Let $A$ denote the class of functions $f$ regular in the unit disc $E$, such that $f(0)=0=f'(0)-1$. Let $k_a(z)= z/(1-z)^a$ where a is a real number. We denote by $K_a(h)$ the class of functions $f\in A$ satisfying $1+\frac{z(k_a\ast f)''(z)}{(k_a\ast f)'(z)}\prec h(z)$, where $h$ is a convex univalent function in $E$ with $h(0)=1$ and Re$a(h(z))>0$. Several properties of the class $K_a(h)$ are investigated. Certain allied classes are also studied.