UNIQUENESS OF $\mathcal{L}$-FUNCTION WITH CERTAIN POLYNOMIAL OF MEROMORPHIC FUNCTION SHARING A SMALL FUNCTION

This study investigates the uniqueness properties of L-functions associ ated with meromorphic functions that share a small function of finite weight. Based on the Value Distribution Theory of Nevanlinna and the uniqueness properties of L-functions, we establish several theorems that demonstrate the conditions under which two L-functions can be considered equivalent, particularly in the context of their shared values and the behavior of associated polynomials. Our results extend, generalize, and improve those of Mandal and Datta [10]. We also provide an ex ample that supports our results and poses open questions regarding the relaxation of conditions in uniqueness theorems.