uniqueness theorem for meromorphic functions concerning ...

MEROMORPHIC FUNCTIONS CONCERNING POLYNOMIALS 255and henceF ∗ = G ∗ + c,where c is a constant. And henceT(r,f) = T(r,g) + S(r,f). (34)Suppose c ≠ 0, by Second Fundamental Theorem(T(r,G ∗ 1) (1)) = (n + 1)T(r,g) < N r,G ∗ + N r,G ∗ + N(r,G ∗ ) + S(r,g)(+ C≤ N r, 1 ) ( )1(+N r,g g n +N(r,g)+N r, 1 ) ( )1+N r,−af f n +S(r,f)− a≤ 5T(r,f) + S(r,f),which is a contradiction to (25), therefore c = 0, and hencef n+1 { 1f n −F ∗ = G ∗ ,f − afn+1n + 1 = g − agn+1n + 1 ,a }n + 1= g n+1 { 1g n −a }.n + 1Now, leth = f g .If h = 1, then f ≡ g.Suppose h ≠ 1, thenh n+1 { 1f n −h n+1 { 1g n h n −a }n + 1a }n + 1= 1g n −= 1g n −hg n − ahn+1n + 1 = 1g n −1(h − 1) =agn n + 1an + 1 ,an + 1 ,an + 1 ,{h n+1 − 1 } ,=⇒ g n = n + 1a{ } h − 1h n+1 − 1and f n = h nn + 1a{ } h − 1h n+1 ,− 1