Classes of convex functions (Q1580308)

Let \[ f(z)=z+ \sum^\infty_{n=2} a_nz^n\tag{1} \] by analytic in \(\Delta=\{z: |z|< 1\}\). We say that the function (1) is in \(UCV\) if \[ Re\left( 1+{zf''(z)\over f'(z)}\right) \geq\left|{zf''(z)\over f'(z)} \right |,\;(z\in\Delta). \] If the function (1) satisfy \(Ref'(z)> \alpha|zf'' (z) |\), \((z\in\Delta)\), \(\alpha\geq 0\), then we say that \(f\in UCD (\alpha)\). Let \(K(\alpha)\) denote the class of functions (1) that satisfy \[ Re\left(1 +{z f''(z)\over f'(z)} \right)> \alpha,\;0\leq\alpha <1. \] Theorems: 1. \(UCD(\alpha) \subset K(1-{1\over \alpha})\), \(\alpha\geq 1\). 2. \(UCD(\alpha) \subset UCV \Leftrightarrow \alpha\geq 2\). 3. If \(\alpha\geq 0\) and \(\sum^\infty_{k=2} k[1+ \alpha (k-1)]|a_k|\leq 1\), then the function (1) is in \(UCD(\alpha)\). Some results known for the function (1) that satisfy \(Ref' (z)>0\) in \(\Delta\) are extended to the class \(UCD(\alpha)\).