This is Theorem V (page 16) from:

A. Beurling, On the spectral synthesis of bounded functions. Acta Math. 81 (1948).

In fact, Beurling proves the stronger statement:

Theorem Let $f(x) = \sum_{n=-\infty}^{\infty} a_n e(nx)$ (with $a_0=0$) have an absolutely convergent Fourier series such that $|a_{\pm n}| \leq a_n^{*}$ where $a_n^{*}$ is a non-increasing sequence in $\ell^{1}$. Moreover, assume that $g(x) = \sum_{n=-\infty}^{\infty} b_n e(nx)$ (with $b_0=0$) is a contraction of $f$. Then the Fourier series of $g(x)$ converges absolutely and $\sum_{n=-\infty}^{\infty}|b_n| \ll \sum a_n^{*}$.

This is Theorem V (page 16) from:

A. Beurling, On the spectral synthesis of bounded functions. Acta Math. 81 (1948).

In fact, Beurling proves the stronger statement:

Theorem Let $f(x) = \sum_{n=-\infty}^{\infty} a_n e(nx)$ (with $a_0=0$) have an absolutely convergent Fourier series such that $|a_{\pm n}| \leq a_n^{*}$ where $a_n^{*}$ is a non-increasing sequence in $\ell^{1}$. Moreover, assume that $g(x) = \sum_{n=-\infty}^{\infty} b_n e(nx)$ (with $b_0=0$) is a contraction of $f$. Then the Fourier series of $g(x)$ converges absolutely and $\sum_{n=-\infty}^{\infty}|b_n| \ll \sum_{n=-\infty}^{\infty} a_n^{*}$.