred Gross

Bio: Fred Gross is an academic researcher from University of Maryland, Baltimore County. The author has contributed to research in topics: Meromorphic function & Entire function. The author has an hindex of 8, co-authored 27 publications receiving 690 citations. Previous affiliations of Fred Gross include United States Naval Research Laboratory.

The fix-points and factorization of meromorphic functions

Abstract: In this paper, we use the Nevanlinna theory of meromorphic functions and a result of Goldstein to generalize some known results in factorization and fixpoints of entire functions. Specifically, we prove (1) Iff and g are nonlinear entire functions such that f(g) is transcendental and of finite order, then f(g) has infinitely many fix-points. (2) If f is a polynomial of degree > 3, and g is an arbitrary transcendental meromorphic function, then f(g) must have infinitely many fix-points. (3) Let p(z), q(z) be any nonconstant polynomials, at least one of which is not c-even, and let a and b be any constants with a or b#0. Then h(z) =q(z) exp (az2 + bz) +p(z) is prime.

On the Nevanlinna characteristic of f(z+η) and difference equations in the complex plane

Abstract: We investigate the growth of the Nevanlinna characteristic of f(z+η) for a fixed η∈C in this paper. In particular, we obtain a precise asymptotic relation between T(r,f(z+η)) and T(r,f), which is only true for finite order meromorphic functions. We have also obtained the proximity function and pointwise estimates of f(z+η)/f(z) which is a discrete version of the classical logarithmic derivative estimates of f(z). We apply these results to give new growth estimates of meromorphic solutions to higher order linear difference equations. This also allows us to solve an old problem of Whittaker (Interpolatory Function Theory, Cambridge University Press, Cambridge, 1935) concerning a first order difference equation. We show by giving a number of examples that all of our results are best possible in certain senses. Finally, we give a direct proof of a result in Ablowitz, Halburd and Herbst (Nonlinearity 13:889–905, 2000) concerning integrable difference equations.

Weighted sharing and uniqueness of meromorphic functions

Abstract: Introducing the idea of weighted sharing of values we prove some uniqueness theorems for meromorphic functions which improve some existing results.

Uniqueness and value-sharing of meromorphic functions

Abstract: Concerning the uniqueness and sharing values of meromorphic functions, many results about meromorphic functions that share more than or equal to two values have been ob- tained. In this paper, we shall study meromorphic functions that share only one value, and prove the following result: For n ≥ 11 and two meromorphic functions f(z) and g(z), if f n fand g n g ' share the same nonzero and finite value a with the same multiplicities, then f ≡ dg or g = c1e cz and f = c2e −cz , where d is an (n + 1)th root of unity, c, c1 and c2 being constants. As an application, we solve some non-linear differential equations.

Zeros of differences of meromorphic functions

Abstract: Let f be a function transcendental and meromorphic in the plane, and define g(z) by g(z) = ?f(z) = f(z + 1) - f(z). A number of results are proved concerning the existence of zeros of g(z) or g(z)/f(z), in terms of the growth and the poles of f. The results may be viewed as discrete analogues of existing theorems on the zeros of f' and f'/f.