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N is a random variable whose sample space is [0,$\infty$). I have an expression in terms of the expectation of this variable and I want to find a tight upper bound on the whole expression. The expression is as below:

$1-\gamma E[\frac{(N+1)}{(1+b*N+\gamma *N+b+\gamma)}]$ where $\gamma >0$ and $b>0$

I don't know the distribution of N but assuming that I know the $E[N]$, can a tight bound be established on the above expression? If so, can you please refer me to theorems/inequalities that would help me find it? I looked at Jensen's inequality but it provides a lower bound since the expression over which the expectation is being taken is concave.

(This expression is actually a bound on the probability mass for a subset of the sample space but I don't know the distribution.)

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  • $\begingroup$ You can simplify the problem considerably by re-expressing it in terms of the random variable $X=1+(b+\gamma)(1+N),$ thereby reducing it to bounding $E[1/X]$ given $E[X]$ and a lower bound on the support of $X.$ $\endgroup$
    – whuber
    Nov 28, 2018 at 18:57
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    $\begingroup$ @whuber, thank you for the reply. Wouldn't [1/X] become convex now and applying Jensen's give a lower bound again for the whole expression? I think the expression would become $1−p+p*E[1/X]$ where X is as defined by you and $p=\frac{\gamma}{b+\gamma}$ $\endgroup$
    – gaganso
    Nov 28, 2018 at 19:23
  • $\begingroup$ you may see this one: stats.stackexchange.com/questions/248142/… $\endgroup$
    – user360007
    Jun 6, 2022 at 15:27
  • $\begingroup$ this link might help: stats.stackexchange.com/questions/248142/… $\endgroup$
    – user360007
    Jun 6, 2022 at 15:31

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