Integral means of certain class of analytic functions (Q687851)

The author studies a class \(S_{\lambda}(\alpha,\beta)\) of functions analytic in the unit disk. A function \(f\) belongs to this class if there exists a function \(g\) starlike of order \(\alpha\) for which \[ \left|{f(z)\over g(z)}- 1\right|< \beta\left|\lambda{f(z)\over g(z)}+1\right|, \] for every \(z\) with \(| z|< 1\). Here \(\lambda\) and \(\beta\) are in \([0,1]\) and \(\alpha\) is in \([0,1)\). The first author shows that \(f/g\) is subordinate to \(p_{\beta,\lambda}= {1+\beta z\over 1-\beta\lambda z}\). Then, using considerations involving star-functions, he shows that integral means \[ \int_{-\pi}^ \pi \Phi\left(\log{| f(re^{i\theta})|\over r}\right) d\theta \] for convex \(\Phi\) and \(0< r<1\) are maximal when \(f= p_ \beta,\lambda\cdot k_ \alpha\), where \(k_ \alpha(z)= \{{z\over (1-z)}\}^{2(2-\alpha)}\). Previously, coefficient problems for a subclass of \(S_ \lambda(\alpha,\beta)\) were considered by S. Owa.