On alpha-close-to-convex functions of order beta (Q1091500)

The author studies a class \(M_{\beta}(\alpha)\) of normalized analytic functions in the unit disc of the complex plane satisfying \[ \int^{\theta_ 2}_{\theta_ 1}Re\{(1-\alpha)(zf'(z)/f(z))+\alpha (zf'(z))'/f'(z)\}d\theta >\beta \pi \] for all \(\theta_ 2>\theta_ 1\geq -\pi\), \(0\leq \beta \leq \alpha\), \(z=re^{i\theta}\), \(0<r<1\). The main result is that if \(f\in M_{\beta}(\alpha)\) then \(f\in M_{\beta}(0)\).