  In this paper, we prove that a closed minimal hypersurface in $\SSS$ with
$\lambda_1<n$ has Morse index at least $n+4$, providing a partial answer to a
conjecture of Perdomo. As a corollary, we re-obtain a partial proof of the
famous Urbano Theorem for minimal tori in $\mathbb{S}^3$: a minimal torus in
$\mathbb{S}^3$ has Morse index at least $5$, with equality holding if and only
if it is congruent to the Clifford torus. The proof is based on a comparison
theorem between eigenvalues of two elliptic operators, which also provides us
simpler new proofs of some known results on index estimates of both minimal and
$r$-minimal hypersurfaces in a sphere.
