A renewal theorem in the finite-mean case

Let $F(.)$ be a c.d.f. on $(0,\infty )$ such that $\overline F(.) \equiv 1-F(.)$ is regularly varying with exponent $-\alpha ,~1<\alpha <2$. Then $U(t)- \frac {t}{\mu } -\frac {1}{\mu ^2} \int _0^t \int _s^\infty \overline F(v) dv ds = O(t^4 \overline F(t)^2 \overline F(t^2\overline F(t)))$ as $t \to \infty$, where $U(t)=EN(t)$ is the renewal function associated with $F(t)$. Moreover similar estimates are given for distributions in the domain of attraction of the normal distribution and for the variance of $N(t).$ The estimates improve earlier results of Teugels and Mohan.