Assume we have a trigonometric series $$ f(x)=\sum_{n=1}^{\infty} a_n\sin nx \in \text{Lip }\alpha, \, 0<\alpha <1. $$ Is there anything we can say about the series $$ g(x)=\sum_{n=1}^{\infty} |a_n|\sin nx, $$ like $g\in $ Lip $\beta$ for some values of $0<\beta<1$ which depend on $\alpha$? For example, consider the following theorems that can be found in any trigonometric series monograph:
$\textbf{Theorem}$. Suppose that there is $0<\alpha <1$ such that $$ \sum_{k=n}^{\infty} |a_n|+|b_n| = O(n^{-\alpha}). $$ Then, the trigonometric series $$ h(x)=\sum_{n=0}^{\infty}a_n \sin nx + b_n \cos nx$$ belongs to the class Lip $\alpha$.
$\textbf{Theorem}.$ If the trigonometric series $h(x)$ belongs to the class Lip $\alpha$ with $1/2<\alpha<1$, then $$ \sum_{k=n}^{\infty} |a_n|+|b_n| = O(n^{-\alpha+1/2}). $$
If we combine both theorems in our setting, we obtain that if $f\in $ Lip $\alpha$, then $g\in $ Lip $\alpha-1/2$. Do you know any way to improve this so that we get something ''better'' than $\beta=\alpha-1/2$, or any reference to some paper? Thanks in advance.