Subclasses of regular and schlicht functions with fixed coefficients (Q810188)

Let \(\alpha\),\(\beta\in [0,1)\), \(\theta\in [0,2\pi]\), n,k\(\in N\) be fixed numbers and let \(E=\{z:| z| <1\}\). Denote by \(P(2\alpha (1- \beta)e^{i\theta},n,n+k)\) the class of functions q holomorphic in E, satisfying in the disc the condition Re q(z)\(>\beta\) and having the form \[ q(z)=1+2\alpha (1-\beta)e^{i\theta}z^ n+2p_{n+k}z^{n+k}+...,\quad z\in E. \] In the paper, the author determines in this class the domain of values of the functional \(I_ 0(q)=q(z)\), \(z\in E\)- fixed. He also gives a few applications of the above-mentioned result.