Zero arithmetic mean: bound on Abel sum?

Question

Let $(a_0, a_1, a_2, \ldots)$ be a bounded sequence in $\mathbb{R}$ with arithmetic means \begin{equation} \frac{a_0 + a_1 + \cdots + a_{n-1}}{n} \end{equation} converging to zero as $n \to \infty$. Is the following statement true: \begin{equation} \liminf_{x \to 1^-} \sum_{n=0}^{\infty} a_n x^n \geq 0 \quad \text{or} \quad \liminf_{x \to 1^-} \sum_{n=0}^{\infty} (-a_n) x^n \geq 0 \quad \text{(or both)}. \end{equation} What I tried: for nonzero means, the partial sums are bounded away from zero for sufficiently large $n$, giving corresponding estimates on the lower/upper Abelian sum. But for the zero mean case, I couldn't figure it out. Tried finding this in Hardy's Divergent Series as well.