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We determine the Bohr radius for the class of all functions $f$ of the form$f(z)=\sum_{k=1}^\infty a_{kp+m} z^{kp+m}$ analytic in the unit disk $|z|<1$and satisfy the condition $|f(z)|\le 1$ for all $|z|<1$. In particular, ourresult also contains a solution to a recent conjecture of Ali, Barnard andSolynin \cite{AliBarSoly} for the Bohr radius for odd analytic functions,solved by the authors in \cite{KayPon1}. We consider a more flexible approachby introducing the $p$-Bohr radius for harmonic functions which in turncontains the classical Bohr radius as special case. Also, we prove severalother new results and discuss $p$-Bohr radius for the class of odd harmonicbounded functions.

Bohr's inequality for analytic functions $\sum_k b_k z^{kp+m}$ and harmonic functions