A note on ‘‘Some inequalities in inner product spaces related to the generalized triangle inequality’’ by S.S. Dragomir et al. Ángel Plaza ⇑ , Kishin Sadarangani Department of Mathematics, University of Las Palmas de Gran Canaria, 35017 Las Palmas de Gran Canaria, Spain article info Keywords: Generalized triangle inequality Inner product spaces abstract In this note, we present an affirmative answer to a question presented in the paper ‘‘Some inequalities in inner product spaces related to the generalized triangle inequality’’ by S.S. Dragomir et al. [Appl. Math. Comput. 217 (18) (2011) 7462–7468]. Ó 2011 Elsevier Inc. All rights reserved. In a recent paper [1], Dragomir et al. proposed the open problem of whether constant 1 2 , appearing in the following two theorems, is the best possible. In this short note, we answer in affirmative sense this question. Theorem 1 [1, Theorem 6]. Let (H; hÁ, Ái) be an inner product space, x i 2 H, for all i 2 {1, ...,n} and p i P 0 with P n i¼1 p i ¼ 1. Suppose there exist constants r i > 0, i 2 {1, ... ,n}, so that x i À X n j¼1 p j x j           6 r i for all i 2 {1, ... ,n}. Then ð0 6Þ X n i¼1 p i kx i kÀ X n i¼1 p i x i           6 1 2 Á P n i¼1 p i r 2 i P n i¼1 p i x i     ð1Þ provided that P n i¼1 p i x i – 0. h Theorem 2 [1, Theorem 7]. Let x i ,p i and r i be as in the statement of previous theorem. Then 0 6 X n i¼1 p i kx i k 2  ! 1 2 À X n i¼1 p i x i           6 1 2 Á P n i¼1 p i r 2 i P n i¼1 p i x i     : à ð2Þ Our example is the following. Let us consider the Euclidean space R 2 ; x 1 ¼ð1; 0Þ and x 2 =(a, b) with kx i k 2 = 1 for i = 1, 2, that is a 2 + b 2 = 1. We choose p 1 ¼ 1 4 and p 2 ¼ 3 4 . Then p 1 kx 1 k + p 2 kx 2 k = 1 and 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.04.019 ⇑ Corresponding author. E-mail address: aplaza@dmat.ulpgc.es (Á. Plaza). Applied Mathematics and Computation 217 (2011) 9497–9498 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc