Doubly close-to-convex functions

We introduce the class L(β, γ ) of holomorphic, locally univalent functions in the unit disk D = {z: |z| < 1}, which we call the class of doubly close-to-convex functions. This notion unifies the earlier known extensions. The class L(β, γ ) appears to be linear invariant. First of all we determine the region of variability {w: w = log f (r), f ∈ L(β, γ )} for fixed z, |z| = r <1, which give us the exact rotation theorem. The rotation theorem and linear invariance allows us to find the sharp value for the radius of close-to-convexity and bound for the radius of univalence. Moreover, it was helpful as well in finding the sharp region for α ∈ R, for which the integral z 0 (f (t ))α dt, f ∈ L(β, γ ), is univalent. Because L(β, γ ) reduces to β-close-to-convex functions (γ = 0) and to convex functions (β = 0 and γ = 0), the obtained results generalize several corresponding ones for these classes. We improve as well the value of the radius of univalence for the class considered by Hengartner and Schober (Proc. Amer. Math. Soc. 28 (1971) 519–524) from 0.345 to 0.577.  2003 Elsevier Inc. All rights reserved.