Let $f\in L^{1}(\mathbb T)$ and define the Fourier coefficient of $f$ : $\hat{f}(n)=\frac{1}{2\pi} \int _{-\pi}^{\pi} f(t) e^{-int} dt; (n\in \mathbb Z)$.Consider the space, $$A(\mathbb T):= \{f\in L^{1}(\mathbb T): \hat{f}\in \ell^{1}(\mathbb Z), \ \text {that is,} \ \sum_{n\in \mathbb Z} |\hat{f}(n)| < \infty \}.$$ $A(\mathbb T)$ is normed by the $L^{1}-$ norm on $\mathbb Z$: $$||f||= \sum_{n\in \mathbb Z} |\hat{f}(n)|; \ \text {for} \ f\in A(\mathbb T). $$ We also note that $A(\mathbb T)$ is a Banach algebra under pointwise addition and multiplication. Let $f_{0}\neq 0 \in A(\mathbb T)$ and fix it; and take $M= 2||f_{0}||$ and put $B_{M}= \{f\in A(\mathbb T): ||f||\leq M \}.$

Let $f, g \in B_{M}.$

My Question: Can we expect $\left\||f|^{2}f-|g|^{2}g\right\|\leq C ||f-g||$ , where $C$ is some constant ? If yes, what can we say about $C$ ?

I guess, the trivial relation: $|f|^{2}f-|g|^{2}g=(f-g)|f|^{2}+g(|f|^{2}-|g|^{2})$; may be useful.

Thanks,