The object of this paper is to show some properties of functions belonging to a subclass $M_p(A,B,b,n)$ (where $b$ is a complex number with $\mathrm{Re}(b)>0$ and $A$ and $B$ are two arbitrary constants with $-1\le B<A\le 1)$. Coefficient estimates and some distortion theorems for this class of functions are found. Radii of close-to-convexity, starlikeness and convexity are derived. An application to fractional calculus is given.