$x$ is a continuous random variable with $pdf$ given by $f$ in the interval $[0,1]$. There is a continuous function $\lambda(x):[0,1]\rightarrow[0,1]$ with $\lambda'(x)>0$ such that its unconditional expectation is constant $E(\lambda(x))=\int_0^1\lambda(x)f(x)dx=\bar{\lambda}$.
Define the conditional expectation $E(\lambda(x)|x<c)=\frac{1}{F(c)}\int_0^c\lambda(x)f(x)dx$.
Under which conditions on $\lambda(\cdot)$, $f$ or $c$: $E(\lambda(x)|x<c)>E(\lambda(x))$.
If $\lambda(x)$ is strictly monotonically decreasing then you will have this property, for any $0 < c < 1$ and any strictly positive $f$. I'm not sure if there is an easy to state more general property under which your desired condition holds.