Sharp estimates for the integral means of three classes of domains. (Q556554)

Let \(\Sigma_0\) and \(\Sigma^*\) be the classes of functions of the form \(f(z)=z+a_0+a_1/z+\cdots\) defined on \(| z| >1\) which maps the exterior of the unit disk \(| z| >1\) onto a convex and starlike set respectively. In this paper, the author has obtained a sharp upper bound for \(\int_{| z| =R} | \psi (F'(z))| ^2d\theta\) \ (\(z=Re^{i\theta}\)) where \(\psi\) is an arbitrary analytic function in \(| z-1| <1\) and \(F\in\Sigma_0\). In particular, the following sharp inequality is obtained: \(\int_0^{2\pi} | F'(e^{i\theta})| ^pd\theta\leq \sqrt{\pi} 2^{1+p} \frac{\Gamma(1/2+p/2) }{\Gamma(1+p/2)}\) (\(p>-1\)) for any function \(F\in \Sigma_0\). Similar estimates for the function \(\Sigma^*\) is also obtained. Also considered is the class of univalent functions whose Taylor coefficients are real.