  We introduce a variable exponent version of the Hardy space of analytic
functions on the unit disk, we show some properties of the space, and give an
example of a variable exponent $p(\cdot)$ that satisfies the $\log$-H\"older
condition such that $H^{p(\cdot)}\neq H^q$ for any constant exponent
$1<q<\infty$. We also consider the variable exponent version of the Hardy space
on the upper-half plane.
