# Convexity of a function with discrete parameters

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.

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