I am studying the concept of Fourier multipliers in $L^p$ and I have a doubt. Suppose $1/m$ is a Fourier multiplier in $L^p$ (with $1<p<\infty$). From what I can see, $1/(1+m)$ must also be a Fourier multiplier, but I don't see how to prove it in a simple way. (I don't think I should do the hard calculation because it is only adding $1$.) Is there any general result to conclude that the function $1/(1+m)$ is also a multiplier? Thanks.
pd: To avoid technical steps or singularities, suppose that $1/m(x)$ and $1+m(x)\neq 0$ for all $x\in\mathbb{R}^n.$