The following theorem is proven: Let F ( z ) F(z) be convex and univalent in Δ = { z : | z | > 1 } , F ( 0 ) = 1 \Delta = \{ z:|z| > 1\} ,F(0) = 1 . Let f ( z ) f(z) be analytic in Δ , f ( 0 ) = 1 , f ′ ( 0 ) = … = f ( n − 1 ) ( 0 ) = 0 \Delta ,f(0) = 1,f’(0) = \ldots = {f^{(n - 1)}}(0) = 0 , and let f ( z ) ≺ F ( z ) f(z) \prec F(z) in Δ \Delta . Then for all γ ≠ 0 \gamma \ne 0 , Re γ ⩾ 0 \gamma \geqslant 0 , \[ γ z − γ ∫ 0 z τ γ − 1 f ( τ ) d τ ≺ γ z − γ / n ∫ 0 z 1 / n τ γ − 1 F ( τ n ) d τ . {\gamma _z}^{ - \gamma }\int _0^z {{\tau ^{\gamma - 1}}f(\tau )d\tau \prec \gamma {z^{ - \gamma /n}}\int _0^{{z^{1/n}}} {{\tau ^{\gamma - 1}}F({\tau ^n})d\tau .} } \] This theorem, in combination with a method of D. Styer and D. Wright, leads to the following Corollary. Let f ( z ) , g ( z ) f(z),g(z) be convex univalent in Δ , f ( 0 ) = f ( 0 ) = g ( 0 ) = g ( 0 ) = 0 \Delta ,f(0) = f(0) = g(0) = g(0) = 0 . Then f ( z ) + g ( z ) f(z) + g(z) is starlike univalent in Δ \Delta . Other applications of the theorem are concerned with the subordination of f ( z ) / z f(z)/z where f ( z ) f(z) belongs to certain classes of convex univalent functions.