uniqueness theorem for meromorphic functions concerning ...

MEROMORPHIC FUNCTIONS CONCERNING POLYNOMIALS 253f ′ (1 − af n ) ≠ 0, g ′ (1 − ag n ) ≠ 0,or if z 0 is a zero of f ′ (1 − af n ) of order m then it is a pole of g ′ (1 − ag n ) of orderm. So,N( )1r,f ′ (1 − af n = N ( r,g ′ (1 − ag n ) ) = N(r,g),)provided z 0 is not a zero of P(z). Now by using Lemma 3, we get that(nT(r,f) < N(r,f) + N r, 1 )+ N(r,g) + S(r,f),fnT(r,f) ≤ 2−{Θ(0,f)+Θ(∞,f)} T(r,f) + [1−Θ(∞,g)] T(r,g)+S(r,f). (31)But(T(r,f ′ (1 − af n P(z) 2 ))) = T r,g ′ (1 − ag n )(= T(r,P(z) 2 ) + T r,(≤ O(log r) + T r,1g ′ (1 − ag n )1g ′ (1 − ag n ))+ log 2)+ log 2≤ T(r,g ′ ) + T(r,(1 − ag n )) + S(r,g),and alsoT(r,f ′ (1 − af n ) ≤ (n + 2)T(r,g) + S(r,g), (32)nT(r,f) = T(r, afn )(≤ T(r,af n ) + S(r,f) = T r, f ′ (1 − af n ))af ′ + c≤ T(r,f ′ ) + T(r,f ′ (1 − af n )) + S(r,f),(n − 2)T(r,f) ≤ T(r,f ′ (1 − af n )) + S(r,f). (33)Using (31), we getSimilarly, we can prove(n − 2)T(r,f) ≤ (n + 2)T(r,g) + S(r,g) + S(r,f).(n − 2)T(r,g) ≤ (n + 2)T(r,f) + S(r,g) + S(r,f).Hence S(r,g) = S(r,f). Now, we claim thatT(r,g) ≤ T(r,f) + S(r,f).