We say that a class $\mathcal{ B}$ of analytic functions $f$ of the form $f(z)=\sum_{ n=0}^{ \infty} a_{ n}z^{ n}$ in the unit disk $\mathbb{ D}:=\{ z\in \mathbb{ C}: |z|<1\}$ satisfies a Bohr phenomenon if for the largest radius $R_{ f}<1$, the following inequality $$ \sum\limits_{ n=1}^{ \infty} |a_{ n}z^{ n}| \leq d(f(0),\partial f(\mathbb{ D}) ) $$ holds for $|z|=r\leq R_{ f}$ and for all functions $f \in \mathcal{ B}$. The largest radius $R_{ f}$ is called Bohr radius for the class $\mathcal{ B}$. In this article, we obtain Bohr radius for certain subclasses of close-to-convex analytic functions. We establish the Bohr phenomenon for certain analytic classes $\mathcal{ S}_{ c}^{ *}(\phi),\,\mathcal{ C}_{ c}(\phi),\, \mathcal{ C}_{ s}^{ *}(\phi),\, \mathcal{ K}_{ s}(\phi)$. Using Bohr phenomenon for subordination classes \cite[Lemma 1]{ bhowmik-2018}, we obtain some radius $R_{ f}$ such that Bohr phenomenon for these classes holds for $|z|=r\leq R_{ f}$. Generally, in this case $R_{ f}$ need not be sharp, but we show that under some additional conditions on $\phi$, the radius $R_{ f}$ becomes sharp bound. As a consequence of these results, we obtain several interesting corollaries on Bohr phenomenon for the aforesaid classes.
Categories: math.CV