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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...

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.