Let $\mathcal{P}_n$ be the set of all polynomials of degree $n$ or smaller, and consider the following subset: \begin{equation} \mathcal{M}_n = \left\{ F \in \mathcal{P}_n: F(0) = 0, F(1) = 1, F(x) \nearrow x \in [0, 1] \right\}. \end{equation} Here, $F(x) \nearrow x \in [0, 1]$ means that $F$ is monotonically increasing on $[0, 1]$. For all $n \in \mathbb{Z}_+$, I would like to show \begin{equation} x^n \le F(x) \le 1 - (1 - x)^n, \quad \forall F \in \mathcal{M}_n, \forall x \in [0, 1]. \end{equation} In fact, it would suffice to prove just one inequality, as the other would follow from symmetry. For context, I use $F$ to denote a member of $\mathcal{M}_n$ because it is a function that could serve as a CDF for a random variable with unit interval support. The result I am trying to prove pertains to first-order stochastic dominance. It is clear to me that for $F \in \mathcal{M}_n$ represented as $F(x) = \sum_{i=0}^n a_i x^i$, the following conditions must hold: \begin{align*} & a_0 = 0 && \text{for}~F(0) = 0, \\ & \textstyle \sum_{k=1}^n a_k = 1 && \text{for}~F(1) = 1, \\ & \textstyle F'(x) = \sum_{k=1}^n a_k\,k\,x^{k-1} \ge 0, \quad x \in [0, 1] && \text{for monotonicity}. \\ \end{align*}
However, it is unclear to me if this is the best characterization to prove (or disprove) the desired result. I tried starting with $\mathcal{M}_1 = \{x\}$ and then applying induction, but I have so far come up empty. Thanks in advance for your ideas.
The result is true for $\mathcal{M}_1 = \{x\}$ and $\mathcal{M}_2 = \{\alpha x + (1 - \alpha) : \alpha \in [0, 1]\}$. However, there are polynomials in $\mathcal{M}_3$ that do not satisfy the desired inequalities.
For example, $\exists x \in [0, 1]$ such that $3.97 x - 6.24 x^2 + 3.27 x^3 > 1 - (1 - x)^3$. Likewise, $\exists x \in [0, 1]$ such that $0.44 x - 1.74 x^2 + 2.30 x^3 < x^3$.