On the radii of convexity and starlikeness for some special classes of analytic functions in a disk

Abstract

We introduce the class O _α, 0≤α≤1, of functions w=ƒ(z), ƒ(0)=0, ƒ′(0)=0,..., ƒ ⁽ⁿ⁻¹⁾₍₀₎ =0, f ⁽ⁿ⁾(0)=(n-l)! analytic in the disk |z|<1 and satisfying the condition

$$\operatorname{Re} \left( {\frac{{1 - 2z^n \cos \Theta + z^{2n} }}{{z^{n - 1} }}f'(z)} \right) > \alpha , 0 \leqslant \Theta \leqslant \pi , n = 1,2,3,... .$$

We establish the radius of convexity in the class Oα and the radius of starlikeness in the class Uα of functions σ(z)=zƒ′(z), ƒ(z)⊂O _α.