Abstract
It is well-known that a normalized analytic function for which the quantity $1+zf''(z)/f'(z)$ lies in the half-plane $\real w>-1/2$ is close-to-convex and hence univalent. In this paper, we show that the derivative of the function $f$ has positive real part if the quantity $1+\alpha zf''(z)/f'(z)$ with $\alpha>0$ lies in the sector $|\arg w| <\arctan( \alpha )$.
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