A question on Gamma function

This might be basic but I have difficulty understanding what exactly goes wrong in the following logic:

Consider the Gammma function

$$\Gamma(z) = \int_0^{\infty} t^{z-1} \, e^{-t}\,dt \quad \textrm{Re}(z) >0.$$

Now let us consider $z=1+iy$ with $y\in \mathbb R$. Then, it seems to me that by triangle inequality there holds:$$ |\Gamma(1+iy)| \leq \int_0^{\infty} |t^{iy}\,e^{-t}|\,dt \leq 1.$$However, the previous inequality is obviously in contradiction with Stirling's asymptotic formula for $y$ large. Where is the flaw in this?