  In this article, we study the connection between the fractional
Moser-Trudinger inequality and the fractional
$\left(\frac{kp}{p-1},p\right)$-Poincar\'e type inequality for any Euclidean
domain and discuss the sharpness of this inequality whose analogous results are
well known in the local case. We further provide sufficient conditions on
domains for fractional $(q,p)$-Poincar\'e type inequalities to hold. We also
derive Adachi-Tanaka type inequalities in the non-local setting.
