Answer
$r^2 < x^2+y^2+z^2 < R^2$
Work Step by Step
"between (but not on)...":
"spheres of radius r and R centered at the origin":
$x^2+y^2+z^2 [< or >] r^2$
$x^2+y^2+z^2 [< or >] R^2$
"r < R":
$r^2 < x^2+y^2+z^2 $
$x^2+y^2+z^2 < R^2$
Combine these statements to get:
$r^2 < x^2+y^2+z^2 < R^2$