Given a function $f(X\_1,\cdots,X\_n,Y)$ on random variables $\{X\_i\}$ and $Y$, which is continuous ,
I want to 
show that $f$ concentrates around its expectation $\operatorname\*{E}[f]$, i.e., a formula like this: 
$\Pr[|f(X\_1,\cdots,X\_n, Y)-\operatorname\*{E}[f(X\_1,\cdots,X\_n, Y)]|\geq t]\leq \exp(-\frac{t^2}{2c^2})$, where $c^2$ is the Lipschitz-type bound on $f$.


The case considered here is different from the traditional one which does not have the additional continous 
random variable $Y$. However, we can still use the traditional way to show the concentration. By the 
Law of total probability, it is equal to bound 
$\operatorname\*{E}\_Y[\Pr[|f(X\_1,\cdots,X\_n, y)-\operatorname\*{E}[f(X\_1,\cdots,X\_n, y)]|\geq t|Y=y]] \qquad (1)$.


Given $y$ , if we have that
$|\operatorname\*{E}[f|X\_1,\cdots,X\_{i-1},X\_i=x\_i,y]-|\operatorname\*{E}[f|X\_1,\cdots,X\_{i-1},X\_i=x'\_i, y]\leq c\_i(y),$
for all $i$ with $1\leq i\leq n$ and any $x\_i,x'\_i$.


then by the stardard use of Azuma's inequality, 
$\Pr[|f(X\_1,\cdots,X\_n, y)-\operatorname\*{E}[f(X\_1,\cdots,X\_n, y)]|\geq t|Y=y]\leq \exp(-\frac{t^2}{2\sum\_{i=1}^n c\_i^2(y)})$.


Thus, from (1),
$\Pr[|f(X\_1,\cdots,X\_n, Y)-\operatorname\*{E}[f(X\_1,\cdots,X\_n, Y)]|\geq t]\leq \operatorname\*{E}[\exp(-\frac{t^2}{2\sum\_{i=1}^n c\_i^2(Y)})] \qquad (2)$


My question is that can the above inequality be improved as:
$\Pr[|f(X\_1,\cdots,X\_n, Y)-\operatorname\*{E}[f(X\_1,\cdots,X\_n, Y)]|\geq t]\leq \exp(-\frac{t^2}{2\sum\_{i=1}^n \operatorname\*{E}[c\_i(Y)]^2}) \qquad (3)$.


P.S. I think that the Jassen's inequality (i.e., $\operatorname\*E[g(Z)]\geq g(\operatorname\*E[Z])$ for convex function
$g$) may be useful here, but I donot see the convexity of the right hand of inequality (2). 



Concentration bound using Azuma's inequality and Law of total probability
See my comment above for some problem in your argument but anyhow (3) is wrong. If the $X\_i$-s are constants then the right hand side of (3) is 0, while the left hand side is not in general. If you don't like to use constant r.v.: if each of the $X\_i$ takes values in a small interval, the right hand side is arbitrarily close to zero while the left hand side not, in general (say, for the function $f(x\_1,...,x\_n,y)=y$ and any reasonable $Y$).


