I am reading Pinelis "[An approach to inequalities for the distributions of infinite -dimensional martingales](http://link.springer.com/chapter/10.1007/978-1-4612-0367-4_9#page-1)" and cannot follow his proof of Theorem 3:


Let $(f\_n)$ be a martingale in a separable Banach space $(\mathcal{X},||~||)$, $\mathcal{X} = L^p, p \geq 2$ and $f^\*=\sup\{||f\_n||\}$.


Theorem 3 says



> 
> \begin{align} P(f^\*>r) \leq 2\exp\big(-r^2/2(p-1)\big), \quad r \geq 0
> \end{align}
> 
> 
> 


and Pinelis writes 



> 
> One can compare the last inequality with
>  \begin{align}
> P(f^\*>r) \leq (r+1)\exp(-r^2/2)
> \end{align}
>  in Kallenberg ans Sztencel (1991) proved for $\mathcal{X} = L^2$.
> 
> 
> 


Does he refer to


\begin{align}
P(f^\*>r) \leq \frac{1+r}{1+rc}\exp\big(-\frac{r}{2c} \ln(1+rc)\big), \quad r \geq 0
\end{align}
from Kallenberg and Sztencel (1991)? If so, I still cannot see the path he takes to prove his theorem.


Kallenberg and Sztencel (1991): [Some dimension-free features of vector-valued martingales](http://link.springer.com/article/10.1007/BF01212560#page-1)



Proof of Pinelis (1992) - Banach space inequalities
As written in my paper [[1](http://link.springer.com/chapter/10.1007/978-1-4612-0367-4_9#page-1)], the inequality 
$$P(f^\*>r) \le 2\exp\big(-r^2/2(p-1)\big)
$$
in Theorem 3 in [[1](http://link.springer.com/chapter/10.1007/978-1-4612-0367-4_9#page-1)] 
for martingales in $\mathcal{X}=L^p$ can be compared with the inequality 
$$
P(f^\*>r) \le C (r+1)\exp(-r^2/2)
$$
with an unidentified absolute constant $C>0$, given 
(as noted in the comment by **Nate Eldredge**) in Theorem 5.3 of [[Kallenberg and Sztencel](http://link.springer.com/article/10.1007/BF01212560#page-1)], proved for $\mathcal{X} = L^2$.


As for the inequality 
$$P(f^\*>r) \le C \frac{1+r}{1+rc}\exp\big(-\frac{r}{2c} \ln(1+rc)\big) 
$$
with an absolute constant $C>0$, 
which is inequality (5.2) of [[Kallenberg and Sztencel](http://link.springer.com/article/10.1007/BF01212560#page-1)], it can be compared with the inequality in Theorem 2 of [[1](http://link.springer.com/chapter/10.1007/978-1-4612-0367-4_9#page-1)]. 


More general results were given in [[3](https://projecteuclid.org/euclid.aop/1176988477)]. 


Also, you wrote: "I still cannot see the path he takes to prove his theorem." Can you specify the steps that seem unclear? 


