Renewal equation with monotone density and hazard rate

Question

Let $f$ be a density function with support $[0,\infty)$ and let $h$ be the associated hazard rate $$h(x) = \frac{f(x)}{\int_x^\infty f(s)\text ds}.$$

I conjecture that if f is decreasing then, if $h$ is increasing (resp. decreasing) then the solution $u$ to the renewal equation $$ u(t) = \int_0^t u(t-s)f(s)\text ds + f(t)$$ is increasing (resp. decreasing).

I'm wondering whether this is true and, if so, whether it is a know result.

Thanks

Answer 1

https://projecteuclid.org/euclid.aop/1176994773 This paper indicates that decreasing hazard rate implies u is decreasing. (But no discussions about increasing hazard rate.)

In detail, decreasing hazard rate implies that the age at time t is stochastically increasing to t, denoted as $A(t)$.

Here $u(t)$ is the renewal density, which has another representation as $$u(t)=(1-F(0))^{-1}E[h(A(t))].$$ Since $h$ is decreasing and $A(t)$ is stochastically increasing, we have $u(t)$ to be decreasing.