Consider the equation $$\sum_{i=1,\ldots,n}p_{i}x_{i}^{1-r} = \sum_{j=1,\ldots,m}q_{j}y_{j}^{1-r}\tag{1}$$ where the "moving part" is $r$, while the $x_{i}, y_{j}, p_{i}, q_{j} \in \mathbb{R}$, are constant parameters with the following conditions:
- $x_{1},\ldots,x_{n}\geq 1 \text{ and } y_{1},\ldots,y_{m}\geq 1$
- $\sum_{i=1,\ldots,n} p_{i} = 1 \text{ and } p_{i} \geq 0 \text{ for all $i$.}$
- $\sum_{j=1,\ldots,m} q_{j} = 1 \text{ and } q_{j} \geq 0 \text{ for all $j$.}$
It is known that there exists $r' \in [0,1)$ such that $\sum_{i=1,\ldots,n}p_{i}x_{i}^{1-r'} < \sum_{j=1,\ldots,m}q_{j}y_{j}^{1-r'}$.
Given all of these assumptions, I have the conjecture that for $r \in [0,1)$, Equation (1) cannot have more than one real root. I have tried a couple of ways to prove it, e.g., an induction proof on the maximum number of summands, but I always fail. How would you go about proving it? Are there some known results that I could exploit? Or is the conjecture incorrect in the end?