Trying to apply Jensen's inequality to a complicated symbolic expression

Question

I have a somewhat complicated inequality expression like this:

ineq =                         i * Log[n/i] + 
                                   Log[r[i, 0]] +
                         (n - i) * Log[(n*(1 - Subscript[p, 0]))/(n - i)] +
      Sum[(n - i)*Binomial[i, k] * Log[1 - Subscript[p, k]], {k, 1, i}] > 0

I would like to bound it be Jensen's inequality: $\log\left(\frac{\sum a_i x_i}{\sum a_i}\right) \ge \frac{\sum a_i \log(x_i)}{\sum a_i}$. Yes, I make sure, that always $a_i >0$, in this case 0 < i < n. So first I need to compute the normalization constant, in this case it is:

z = i + 1 + (n - i) + Sum[(n - i)*Binomial[i, k], {k, 1, i}]

So, how can I compute this z in this symbolic case when the $a_i$ can be 1 or a Sum like here?

Then I suppose I would get the desired bound by:

ApplySides[#/z&, ineq]
With[{lhs = %[[1]]}, 
  Log[lhs /. Log -> Identity] >= lhs
]

So basically my question reduces to: how to compute this normalization constant symbolically?

Answer 1

OK, I guess I figured it myself, the following seems to do what I need:

JensenCoef[a_ Log[b_]] := a
JensenCoef[Log[b_]] := 1
JensenCoef[Sum[Times[a__, Log[b_]], i_]] := Sum[Times[a], i]
JensenUpperBound[expr_] := Log[(expr /. Log -> Identity)/(JensenCoef /@ expr)]