uniqueness theorem for meromorphic functions concerning ...
uniqueness theorem for meromorphic functions concerning ...
uniqueness theorem for meromorphic functions concerning ...
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
MEROMORPHIC FUNCTIONS CONCERNING POLYNOMIALS 251There<strong>for</strong>e (19) becomesT(r,F ∗ ) ≤ T(r,F) + 2T(r,f) + O(log r) + S(r,f). (20)Let H be defined as in Lemma 6 and let H ≠ 0.Then by using Lemma 6 and equation (14), we obtain that(N 2 (r,F) + N 2 r, 1 )(1)≤ 2N(r,f) + N r,F1 − af n + N(r, 1 )f ′ + O(log r), (21)(N 2 (r,G) + N 2 r, 1 )(≤ 2N(r,g) + N r,GNow using (14)-(22) in (20) we have1)1 − ag n + N(r, 1 g ′ )+ O(log r). (22)(T(r,F ∗ ) ≤ N 2 (r,F) + N 2 r, 1 ) (+ N 2 (r,G) + N 2 r, 1 )+ 2T(r,f) + O(log r)FG+S(r,f)≤ 2N(r,f) + N(r,1)(1 − af n + 2N(r,g) + N r,+N(r, 1 )f ′ + 2T(r,f) + O(log r),1)1 − ag n + N(r, 1 )g ′and N(r, 1 g ′ ) ≤ T(r, 1 g ′ ) = T(r,g ′ ) + o(1) = 2T(r,g) + S(r,g). There<strong>for</strong>eN(r, 1 g ′) ≤ 2T(r,g) + S(r,g), (23)(T(r,F ∗ 1) (1)) ≤ 2N(r,f) + 2N(r,g) + N r,1 − ag n + N r,1 − af n + 2T(r,g)+2T(r,f) + 2T(r,f) + O(log r) + S(r,f). (24)By (16), we have((n + 1)T(r,f) ≤ 2N(r,f) + 2N(r,g) + N r,Similarly, we can prove1)1 − ag n+2T(r,g) + 2T(r,f) + 2T(r,f),(+ N r,1)1 − af n=⇒ (n − 6)T(r,f) ≤ 5T(r,g) + O(log r) + S(r,f). (25)(n − 6)T(r,g) ≤ 5T(r,f) + O(log r) + S(r,f), (26)
Change language