**1. Introduction**

It was proven by V. Borrelli, B.-Y. Chen and J. M. Morvan [1], and independently by A. Ros and F. Urbano [2], that if *M* is a Lagrangian submanifold, with dim(*M*) = *m*, of C*<sup>m</sup>*, with mean curvature vector *H* and scalar curvature *τ*, then -*H*-2 ≥ <sup>2</sup>(*m* + 2) *m*<sup>2</sup>(*m* − 1) *τ*, with equality if and only if *M* is either totally geodesic or a (piece of a) Whitney sphere. Moreover, they proved that *M* satisfies the equality case at every point if and only if its second fundamental form *σ* is given by

$$
\sigma(\mathbf{X}, \mathbf{Y}) = \frac{m}{m+2} \{ \mathbf{g}(\mathbf{X}, \mathbf{Y}) \mathbf{H} + \mathbf{g}(\mathbf{J} \mathbf{X}, \mathbf{H}) \mathbf{J} \mathbf{Y} + \mathbf{g}(\mathbf{J} \mathbf{Y}, \mathbf{H}) \mathbf{J} \mathbf{X} \}, \tag{1}
$$

for any tangent vector fields *X* and *Y*. Thus, they found a simple relationship between one of the main intrinsic invariants, *τ*, and the main extrinsic invariant *H*.

It was also proven in [2], that the Maslov form, which is a closed form for a Lagrangian submanifold of C*<sup>m</sup>*, is a conformal form if and only if *M* satisfies (1).

Later, D. E. Blair and A. Carriazo [3] established an analogue inequality for anti-invariant submanifolds in R2*m*+<sup>1</sup> with its standard Sasakian structure and characterized the equality case with a specific expression of the second fundamental form, similar to Equation (1). In a previous paper [4], we studied the corresponding inequality for slant submanifolds of generalized Sasakian space forms; we also characterized the equality case with an specific expression of the second fundamental form; and finally, we presented some examples satisfying the equality case.

Both B.-Y. Chen, [5] and A. Carriazo, [6], have studied the existence of closed forms for slant submanifolds in different environments. The existence of closed forms is particularly interesting, as they provide conditions about submanifolds admitting an immersion in a certain environment.

The purpose of this paper was to obtain some results similar to those of [2] for slant submanifolds of a generalized Sasakian space form. After a section with the main preliminaries, we show that for a slant submanifold of a generalized Sasakian manifold, the Maslov form is not always closed. Therefore, in the following section, we present a form that is always closed for a slant submanifold, so it really plays the role of the Maslov form in the cited papers. Later, if the submanifold satisfies the equality case in the corresponding inequality, that is, if the second fundamental form takes a particular expression [4], we study if the vector field associated with the given form is a conformal vector field.
