Let f be analytic on the unit disk, and assume that 1. left bracketing bar f(z) right bracketing bar <=1 for all z, and 2. f(a) = b for some a, b element D(0, 1), the unit disk. Then left bracketing bar f'(a) right bracketing bar <=(1 - ( left bracketing bar b right bracketing bar )^2)/(1 - ( left bracketing bar a right bracketing bar )^2). Furthermore, if f(a_1) = b_1 and f(a_2) = b_2, then left bracketing bar (b_2 - b_1)/(1 - b^__1 b_2) right bracketing bar <= left bracketing bar (a_2 - a_1)/(1 - a^__1 a_2) right bracketing bar , where z^_ is the complex conjugate.