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Hello everyone,
I've been working on an inequality involving sums and products over certain functions and indices, and I've come up with a proof sketch. However, I'm not entirely sure about its correctness and would appreciate any insights on how to formalize it better or even just to verify if the inequality holds.
Inequality Statement
Consider the following inequality:
Left-Hand Side (LHS): $$ \sum_{P_1 * P_2 * \ldots * P_n = R} \left( \omega(t_1)^{\frac{n_1(P_1)}{p_1}} \cdots \omega(t_n)^{\frac{n_1(P_n)}{p_1}} \right) \cdots \left( \omega(t_1)^{\frac{n_k(P_1)}{p_k}} \cdots \omega(t_n)^{\frac{n_k(P_n)}{p_k}} \right) $$
Right-Hand Side (RHS) in Ellipsis Notation: $$ \left( \sum_{i_1, \ldots, i_n \in \mathbb{N} : i_1 + \cdots + i_n = n_1(R)} \omega(t_1)^{\frac{i_1}{p_1}} \cdots \omega(t_n)^{\frac{i_n}{p_1}} \right) \cdots \left( \sum_{i_1, \ldots, i_n \in \mathbb{N} : i_1 + \cdots + i_n = n_k(R)} \omega(t_1)^{\frac{i_1}{p_k}} \cdots \omega(t_n)^{\frac{i_n}{p_k}} \right) $$
Proof Sketch
Consider a specific term in the LHS summation defined by a fixed sequence of words $P_1, P_2, ..., P_n$ that concatenate to form $R$. For simplicity, let's focus on the number 1 and its corresponding counts in the words $P_i$.
The contribution of each word $P_i$ to the count of the number 1 in $R$ is given by $n_1(P_i)$. The sum of these counts, $n_1(P_1) + \cdots + n_1(P_n)$, equals $n_1(R)$, as it enumerates the occurrences of 1 in $R$.
In the RHS, the first term (for $\ell = 1$) sums over all tuples $(i_1, ..., i_n)$ in the naturals that add up to $n_1(R)$. The LHS term for $P_1, P_2, ..., P_n$ corresponds to one such tuple in this RHS summation.
Extending this reasoning to all $k$ terms in the RHS, we see that each term in the LHS can be matched to a specific product of terms in the RHS summations. Consequently, the LHS is bounded above by the RHS, as it is a sum of such terms.
Definitions
- $\omega(t_i)$: Positive numbers representing specific functions of $t_i$.
- $n_\ell(P_i)$: A function counting the occurrences of the number $\ell$ in the word $P_i$.
- $R = (r_1, r_2, ..., r_m)$: A tuple consisting of $m$ elements, each being a number ranging from 1 to $k$.
- Words $P_1, P_2, ..., P_n$: Words that concatenate to form $R$, including the possibility of empty words.
Question
Is there a way to formalize this proof sketch better? Additionally, I'm not completely certain if this inequality is true, but I have a strong suspicion that it is. Any insights or suggestions would be greatly appreciated!