Abstract
This paper is aimed at establishing new upper bounds for the first positive eigenvalue of the
ϕ
-Laplacian operator on Riemannian manifolds in terms of mean curvature and constant sectional curvature. The first eigenvalue for the
ϕ
-Laplacian operator on closed oriented
m
-dimensional slant submanifolds in a Sasakian space form
M
~
2
k
+
1
ε
is estimated in various ways. Several Reilly-like inequalities are generalized from our findings for Laplacian to the
ϕ
-Laplacian on slant submanifold in a sphere
S
2
n
+
1
with
ε
=
1
and
ϕ
=
2
.