What about an approximation of the Hölder's constant associated to $\sum_{n=0}^\infty\gamma^n\cos(11^n\pi x)$, where $\gamma$ is the Euler's constant?

Question

With $a=\gamma$ the Euler-Mascheroni constant and $b=11$ I've created this Weierstrass function, see this Wikipedia if your need it, $$W(x)=\sum_{n=0}^\infty\gamma^n\cos(11^n\pi x).\tag{1}$$

I was exploring this definition proving claims like that (using uniform convergence) $$\int_0^{1/2}W(x)dx=\frac{1}{\pi}\frac{1}{1+\gamma}.$$ After I wondered about an independet question: since our function is Hölder continuous I'm curious to know what about the associated constant $C$.

Question. What about an estimation of our constant $C$? I am asking what can we say about the size of this constant $C$. Many thanks.

I can not say anything about the size of this constant $C$, for our function with the corresponding associated exponent. Is it possible deduce a lower and/or an upper bound for $C$?

Answer 1

Let $a=-\log\gamma/\log b$. Then $\gamma=b^{-a}$and $0<a<1$. We have $$ |\cos(b^n\,\pi\,x)-\cos(b^n\,\pi\,y)|\le\min(2,b^n\,\pi\,|x-y|). $$ Let $N$ be the largest integer such that $b^N\,\pi\,|x-y|\le2$. Then \begin{align} |W(x)-W(y)|&\le\sum_{n=0}^Nb^{-na}\,b^n\,\pi\,|x-y|+2\sum_{n=N+1}^\infty b^{-na}\\ &=\frac{\pi\,b^{1-a}\bigl(b^N|x-y|\bigr)^{1-a}}{b^{1-a}-1}|x-y|^a+2\,\frac{b^{-(N+1)a}}{b^{1-a}-1}\\ &\le\frac{\pi\,b^{1-a}(2/\pi)^{1-a}}{b^{1-a}-1}|x-y|^a+\frac{2\,(2/\pi)^{1-a}}{b^{1-a}-1}|x-y|^a. \end{align} Thus $$ C\le\frac{(2/\pi)^{1-a}}{b^{1-a}-1}(\pi\,b^{1-a}+2). $$