For $f(z) = \sum_{ n=0}^{ \infty} a_n z^n$ and a fixed $z$ in the unit disk, $|z| = r,$ the Bohr operator $\mathcal{ M}_r$ is given by \[\mathcal{ M}_r (f) = \sum_{ n=0}^{ \infty} |a_n| |z^n| = \sum_{ n=0}^{ \infty} |a_n| r^n.\] This papers develops normed theoretic approaches on $\mathcal{ M}_r$. Using earlier results of Bohr and Rogosinski, the following results are readily established: if $f(z)=\sum_{ n=0}^{ \infty} a_{ n}z^{ n}$ is subordinate (or quasi-subordinate) to $h(z)=\sum_{ n=0}^{ \infty} b_{ n}z^{ n}$ in the unit disk, then \[\mathcal{ M}_{ r}(f) \leq \mathcal{ M}_{ r}(h), \quad 0 \leq r \leq 1/3,\] that is, \[\sum_{ n=0}^{ \infty} \ | a_{ n}\ | |z|^{ n} \leq \sum_{ n=0}^{ \infty} \ | b_{ n}\ |t |z|^{ n}, \quad 0 \leq |z| \leq 1/3. \] Further, each $k$-th section $s_k(f) = a_0 + a_1 z + \cdots + a_kz^k$ satisfies \[\ | s_k(f)\ | \leq \mathcal{ M}_r \ ( s_k(h)\ ), \quad 0 \leq r \leq 1/2,\] and \[\mathcal{ M}_{ r}\ ( s_{ k}(f) \ ) \leq \mathcal{ M}_{ r}(s_{ k}(h)), \quad 0 \leq r \leq 1/3.\] A von Neumann-type inequality is also obtained for the class consisting of Schwarz functions in the unit disk.
Categories: math.CV