Prove Holdercontinuity for $\mu_\lambda (x) = \sum\limits_{n=1}^\infty \frac{ \cos(2^n x)}{2^{n \lambda} }$
Consider $\lambda \in (0,1]$ and the function $\mu_\lambda$ defined by $\mu_\lambda (x) = \sum\limits_{n=1}^\infty \frac{ \cos(2^n x)}{2^{n \lambda} }$ for $x \in [0,\pi]$. Show that $\mu_\lambda$ is $\alpha$Holdercontinuous on $[0, \pi]$ for all $0 < \alpha < \lambda$.
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