Hello Wolfram Community,

I am working on a problem in Mathematica where I need to define a series of assumptions for an array of variables a[i], where the size of the array m is symbolic (i.e., not a fixed integer). My goal is to use these assumptions in conjunction with FullSimplify and Reduce to check the validity of a certain inequality.

The specific challenge I'm facing is in correctly defining these assumptions for an array of symbolic size. Here is the inequality I am working with:

$${\frac{a_{1}}{\sqrt[n]{a_{1}\ldots a_{n}}}+\cdots+\frac{a_{n}}{\sqrt[n]{a_{1}\ldots a_{n}}}\geq n}$$

Where all the $a_i$s should be assumed positives.

My initial approach was to use Array[a, m] > 0 within Assuming, but this resulted in an error indicating that a list of non-negative machine-sized integers was expected at a certain position. I then tried to use And @@ Table[a[i] > 0, {i, 1, m}], but this also did not compile as intended.

Here is the code snippet that I am currently struggling with:

Clear[a, m];
Assuming[
{And @@ Table[a[i] > 0, {i, 1, m}], m > 0, Element[m, Integers]},
FullSimplify[
Reduce[
Sum[a[i]/Product[a[j], {j, 1, m}]^(1/m), {i, 1, m}] >= m
]
]
]