We introduce the class $R(\alpha)$ of functions of the form $$ f(z)= z- \sum_{k=2}^\infty a_kz^k \qquad (a_k\geq 0) $$ satisfying the condition $$ \text{Re}\{D^{\alpha+1}f(z)/D^\alpha f(z)\}>\alpha/(\alpha+1) $$ for some $(\alpha\geq 0)$ and for all $z\in U=\{z: |z|< 1\}$, where $D^\alpha f(z)$ denotes the Hadamard product of $z/(1-z)^{\alpha+1}$ and $f(z)$. The object of the present paper is to prove some distortion and closure theorems for functions $f(z)$ in $R(\alpha)$, and to give the result for the modified Hadamard product of functions $f(z)$ belonging to the class $R(\alpha)$. Furthermore, we determine the radii of starlikeness and convexity of functions $f(z)$ in the class $R(\alpha)$.