Kragujevac Journal of Mathematics Volume 41(1) (2017), Pages 81–92. SOME REVERSES OF THE CAUCHY-SCHWARZ AND TRIANGLE INEQUALITIES IN 2-INNER PRODUCT SPACES MOHSEN ERFANIAN OMIDVAR 1 , HAMID REZA MORADI 2 , SILVESTRU SEVER DRAGOMIR 3 , AND YOEL JE CHO 4,5 Abstract. In this paper, we give some reverses of the Cauchy-Schwarz inequality and triangle inequality in 2-inner product spaces. Applications for determinantal integral inequalities are also provided. 1. Introduction The Cauchy-Schwarz inequality plays an important role in the theory of inner product spaces (see, for instance, [22, 23]), which is one of the classical inequalities. It is well known that, in a semi-inner product space (X , 〈·, ·〉), the Cauchy-Schwarz inequality has the form |〈x, y〉| 2 ≤〈x, x〉〈y,y〉 , for all x, y ∈ X . In recent years, many authors have studied some related topics such as the reverse of the Cauchy-Schwarz inequality, the triangle and Bessel inequality as well as Grüss inequality (see [7, 10, 11, 18]). The probably first reverse of the Cauchy-Schwarz inequality for positive real numbers was obtained by Pólya and Szegö in 1925 (see [20, p. 57 and 213–214] and [21, p. 71–72 and 253–255]). Since then, there exist a lot of generalizations of the reverse of the Cauchy-Schwarz inequality. For example, in 2007, Dragomir [6, Chapter 2] contributed much to the reverses of the Cauchy- Schwarz inequality and also similar results for integrals, isotonic functionals as well as generalizations in the setting of inner product spaces are well-studied and understood Key words and phrases. Cauchy-Schwarz inequality, triangle inequality, 2-inner product, 2-norm. 2010 Mathematics Subject Classification. Primary: 26D15. Secondary: 46C05, 46C50, 26D10. Received: June 3, 2016. Accepted: August 6, 2016. 81