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We obtain refined estimates of the triangle inequality in a normed space using integrals and the Tapia semi-product. The particular case of an inner product space is discussed in more detail. MSC: 46B99, 26D15, 46C50, 46C05.

The aim of this presentation is to show several integral inequalities. Among these inequalities we have the inequality varh(f)≤(Γ1−Mh[f])(Mh[f]−γ1), where varh(f) denotes the h-variance of f, which is a bounded function defined on [a,b] with γ1≤f(x)≤Γ1, and γ1, Γ1 are two constants. This inequality is important because it proves a generalized form of the Grüss type inequality. This ...

We give several inequalities on generalized entropies involving Tsallis entropies, using some inequalities obtained by the improvements of Young’s inequality. We also give a generalized Han’s inequality.MSC: 26D15, 94A17.

The objective of this article is to show a refinement of Sándor-Tóth's inequality related to the arithmetic functions which use unitary divisors. A new estimate of the average order of the arithmetic function given by Sándor-Tóth's inequality is suggested.