  In this paper, we determine the $L^p(\mathbb{R})\times
L^q(\mathbb{R})\rightarrow L^r(\mathbb{R})$ boundedness of the bilinear Hilbert
transform $H_{\gamma}(f,g)$ along a convex curve $\gamma$
$$H_{\gamma}(f,g)(x):=\mathrm{p.\,v.}\int_{-\infty}^{\infty}f(x-t)g(x-\gamma(t))
\,\frac{\textrm{d}t}{t},$$ where $p$, $q$, and $r$ satisfy
$\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$, and $r>\frac{1}{2}$, $p>1$, and $q>1$.
Moreover, the same $L^p(\mathbb{R})\times L^q(\mathbb{R})\rightarrow
L^r(\mathbb{R})$ boundedness property holds for the corresponding (sub)bilinear
maximal function $M_{\gamma}(f,g)$ along a convex curve $\gamma$
$$M_{\gamma}(f,g)(x):=\sup_{\varepsilon>0}\frac{1}{2\varepsilon}\int_{-\varepsilon}^{\varepsilon}|f(x-t)g(x-\gamma(t))|
\,\textrm{d}t.$$
