Abstract. Let $\cal T(\beta,b)$, $\beta(\beta\geq0)$ and $b\in\Bbb C$ denote the class of analytic functions $f(z)$ in the open unit disk which satisfy the condition $\operatorname{Re}\{f'(z)+\beta zf''z)\}>1-|b|$. Inclusion relations of functions in the class $\cal T(\beta,b)$ are given. Lower bounds are also obtained for the $n$-th partial sums $F_n(z)$ of the Libera integral operator $F(z)$ and the $n$-th partial sums of $f(z)$. Furthermore, some convolution properties of functions in $\cal T(\beta,b)$ are shown.