Let $f\in L^{1}(\mathbb R)$ and it [Fourier transform](http://en.wikipedia.org/wiki/Fourier_algebra), $\hat{f} (y) : = \int _ {\mathbb R} f(x) e^{-2\pi i x\cdot y} dx ; y \in \mathbb R ;$  and consider Fourier algebra
$$A(\mathbb R):=  \{f\in L^{1}(\mathbb R): \hat{f}\in L^{1}(\mathbb R) \}$$
$A(\mathbb R)$ is normed by:
$$\left\|f\right\|:= \left\|\hat{f}\right\|_{L^{1}(\mathbb R)}=\int_{\mathbb R}|\hat{f}(\xi)| d\xi; \ (f\in A(\mathbb R)).$$
We note that $A(\mathbb R)$ is a [Banach algebra](http://en.wikipedia.org/wiki/Banach_algebra) under point wise addition and multiplication.

Let $M>0$ and fix it; and consider $B_{M}= \{f\in A(\mathbb R): f|f|\in A(\mathbb R) \ \text{and}  \ ||f||\leq M \}.$


>**My Question**: Can we expect, $\left\|f|f|- g|g|\right\|\leq C \left \|f-g\right \|$; for $f,g \in B_{M}$ , where $C$ is some Constant ? If yes, what can we say about $C$ ? If not, can we produce counter example ?

Thanks,