Bull. Korean Math. Soc. 53 (2016), No. 4, pp. 1095–1103 http://dx.doi.org/10.4134/BKMS.b150523 pISSN: 1015-8634 / eISSN: 2234-3016 GEOMETRIC INEQUALITIES FOR SUBMANIFOLDS IN SASAKIAN SPACE FORMS Ileana Presur˘ a Abstract. B. Y. Chen introduced a series of curvature invariants, known as Chen invariants, and proved sharp estimates for these intrinsic invari- ants in terms of the main extrinsic invariant, the squared mean curvature, for submanifolds in Riemannian space forms. Special classes of submanifolds in Sasakian manifolds play an impor- tant role in contact geometry. F. Defever, I. Mihai and L. Verstraelen [8] established Chen first inequality for C-totally real submanifolds in Sasakian space forms. Also, the differential geometry of slant submanifolds has shown an increasing development since B. Y. Chen defined slant submanifolds in complex manifolds as a generalization of both holomorphic and totally real submanifolds. The slant submanifolds of an almost contact metric manifolds were defined and studied by A. Lotta, J. L. Cabrerizo et al. A Chen first inequality for slant submanifolds in Sasakian space forms was established by A. Carriazo [4]. In this article, we improve this Chen first inequality for special contact slant submanifolds in Sasakian space forms. 1. Preliminaries A (2m + 1)-dimensional Riemannian manifold  M it said to be a Sasakian manifold if it admits an endomorphism φ of its tangent bundle T  M, a vector field ξ and a 1-form η, satisfying φ 2 = −I + η ⊗ ξ, η(ξ )=1,η ◦ φ =0, g(φX,φY )= g(X,Y ) − η(X )η(Y ),η(X )= g(X,ξ ), (  ∇ X φ)Y = g(X,Y )ξ − η(Y )X,  ∇ X ξ = −φX, for any vector fields X,Y on T  M, where  ∇ denotes the Riemannian connection with respect to g. A plane section π in T  M is called a φ-section if it is spanned by X and φX, where X is a unit tangent vector orthogonal to ξ. The sectional curvature of a Received July 1, 2015; Revised December 8, 2015. 2010 Mathematics Subject Classification. 53C25, 53B21, 53C40. Key words and phrases. Sasakian space forms, special contact slant submanifolds, Chen invariants. c 2016 Korean Mathematical Society 1095