Close-to-Convex Functions

A function \(f\in \mathcal {A}\) is said to be close-to-convex if there exists a function \(g\in \mathcal{C}\mathcal{V}\) and \(\alpha \in (-\pi /2,\pi /2)\) such that \(\operatorname {Re}\left( \text {e}^{\text {i}\alpha } f'(z)/g'(z)\right) >0 \) for all \(z\in \mathbb {D}\) . Every close-to-convex function is shown to be univalent. A characterization involving only the function f is also established. Several subclasses of close-to-convex functions are duly investigated.