Fourier transform of an integrable odd function

Question

I'm trying to prove a proposition about the Fourier transform of an odd function.

Let $f\in L^1(\mathbb{R})$ be an odd function. Then there is $M>0$ such that for any $a,A>0$,

$\displaystyle{\Bigg|\int_a^A\frac{\hat{f}(\alpha)}{\alpha}d\alpha\Bigg|\leq M}$.

Answer 1

If $f(t)$ is an odd function, then $$|\int_a^A{\frac{d \alpha}{\alpha}\int_R{f(t)e^{i\alpha t}dt}}| = |\int_a^A{\frac{d \alpha}{\alpha}\int_{R_+}{2f(t)sin(\alpha t)dt}}| \le M_{L_1(f)} \int_a^A{\frac{sin(\alpha t)d\alpha t}{\alpha t}} \lt M_{L_1(f)} \frac{\pi}{2}$$ Where $M_{L_1(f)} $ is $L_1$ measure of $f(t)$