About Bergström's inequality

Abstract

Abstract. In this paper, we generalize identity (3), from where we obtain a rafinement of inequalities (1) and (2). IntroductionThe inequality from Theorem 1 is called in the literature Bergström's inequality (see [2]with equality if and only ifThe generalization of Bergström's inequality is contained in the following theorem (see [5]). THEOREM 2. If x k ∈ R and a k > 0 , k ∈ {1, 2,...,n} , thenBy particularizations in Theorem 2, in paper [5] the refinements are obtained of Cauchy-Schwarz's inequality.For the complex numbers, it is well-known the identity:Mathematics subject classification (2000): 26D15.

About Aczél Inequality and Some Bounds for Several Statistical Indicators

“…for any n ≥ 2, with equality if and only if the sequences a and b are proportional. In [5], Pop showed an improvement of inequality (2). Two consequences can be obtained from above inequality, namely: for arbitrary sequence a = (a 1 , a 2 , .…”

A refinement of the Bergström inequality

“…Bergström inequality has stimulated several mathematicians' interest, and various extensions, refinements, and proofs of the inequality have been provided. We refer to [7,9,14,17,28,29] and the references given therein.…”

Some inequalities obtained for positive power series

“…If x i ∈ R + then a particularization of a theorem given in [10] can be formulated as below and will be used in next section. Theorem 1.…”

“…Theorem 1. ( [10]) If n ∈ N, n ≥ 2, x 1 , x 2 , ..., x n ∈ R + , and a 1 , a 2 , ..., a n ∈ R \ {0} with a 1 + a 2 + ... + a n = 0 then, (a i x j − a j x i ) 2 a i a j .…”