An averaging operator and a subordination problem for convex functions of order \(\alpha\) (Q1282010)

Here, the author establishes the following average formula \[ 1/g(z)\int^x_0 F(t) g(t)/t dt\in C_\alpha \overline{\text{co}}(F(D)),\quad z\in D, \] where \(g\) is a starlike function of order \(\alpha\), \(\alpha\in [0,1]\), \(F\) is an analytic function in the unit disk \(D\), \(F(0)= 0\) and \[ C_\alpha= -2\min\text{Re } \int^1_0 t\zeta(1- t\zeta) [(1- z)/(1- tz)^{2-2\alpha} dt\in (0,1],\quad \alpha\in[0,1]. \] Further, he obtains \(f^2(z)/(2zf'(z))< C_\alpha f(z)\), for each function \(f\) which is convex of order \(\alpha\).