Suppose that we have $q$ positive integers $a\_1, \ldots, a\_q$ satisfying $a\_1 \leq \cdots \leq a\_q \leq q$. I'm interested in the possible types of behaviors for the function given by $$f(x) = (a\_1^{x-1} + \cdots + a\_q^{x-1})^{1/x},$$ where $x \in [2,\infty)$. In particular, I'm interested in the behavior at integer values of $x$ in that range, but I think that the continuous version of $f$ might be easier to handle.


It isn't hard to see that $\lim\_{x \to \infty} f(x) = a\_q$. I can show that if there is an $x\_0 \in (2,\infty)$ with $f(x\_0) > a\_q$, then $f$ is decreasing at $x\_0$, and furthermore I can show that $f$ is either eventually decreasing, increasing, or constant (this almost entirely comes from the number of $i$ with $a\_i = a\_q$). 


Each example $(a\_1,\ldots,a\_q)$ that I've used also has the nice property that it is either always decreasing, always constant (this in fact only occurs if $q=a\_q$ and $a\_i = a\_q$ for each $i$), or decreasing on $[2,x\_0)$ and increasing on $(x\_0,\infty)$ for some $x\_0$ which depends on the $a\_i$. Does anyone know of either a proof or counterexample of this property? Is any other behavior possible? Simply computing the derivative of $f$ doesn't seem to be too helpful, but perhaps I'm missing the correct viewpoint on this derivative.


**Edit**: added the restriction $a\_q \leq q$. Without this restriction, $f(x)$ can be strictly increasing as well, e.g. $a\_1=3,a\_2=4$. An example of the decreasing/increasing is $a\_1=1, a\_2=2$.



Sum of exponential functions
As @David points out, the function is log-convex -- this is a general fact about $L^p$ norms (as pointed out by @Tom, this is a special case of an $L^p$ norms where the measure of a point $x\_i$ is proportional to $1/a\_i.$) Not one, not two, but *three* proofs of this fact is given in [Terry Tao's excellent as usual blog post.](http://terrytao.wordpress.com/2009/03/30/245c-notes-1-interpolation-of-lp-spaces/) Since a convex function has at most one minimum, that seems to resolve your conjecture. 


