Inequalities for functions regular in the unit circle

Abstract

Let f(z) = u(z) + iv(z) be a function regular for |z| < 1 with real and imaginary parts u(z) and u(z) respectively. The following theorems have been proved by Miss M. L. Cartwright(1) and (2).Theorem A. if v(0) = 0, 0 < α < ∞ andfor 0 ≤ r < 1, 0 ≤ θ < 2π thenfor 0 ≤ r < 1, 0 ≤ θ < 2π