21. (12 points) As shown in Figure 5, the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ has its left focus at $F$. A line passing through point $F$ intersects the ellipse at points $A$ and $B$. The maximum value of $A F$ is $M$, and the minimum value of $B F$ is $m$, satisfying $M m=\frac{3}{4} a^{2}$.
(1) Find the eccentricity of the ellipse;
(2) Let the midpoint of segment $A B$ be $G$, and the perpendicular bisector of $A B$ intersect the $x$-axis and $y$-axis at points $D$ and $E$, respectively. $O$ is the origin. Denote the area of $\triangle G F D$ as $S_{1}$ and the area of $\triangle O E D$ as $S_{2}$. Find the range of $\frac{2 S_{1} S_{2}}{S_{1}^{2}+S_{2}^{2}}$.
21. (1) Let $F(-c, 0)(c>0)$. Then, according to the properties of the ellipse, we have
$$
M=a+c, m=a-c .
$$

And $M m=\frac{3}{4} a^{2} \Rightarrow a^{2}-c^{2}=\frac{3}{4} a^{2}$
$$
\Rightarrow a^{2}=4 c^{2} \Rightarrow a=2 c \text {. }
$$

Therefore, the eccentricity of the ellipse is $e=\frac{c}{a}=\frac{1}{2}$.
(2) From (1), we know
$$
a=2 c, b=\sqrt{a^{2}-c^{2}}=\sqrt{3} c \text {. }
$$

Thus, the equation of the ellipse is $\frac{x^{2}}{4 c^{2}}+\frac{y^{2}}{3 c^{2}}=1$.
Let $A\left(x_{1}, y_{1}\right), B\left(x_{2}, y_{2}\right)$.
According to the condition, the slope of line $A B$ must exist and be non-zero. Let $l_{A B}: y=k(x+c)$, substituting into the ellipse equation gives
$$
\left(4 k^{2}+3\right) x^{2}+8 c k^{2} x+4 k^{2} c^{2}-12 c^{2}=0 \text {. }
$$

Then $x_{1}+x_{2}=-\frac{8 c k^{2}}{4 k^{2}+3}$,
$$
y_{1}+y_{2}=k\left(x_{1}+x_{2}+2 c\right)=\frac{6 c k}{4 k^{2}+3} \text {. }
$$

Thus, $G\left(-\frac{4 c k^{2}}{4 k^{2}+3}, \frac{3 c k}{4 k^{2}+3}\right)$.
Since $D G \perp A B$, we have
$$
\frac{\frac{3 c k}{4 k^{2}+3}}{-\frac{4 c k^{2}}{4 k^{2}+3}-x_{D}} \cdot k=-1 \Rightarrow x_{D}=-\frac{c k^{2}}{4 k^{2}+3} \text {. }
$$

By the similarity of right triangles $\triangle F G D \sim \triangle E O D$, we know
$$
\begin{array}{l}
\frac{S_{1}}{S_{2}}=\frac{G D^{2}}{O D^{2}} \\
=\frac{\left(-\frac{4 c k^{2}}{4 k^{2}+3}+\frac{c k^{2}}{4 k^{2}+3}\right)^{2}+\left(\frac{3 c k}{4 k^{2}+3}\right)^{2}}{\left(-\frac{c k^{2}}{4 k^{2}+3}\right)^{2}} \\
=9+\frac{9}{k^{2}}>9 . \\
\text { Let } \frac{S_{1}}{S_{2}}=t . \text { Then } t>9 . \\
\text { Hence } \frac{2 S_{1} S_{2}}{S_{1}^{2}+S_{2}^{2}}=\frac{2}{t+\frac{1}{t}}<\frac{2}{9+\frac{1}{9}}=\frac{9}{41} .
\end{array}
$$

Therefore, the range of $\frac{2 S_{1} S_{2}}{S_{1}^{2}+S_{2}^{2}}$ is $\left(0, \frac{9}{41}\right)$.