1

How to prove convexity of this function?I want to prove the convexity of the following function:

$f(\gamma)=$\begin{equation} \begin{aligned} log(\sum^{n+1}_{i=1} i^{-\gamma})-log(\sum^{n}_{i=1} i^{-\gamma}) \end{aligned} \end{equation}

I've tried to calculate the second deriviate, but it's really complicate and hard to tell whether it's positive or not.

Can anyone help me with that? Thank you very much.

taiat
  • 1,137
  • 1
    Can you provide some context? Where does that function come from? – Martin R Jan 24 '22 at 08:35
  • @MartinR At first I want to prove the convexity of the function \begin{equation} \begin{aligned} log(\sum^{n+1}{i=1} i^{-\gamma})-log(\sum^{n}{i=1} i^{-\gamma}) \end{aligned} \end{equation} for any given n. I'm still not sure whether this is correct. But I have checked $n\le 1500$. And let $c_i=\frac {n+1} {k}$, I got the function in the question. At first I think that it's true for arbitrary $c_i$. But as you said, this is wrong. So can you help me with the original function? Thanks again for your time. – taiat Jan 24 '22 at 09:30

0 Answers0