Let $x_1, x_2, \dots, x_n$ be real numbers such that $x_n \geq x_{n-1} \geq \dots \geq x_1$ and $0 < x_i \leq \frac{1}{n}$, for any $i$. Define $t = \left(x_1 \cdot x_2 \cdot \dots \cdot x_n\right)^{\frac{1}{n}}$. Prove that$$(1 + x_n(n-1))(1 - x_1)(1 - x_2) \dots (1 - x_{n-1}) \geq (1 + t(n-1))(1 - t)^{n-1}.$$
I tried running a computer simulation, and it seems like the inequality holds true.
Additionally, it's worth noting that while the simulation suggests the inequality holds, there is a possibility it might be incorrect, but probably it is possiblity correct in cases when $x_i$ is smaller than $O\left(\frac{1}{n^s}\right)$ for some $s$.
I'm attaching the code I used for the simulation for reference.
import numpy as npdef pick_parameters(n): # Generate n random numbers between 0 and 1 and sort them x = sorted(np.random.uniform(0,1/(n), n)) return xdef compute_t(x): n = len(x) t = np.prod(x) ** (1 / n) return tdef check_inequality(x): n = len(x) t = compute_t(x) left_side = (1 + (n - 1) * x[n-1]) * np.prod([(1 - x[i]) for i in range(n - 1)]) right_side = (1 + t * (n-1)) * ((1 - t) ** (n - 1)) return left_side >= right_side, left_side, right_sidedef main(): n = int(input("Enter the number of parameters n: ")) x = pick_parameters(n) print(f"Picked parameters x: {x}") t = compute_t(x) print(f"Computed t: {t}") inequality_holds, left_side, right_side = check_inequality(x) print(f"Inequality holds: {inequality_holds}") print(f"Left side of the inequality: {left_side}") print(f"Right side of the inequality: {right_side}")if __name__ == "__main__": main()