Remark 2.2 Combining Lemma 2.3 and applying it to 1/g(z), it is clear that for any given [epsilon] > 0, there exists a set [E.sub.4] [subset] (1, +[infinity]) that has finite logarithmic measure, such that

Then for any given [epsilon] > 0, there exists a set [E.sub.5] [subset] (1, +[infinity]) having finite logarithmic measure such that for any [theta] [member of] [0,2[pi])\H (H = {[theta] [member of] [0,2[pi]): [delta] (P, [theta]) = 0}) and for [absolute value of z] = r [not member of] [0,1] [union] [E.sub.5], r [right arrow] +[infinity], we have

By Remark 2.2, for any given [epsilon] (0 < [epsilon] < n - [lambda]), there exists a set [E.sub.5] [subset] (1, +[infinity]) that has finite logarithmic measure, such that for [absolute value of z] = r [not member of] [0,1] [union] [E.sub.5], r [right arrow] +[infinity]

By Lemma 2.4 and [sigma]([A.sub.j][f.sup.(j)]) < n (j = 0,1, ..., k - 1), for the above [epsilon], there exists a set [E.sub.5] [subset] (1, +[infinity]) having finite logarithmic measure such that for all z with arg z = [theta] [member of] [0,2[pi])\H, [absolute value of z] = r [not member of] [0,1] [union] [E.sub.5], r [right arrow] +[infinity], we have

By Lemma 2.3, for any given [epsilon](0 < 2[epsilon] < min {1 - c/1 + c, n - [alpha]}), there exists a set [E.sub.3] [subset] (1, +[infinity]) having finite logarithmic measure such that for all z with [absolute value of z] = r [not member of] [0,1] [union] [E.sub.3], r [right arrow] +[infinity], we have (3.1) and

By Lemma 2.3, for any given [epsilon] (0 < 2[epsilon] < min {1, n - [alpha]}) there exists a set [E.sub.3] [subset] (1, +[infinity]) having finite logarithmic measure such that for all z with [absolute value of z] = r [not member of] [0,1] [union] [E.sub.3], r [right arrow] +[infinity], we have (3.1), (3.2) and (3.3).

By Wiman-Valiron theory, there exists a set [E.sub.2] [subset] (0, [infinity]) of finite logarithmic measure, such that

By Lemma 2.1, we see that there exists a subset E [subset] (1, [infinity]) with a finite logarithmic measure and a constant B > 0, such that for all [absolute value of z] [not member of] (0, 1) [union] E, by calculating carefully, we have