This paper mainly uses the nonnegative continuous function ${zeta_n(r)}_{n=0}^{infty}$ to redefine the Bohr radius for the class of analytic functions satisfying $real f(z)<1$ in the unit disk $|z|<1$ and redefine the Bohr radius of the alternating series $A_f(r)$ with analytic functions $f$ of the form $f(z)=sum_{n=0}^{infty}a_{pn+m}z^{pn+m}$ in $|z|<1$. In the latter case, one can also get information about Bohr radius for even and odd analytic functions. Moreover, the relationships between the majorant series $M_f(r)$ and the odd and even bits of $f(z)$ are also established. We will prove that most of results are sharp.
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