Uniqueness of meromorphic functions sharing two sets with least possible cardinalities

Let f and g be two nonconstant meromorphic functions sharing two finite sets, namely S ⊂ ℂ and {∞}. We prove two uniqueness theorems under weaker conditions on ramification indices, reducing the cardinality of the shared set S and weakening the nature of sharing of the set {∞} which improve results of Fang-Lahiri [7], Lahiri [17], Banerjee -Majumder-Mukherjee [5] and others.