Define $Z(t)$ to be the forward recurrence time at $t$ for a renewal process with interarrival time distribution, $F$, which is assumed to be IMRL (increasing mean residual life). It is shown that $E\phi(Z(t))$ is increasing in $t \geq 0$ for all increasing convex $\phi$. An example demonstrates that $Z(t)$ is not necessarily stochastically increasing nor is the renewal function necessarily concave. Both of these properties are known to hold for $F$ DFR (decreasing failure rate).
"Further Monotonicity Properties for Specialized Renewal Processes." Ann. Probab. 9 (5) 891 - 895, October, 1981. https://doi.org/10.1214/aop/1176994317