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Let $k\in\mathbb N_{>0}$, $n_1,\dots,n_k\in\mathbb N_{>0}$ and $N=\sum_{i=1}^kn_i$. I would like to know if the following function is convex: $$ r_{\alpha,\theta}(Z) = \overbrace{\sum_{i=1}^{n-1}\ln(\alpha+i)}^\beta - \overbrace{\sum_{i=1}^{k-1}\ln(\alpha+i\theta)}^{\gamma(k)} - \overbrace{\sum_{c=1}^k\sum_{i=1}^{n_c-1}\ln(i-\theta)}^{\delta(Z)}. $$ where $\alpha$ and $\theta$ are constants. I am not sure how I can do this. At first, I was thinking about relaxing the discreteness of $k$ and $n_i$ but it just doesn't make sense.

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  • $\begingroup$ So the variables are $n_i$? $\endgroup$ – Samrat Mukhopadhyay Sep 30 '15 at 9:39
  • $\begingroup$ $n_i$ and $k$, yes. $\endgroup$ – davcha Sep 30 '15 at 9:56
  • $\begingroup$ I don't know how to define convex functions over a domain which is a countable set. Do you have a definition for this? $\endgroup$ – Samrat Mukhopadhyay Sep 30 '15 at 13:26
  • $\begingroup$ I tried something like replacing sums by integrals, but I don't know if it really makes sense. $\endgroup$ – davcha Sep 30 '15 at 17:26

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