Solve for $k$ for condition to hold.

Question

$C$ is a $n$-tuple consisting of positive reals, so $C\subset\mathbb{R^+}$ and I want this condition to be true

$$(1+\sum\limits_{c\in C}c)^k>\sum\limits_{c\in C}(1+c)^k$$

or

$$1>\frac{\sum\limits_{c\in C}(1+c)^k}{(1+\sum\limits_{q\in C}q)^k}=\sum\limits_{c\in C}\left(\frac{1+c}{1+\sum\limits_{q\in C}q}\right)^k$$

But every time I try something I get an equation of the form $a^k-b^k=1$ and I don't know how to solve for $k$.

Answer 1

Even for $n = 2$, you won't get a closed-form necessary and sufficient condition. If you want a sufficient condition, perhaps this will do: if $m = \max_{c \in C} c$ and $s = \sum_{c \in C} c$, then $$ \sum_{c \in C} (1+c)^k \le n (1+m)^k < (1+s)^k\ \text{if}\ k > \frac{\log n}{\log (1+s)-\log(1+m)}$$