On the uniqueness of two different classes of meromorphic functions under the sharing of two sets of rational functions

We study the uniqueness problem of two special classes of meromorphic functions sharing two sets of rational functions. One of the considered classes has the property to include the Selberg class L functions, while the other class is comprising of arbitrary meromorphic functions having finitely many poles. We obtain a number of results which extend and improve a number of earlier results such as Li [Proc. Amer. Math. Soc. 138 (2010), 2071-2077], Lin and Lin [Filomat 30 (2016), 3795-3806] and others. We have also been able to replace the strict CM (IM) sharing of the sets in our theorems to almost CM (almost IM) sharing.