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The answer is no. E.g., let $f_0=0$, $f_1=1$, $\alpha=1$, $\beta=0$, $F(x)\equiv x/2+3x^2/2$, $I=(0,1)$. Then \begin{equation} F(x)=x/2+3x^2/2\le x+x^2+\dots= \dfrac{x}{1-x}=\dfrac{f_0 + xf_1 - \alpha x f_0}{1-\alpha x - \beta x^2} \end{equation} for $x\in I$, whereas $3/2\not\le1$.