I have two integrals $I_1$ and $I_2$ that are almost similar : $$I_1=\int_K^{+\infty}(x-K)f_1(x)dx$$ $$I_2=\int_K^{+\infty}(x-K)f_2(x)dx$$ with$f_1$ and $f_2$ being two equivalent densities (so they are always positive or null, both integrate to 1 over their domain, and are "good" functions :their limit value when x goes to $-\infty$ or to $+\infty$ is null).
What conditions must respect $f_1$ and $f_2$ to have $I_1 < I_2$ whatever the value of K is ?
Unless I misunderstand your question, why not $f_2(x)=f_1(x-c)$ where $c\gt 0$?