I'm not very well-versed with coding for computation, just mark-up like LaTeX, but I would like to find a code that can approximate the Hölder Condition of a given function.
For what I am working on specifically, a Lip(1/2) function is a function $\lambda:[0, T] \rightarrow \mathbb{R}$ for which there exists some $M>0$ such that for all $s, t \in [0,T]$, $|\lambda(s)-\lambda(t)| \leq M \sqrt{|s-t|}$. More importantly, for a Lip(1/2) function $\lambda(t)$, the Lip(1/2) norm of $\lambda$ is the smallest $M > 0$ such that the equation above holds for all $s,t \in [0,T]$. It's denoted by $||\lambda||_{1/2}$.
For example, given $\lambda(t)=c \sqrt{t}$, for $c \in \mathbb{R}$, one can show that there exists $M>0$ such that for all $s,t \in [0,T], |\lambda(s)-\lambda(t)| \leq M \cdot |t-s|^{1/2}$. Also, one can show that $||\lambda||_{1/2}=c$.
As another example, for a complex-valued function $\lambda_r(t)=re^{2\pi it}$, one can show that $||\lambda||_{1/2}\approx 2.224$ by optimizing $f(t)=\frac{sin^2(\pi (s-t))}{t}$ to find the maximum on $[0,1/2]$. I left out a lot of work here, but the idea is there.
What I am interested in is a code or program that can approximate $||\lambda||_{1/2}$ for me, for much trickier equations such as any equation of the form $\lambda(t)=r\cdot cis(t)$ for $r=3cos(3t)$, $r=\sqrt{8cos(2t)}$, etc. Any point in the right direction would be greatly appreciated!