A proof of the three geometric inequalities conjectured by Yu-Dong Wu and H. M. Srivastava

Abstract

Abstract. In this short note the authors give answers to the three open problems formulatedby Wu and Srivastava [ Appl. Math. Lett. 25 (2012), 1347–1353 ]. We disprove the Problem1, by showing that there exists a triangle which does not satisfies the proposed inequality. Weprove the inequalities conjectured in Problems 2 and 3. Furthermore, we introduce an optimalrefinement of the inequality conjectured on Problem 3. 1. Introductionand conjectures of Yu-DongWuand H.M.SrivastavaThe geometric inequalities are relevant in several areas of the science and engineering [1, 2, 3, 4,5, 6, 7]. The methodologies to prove the geometric inequalities is disperse see for instance [8, 9, 10,11]. In a broad sense, there exist some of methodologies which are based on analytical methods,other in integral and differential calculus, and other on geometric methods. The methodology ofthis paper, in spite of the basic arguments, can be considered belongs to the analytical methodssince the results are strongly dependent on the analytical methodology introduced in [5, 12].The focus of this short note is the open problems given in [5, 12]. Indeed, we introduce somenotation and then we recallthe conjectures. Let us considera triangle △ABC with anglesA,B andC, we denote by a,b,c,s and r, the lengths of the corresponding opposite sides, the semiperimeterand the inradius, respectively. Then, using the symbolPto denote a cyclic sum, i. e.Xf(b,c) = f(a,b)+f(b,c)+f(c,a),we have that, the following geometric inequality2√2 s ≤Xpa