Let $p>2$ be a real number.
In this blog post by George Lowther, a proof of the right-hand Khintchine inequality is given where $$m_p:=\min_{x>0}\;x^{-p}\cosh(x)$$ comes into play.
Question: Is it possible to express $m_p$ in closed form?
Remark: According to the proof given in this post, the best right-hand constant in Khintchine inequality is bounded above by $$K_p:=\frac{p^{-p/2}e^{p/2}}{m_p}.$$ So what I'm really interested in is the value of $K_p$ for $p>2$.