= Solution
Set $\xi=\tanh(z/H)$, so $h=1-\xi^2$ and $d/dz=(1-\xi^2)H^{-1}d/d\xi$. The vertical equation becomes
$$
\frac{1-\xi^2}{H^2}\frac{d}{d\xi}\left[(1-\xi^2)\frac{dF}{d\xi}\right]+k^2(1-\xi^2)F=0.
$$
For $|\xi|<1$, division by the common factor gives the <Legendre differential equation> with eigenvalue $k^2H^2$. Requiring boundedness at both surfaces $\xi=\pm1$ selects
$$
\boxed{F_n(z)=P_n(\tanh(z/H)),\qquad k_n=\frac{\sqrt{n(n+1)}}H,\quad n=0,1,2,\ldots.}
$$
The second independent solution is unbounded at an endpoint. These are the <bounded vertical modes of a sech-squared magnetized disk>. The division by $H$ is outside the square root, as in the original PDF; the converted TeX places it incorrectly. The magnetic perturbation involves $F'_n$, which tends to zero at either surface.
The constant $n=0$ function has $k=0$. The specified magnetic ansatz contains $1/(ik)$ and must not be applied to it. A uniform horizontal velocity with no magnetic perturbation is an <epicyclic motion>, not a growing magnetic mode; the zero-wavenumber formal degeneracy does not provide growing MRI at arbitrary field strength. Nontrivial magnetically coupled vertical modes have $n\geq1$.
For $A>0$, the <magnetorotational instability> grows when $A<3\Omega^2$. Indeed, the quadratic in $s^2$ then has a negative constant term, and one positive root. For $A\geq3\Omega^2$, the two roots are nonpositive, since their sum is negative, their product nonnegative, and their discriminant is $\Omega^4+16\Omega^2A>0$. The smallest nonzero vertical eigenvalue is $k_1^2=2/H^2$, so all vertical modes are stable if
$$
\frac{2v_A^2}{H^2}\geq3\Omega^2
\quad\Longleftrightarrow\quad
\boxed{\beta=\frac{2\Omega^2H^2}{v_A^2}\leq\frac43.}
$$
The strict inequality requested gives stability, and equality is marginal for $n=1$. A sufficiently strong field increases <magnetic tension>. The unstable MRI needs a sufficiently long vertical wavelength to exchange angular momentum without excessive restoring tension; bounded vertical structure imposes a smallest nonzero effective wavenumber. Beyond the <finite-thickness magnetorotational instability criterion>, no allowed magnetic mode is long enough to remain unstable.
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