Ranges of values of some functionals on classes of regular functions

Let $P_{k, n}(\lambda,\beta)$ be the class of functions $g(z)=1+\sum^\infty_{\nu=n}c_\nu z^\nu$, regular in $B|z|<1$ and satisfying the condition
$$ \int^{2\pi}_0\left|\operatorname{Re}\left[e^{i\lambda}g(z)-\beta\cos\lambda\right]\Bigm/(1-\beta)\cos\lambda\right|d\theta\leq k\pi,\quad z=re^{i\theta}, $$
$0<r<1$ ($k\geq2$, $n\geq1$, $0\leq\beta<1$, $-\pi/2<\lambda<\pi/2$); $M_{k,n}(\lambda,\beta,\alpha)$, $n\geq2$, is the class of functions $f(z)=z+\sum^{\infty}_{\nu=n}a_\nu z^\nu$, regular in $|z|<1$ and such that $F_\alpha(z)\in P_{k,n-1}(\alpha,\beta)$, where $F_\alpha(z)=(1-\alpha)\frac{zf^\prime(z)}{f(z)}+\alpha\Bigl(1+\frac{zf^{\prime\prime}(z)}{f^\prime(z)}\Bigr)$ ($0\leq\alpha\leq1$). Onr considers the problem regarding the range of the system $\{g^{(\nu-1)}(z_\ell)/(\nu-1)!\}$, $\ell=1,2,\dots,m$, $\nu=1,2,\dots,N_\ell$, on the class $P_{k,1}(\lambda,\beta)$. On the classes $P_{k,n}(\lambda,\beta)$, $M_{k,n}(\lambda,\beta,\alpha)$ one finds the ranges of $c_\nu$, $\nu\geq n$, $a_m$, $n\leq m\leq2n-2$, and $g(\zeta)$, $F_\alpha(\zeta)$, $0<|\zeta|<1$, $\zeta$ is fixed.