Ranges of values of some functionals on classes of regular functions

E. G. Goluzina

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 6, Tome 144 (1985), pp. 46-50

Résumé

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.