# Tight upper bound on the expectation of a concave function

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.)

• 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 $X.$
– whuber
Nov 28, 2018 at 18:57
• @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 $p=\frac{\gamma}{b+\gamma}$ Nov 28, 2018 at 19:23
• you may see this one: stats.stackexchange.com/questions/248142/… Jun 6, 2022 at 15:27
• this link might help: stats.stackexchange.com/questions/248142/… Jun 6, 2022 at 15:31