Convex functions of bounded \(\alpha\)-type (Q1330823)

The author defines the radius of \(\alpha\)-curvature of the image of \(| z|= r< 1\) under an analytic and locally univalent function in the open unit disc \(U\) of the complex plane at the point \(f(z)\) as \(\rho(f,z)= 1/k_ \alpha(f, z)\), where \[ k_ \alpha(f,z)= [\text{Re}(1+ zf''(z)/(1- \alpha) f'(z))]/[| z| | f(z)|^{1/(1- \alpha)}. \] He also defines a class \(CV_ \alpha(R_ 1, R_ 2)\) as follows: \(f\) which is locally univalent and analytic in \(U\) belongs to \(CV_ \alpha(R_ 1, R_ 2)\) if \[ \rho_ *(\alpha,r)= \min_{| z|= r} \rho_ \alpha(f,z),\quad\rho^*(\alpha,r)= \max_{| z|= 1} \rho_ \alpha(f, z), \] \[ R_ *(\alpha)= \liminf_{r\to 1-} \rho_ *(\alpha, r),\quad R^*(\alpha)= \limsup_{r\to 1-} \rho^*(\alpha, r) \] satisfy \(R_ 1\leq R_ *(\alpha)\) and \(R^*(\alpha)\leq R_ 2\) with \(0< R_ 1\leq R_ 2< \infty\), \(0< \alpha< 1\). The functions in this class are called convex functions of bounded \(\alpha\) type. For this class the author obtains various distortion theorems. The relationship between the class \(CV_ 1(R_ 1, R_ 2)\) and \(CV_ \alpha(R_ 1, R_ 2)\) through integral operators is also studied.