We prove that in variable exponent spaces $L^{p(\cdot)}(Ω)$, where $p(\cdot)$ satisfies the log-condition and $Ω$ is a bounded domain in $\mathbf R^n$ with the property that $\mathbf R^n \backslash \barΩ$ has the cone property, the validity of the Hardy type inequality $$| 1/δ(x)^α\int_Ωϕ(y) dy/|x-y|^{n-α}|_{p(\cdot)} \leqq C |ϕ|_{p(\cdot)}, \quad 0