Abstract: Let f(z) be a meromorphic function in|z|<1, then for ε>0 and o ≤ r<1, we have#br#T(r,f) ≤ C(h(r)/1-r)^1+εT(R,f'),#br#where h(r) is a positive function and h(r) ≥ 1, and R=(1+rh(r))/(1+h(r)). we also apply the above result to study the common Borel points of a meromorphic function of infinite order and its differential polynomials.

Key words: Meromorphic function, Characteristic function, Borel point