uniqueness theorem for meromorphic functions concerning ...

248 SUBHAS S. BHOOSNURMATH, MILIND N. KULKARNI AND VAISHALI PRABHULemma 7.([8]) If H ≡ 0, andN(r, 1 Flim) + N(r, 1 G) + N(r,F) + N(r,G)r→∞T(r,F)< 1,r ∈ I where I is the set with infinite linear measure then, FG ≡ 1 or F ≡ G.Lemma 8.([4]) If f is non constant meromorphic function, thennT(r,f) < ¯N(r,f)(+ T r, 1 )+f¯N(1)r,f ′ (1 − af n + S(r,f).)Lemma 9. Let f and g be two non constant meromorphic functions, n ≥ 7,be a positive integer, α(z) be a small meromorphic function such that T(r,α(z)) =o(T(r,f)) and α ≠ 0, ∞ and letF = f ′ (1 − af n ), G = g ′ (1 − ag n ),where a is (n −1) th root of unity. If F, G share α(z) CM, then S(r,f) = S(r,g).Proof. By Lemma 4, we have(nT(r,f) = T r, f ′ (1 − af n ))f ′ + O(1)≤ T(r,F) + T(r,f ′ ) + O(1),nT(r,f) ≤ T(r,F) + 2T(r,f) + O(1),(n − 2)T(r,f) ≤ T(r,F) + O(1), (12)and by Second Fundamental Theorem, and using the hypothesis that F and Gshare α(z) CM, we haveorAsN(r,T(r,F) ≤ N(r,F) + N(T(r,F) ≤ N(r,f)+N r,1) (1 − af n ≤ N r,(r, 1 ) (1)+ N r, + S(r,f),F F − α1) (1 − af n ≤ T r,1)1 − af n +N(r, 1 ) (f ′ +N r,T(r,F) ≤ T(r,f) + T(r,f) + T(r,f ′ ) + N1)≤ T(r,f) + O(1),af(r,1)+S(r,f).G − α1)+ S(r,f),G − α