In this paper we generalize the results of Libera and McGregor concerning argument property of analytic functions. We use the result in 3 to prove the following: Theorem. Let f(z) = z vertical bar Sigma(infinity)(n=p+K) a(n)z(n), g(z) = z vertical bar Sigma(infinity)(n=p+K) b(n)z(n) be analytic in Delta, f(z) not equal 0 in 0 < vertical bar z vertical bar < 1, and suppose that for some alpha, beta (0 < alpha < 1, 0 < beta < 1) vertical bar arg (f'(z)/g'(z)vertical bar < pi/2 alpha + Tan(-1) 2 alpha beta/1-beta(2) - Tan(-1) 2 alpha beta/(1-beta(2))root 1+alpha(2) in Delta, and taht g'(z)/zg(z) (sic) 1+beta z/1-beta z where (sic) means subordination. Then we have vertical bar arg(f(z)/g(z)vertical bar < pi/2 alpha in Delta.
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