given [x] the 'integer function' would be the following bound valid ??

$$\int_{0}^{\infty}dt [g(x/t)]f(t) \le \int_{0}^{\infty}dt g(x/t)f(t)$$

for a given functions f(t) and g(u) u=x/t , here g is a non-decreasing positive function for positive arguments (and real) of parameter u=x/t

$$f(t) = f^{+}(t) - f^{-}(t)$$,
where $$f^{+}(t) and f^{-}(t)$$ denotes max(f(t),0) and max(-f(t),0) respectively