Is the integral $\int_1^{\infty} {A(t)}{t^{-s-1}}dt$ a holomorphic function of $s$?

Question

My question is whether, for $Re(s)=\sigma > 3/4$, $$s\int_1^{\infty} \dfrac{A(t)}{t^{s+1}}dt$$ is holomorphic, where $A(x)=O(x^{3/4})$.

Under absolute value, it is easy to see that the integral converges to some real number. But I am not sure if without it.