I was reading about the property
$$|B(a,b)|\le B(\Re(a),\Re(b))$$
for any $a,b\in\mathbb{C}$ with $\Re(a),\Re(b)>0$ (as principal-ideal-domain mentioned in the comments), where $B$ denotes the Beta function
$$ B(a,b) = \int_0^1x^{a-1}(1-x)^{b-1}dx. $$
Is this just done by using the triangle inequality?
$$ \begin{align} |B(a,b)|&\le \int_0^1|x^{a-1}|\ |(1-x)^{b-1}|\ dx \\ &=\int_0^1 |e^{\log(x) (\Re(a-1)+i\Im(a-1))}|\ |e^{\log(1-x) (\Re(b-1)+i\Im(b-1))}|\ dx\\ &=\int_0^1 e^{\log(x) \Re(a-1)}\ e^{\log(1-x) \Re(b-1)}\ dx\\ &=\int_0^1 x^{\Re(a-1)}\ (1-x)^{\Re(b-1)}\ dx\\ &=\int_0^1 x^{\Re(a)-1}\ (1-x)^{\Re(b)-1}\ dx\\ &=B(\Re(a),\Re(b)) \end{align}$$