On a radius problem concerning a class of close-to-convex functions

The problem of estimating the radius of starlikeness of various classes of close-to-convex functions has attracted a certain number of mathematicians involved in geometric function theory ([7], volume 2, chapter 13). Lewandowski [11] has shown that normalized close-to-convex functions are starlike in the disc ${\displaystyle \left|z\right|<4\surd 2-5}$. Krzyż [10] gave an example of a function ${\displaystyle f\left(z\right)=z+{\sum }_{n=2}^{\infty }{a}_n{z}^n}$, non-starlike in the unit disc , and belonging to the class H = {f | f'() lies in the right half-plane.} More generally let H* = {f | f'() lies in some half-plane not containing 0.} To the best of our knowledge, the radii of starlikeness of both H and H* are still unknown, in spite of the fact that corresponding extremal functions can be described in a relatively simple way (by using, for example, Ruscheweyh's duality theory [15]). This paper is a survey of recent results concerning the radius of starlikeness of K = {f ∈ H | |f'(z)-1| < 1, z ∈ }.