Find the criteria on a variable

Question

What would the criteria on the variable $v$ be such that $f\left( t\right) $ is always negative . $$f\left( t\right) =\int_{\mathbb{R}^{+}}\frac{\cos \left( xt\right) }{x^{v}}% dx=\frac{\Gamma \left( k+v\right) }{\Gamma \left( v\right) }\frac{1}{t^{k}}% \int_{\mathbb{R}^{+}}\frac{\cos \left( xt-\frac{k\pi }{2}\right) }{x^{v+k}}dx $$