Bohr operator on analytic functions

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.