LetM (α) denote the class of α-convex functions, α real, that is the class of analytic functionsƒ (z) =z + Σn=2/∞
 a
 n
 z
 n
 in the unit discD = {z: |z | < 1} which satisfies inD the condition ƒ′ (z) ƒ(z)/z ≠ 0 and
 
$$\operatorname{Re} \left\{ {(1 - a) \frac{{z f'(z)}}{{f (z)}} + a \left( {1 + \frac{{z f''(z)}}{{f' (z)}}} \right)} \right\} > 0. Let W (a) $$

 denote the class of meromorphic α-convex functions. α real, that is the class of analytic functions ϕ (z) =z
 −1 + Σn=0/∞
 b
 n
 z
 n
 inD* = {z: 0 < |z | < 1} which satisfies inD* the conditionszϕ′(z)/ϕ(z) ≠ 0 and
 
$$\operatorname{Re} \left\{ {(1 - a) \frac{{z\phi ' (z)}}{{\phi (z)}} + a \left( {1 + \frac{{z\phi ''(z)}}{{\phi ' (z)}}} \right)} \right\}< 0. $$

 In this paper we obtain the relation betweenM (a) and W(α). The radius of α-convexity for certain classes of starlike functions is also obtained.

On the radius of α-convexity of certain classes of starlike functions

Closed convex hull and extreme points are obtained for the classR (α) of univalent functions ƒ inU = {z:¦z ¦ α, 0 ≤ α β, 0 ⩽ β< 1, wheref ∈ R (α). Integral representations are given for the hulls of these two classes in terms of probability measures on suitable sets. These results are used to solve extremal problems. For example the upper bounds are determined for the coefficients of a function subordinate to some function in R(α) when 1/2 ≤ α < 1.

Mostafa A. Nasr

Papers

On the radius of α-convexity of certain classes of starlike functions

Convex hull and extreme points of two families of univalent functions