In the paper we present a uniqueness polynomial for a class of meromorphic functions having the same set of poles.
Introduction, definitions and results
Let f be a non-constant meromorphic function in the open complex plane C and S be a set of distinct elements of C [ ΒΉ1ΒΊ. We put E f .S / D S a2S ΒΉz W f .z/ a D 0ΒΊ; where zeros are counted with multiplicities.
F. Gross [6] exhibited the existence of three finite sets S j .j D 1; 2; 3/ such that for any two non-constant entire functions f and g, E f .S j / D E g .S j / .j D 1; 2; 3/, implies f Γ g. In 1982 F. Gross and C. C. Yang [7] found an infinite set S of complex numbers such that for two non-constant entire functions f and g, E f .S / D E g .S / implies f Γ g.
Gross and Yang [7] called a set S a unique range set for entire functions (URSE in short) if E f .S / D E g .S / implies f Γ g for any pair of entire functions. In an analogous manner a unique range set for meromorphic functions (URSM in short) is defined.
It is seen that the finite URSE and URSM are zero sets of some polynomials. This observation led P. Li and C. C. Yang [13] to introduce the idea of uniqueness polynomial for entire and meromorphic functions. Let P .z/ be a polynomial in C. If P .f / Γ P .g/ implies f Γ g for any two non-constant meromorphic (entire) functions f and g, then P .z/ is called a uniqueness polynomial for meromorphic (entire) functions. We say P is a UPM (UPE) in brief.
On the other hand, H. Fujimoto [5] introduced the idea of strong uniqueness polynomial for meromorphic (entire) functions (this terminology is used by T. T. H. An, J. T. Wang and P. Wang [1]) as a polynomial P .z/ in C such that P .f / Γ cP .g/ implies f Γ g for any pair of non-constant meromorphic (entire) functions, where c is a suitable non-zero constant. We say P is a SUPM (SUPE) in brief.