The branch of complex analysis that asks how little information about two entire or meromorphic functions forces them to be identical. Two functions share a value a when f-a and g-a have exactly the same zeros, either counting multiplicity (CM) or ignoring it (IM); they share a finite set S when the preimage of S is the same for both. Nevanlinna's five-value theorem is the founding result: sharing five values IM forces f = g. The theory then asks how the count of five can be reduced by constraining the pair, for instance by taking g to be a derivative, a power, or a differential polynomial of f (the Bruck conjecture and its generalizations), and by replacing shared values with shared sets. A finite set whose sharing already forces equality is a unique range set, and the polynomials whose zero sets produce such sets are uniqueness polynomials and strong uniqueness polynomials. Proofs work by bounding Nevanlinna counting functions of auxiliary expressions until the Second Fundamental Theorem is contradicted.
A PhD thesis (University of Kalyani, November 2017; advisor Abhijit Banerjee) on uniqueness theory for entire and meromorphic functions. Two meromorphic functions share a value a when f-a and g-a hav…