  For discrete martingale-difference sequences $d=\{d_1,\ldots,d_n\}$ we
consider Khintchine type inequalities, involving certain square function
$\mathfrak S (d)$ considered by Chang-Wilson-Wolff in 1982. In particular, we
prove
  \begin{equation}
  \left\|\sum_{k=1}^nd_k\right\|_p\le
2^{1/2}\big(\Gamma((p+1)/2))/\sqrt{\pi}\big)^{1/p}\|\mathfrak
S(d)\|_\infty,\quad p\ge 3,
  \end{equation}
  where the constant on the right hand side is the best possible and the same
as known for the Rademacher sums $\sum_{k=1}^na_kr_k$. Moreover, for a fixed
$n$ the constant in the inequality can be replaced by
$\sum_{k=1}^nr_k/\sqrt{n}$. We apply a technique, reducing the general case to
the case of Haar and Rademacher sums, that allows also establish a sub-Gaussian
estimate
  \begin{equation}
  {\bf E}\left[\exp\left(\lambda\cdot \left(\frac{\sum_{k=1}^nd_k}{\|\mathfrak
S(d)\|_\infty}\right)^2\right)\right]\le \frac{1}{\sqrt{1-2\lambda}},\quad
0<\lambda<1/2,
  \end{equation}
  where the constant on the right hand side is the best possible.
