Hölder and Minkowski type inequalities with alternating signs

Abstract

Abstract. We obtain new inequalities with alternating signs of Hölder and Minkowski type.Mathematics subject classification (2010): 26D15.

On Hölder and Minkowski Type Inequalities

“…In this section we prove precise Minkowski type inequalities with our weights. As we have already mentioned, these generalize both the case of weight with alternating signs and the case of non-negative weights (see [1]).…”

“…Remark 1. From Theorem 1, it is seen that the constant in the right hand side of (4) tends to infinity as a → 0 or b → 0 (note that this constant is better than in [1]). Now we give an example of sequences confirming this [1].…”

“…But we prove even more, namely, that there exist no positive constants bounding F H from below. Following [1], let p k = (−1) k+1 , k = 1, . .…”

“…Recently Chunaev [1] obtained Hölder and Minkowski type inequalities with alternating signs. His results are a supplement to Jensen type inequalities with alternating signs obtained earlier by Szegő [2], Bellman [3,4], Brunk [5], and others (see [6][7][8][9][10][11], [12, §5.38] and also Remark 2).…”

“…In this paper, we intend to give inequalities of Hölder and Minkowski type with more general weights, including both the case of weights with alternating signs and the classical case of non-negative weights (see, for instance, [12, §4.2] and [1,13]). Namely, weights p k , k = 1, .…”