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We show that for real numbers p, q with q < p, and the related power means Pp, Pq, the inequality holds for every vector x of positive reals. Moreover we prove that, for all such pairs (p,q), the constant p-q 8 is sharp. [ABSTRACT FROM AUTHOR]

We introduce a variant of the Rényi entropy definition that aligns it with the well-known Hölder mean: in the new formulation, the r-th order Rényi Entropy is the logarithm of the inverse of the r-th order Hölder mean. This brings about new insights into the relationship of the Rényi entropy to quantities close to it, like the information potential and the partition function of statistical mechanics. We also provide expressions that allow us to calculate the Rényi entropies from the Shannon cross-entropy and the escort probabilities. Finally, we discuss why shifting the Rényi entropy is fruitful in some applications.

In this paper we investigate continuity properties of functions $${f : \mathbb {R}_+ \to \mathbb {R}_+}$$ that satisfy the ( p, q)-Jensen convexity inequality where H stands for the pth power (or Hölder) mean. One of the main results shows that there exist discontinuous multiplicative functions that are ( p, p)-Jensen convex for all positive rational numbers p. A counterpart of this result states that if f is ( p, p)-Jensen convex for all $${p \in P \subseteq \mathbb {R}_+}$$ , where P is a set of positive Lebesgue measure, then f must be continuous. [ABSTRACT FROM AUTHOR]

In this paper we establish a Tauberian condition under which convergence follows from Holder summability of sequences of fuzzy numbers. [ABSTRACT FROM AUTHOR]

For a class of partially ordered means, we introduce a notion of the (nontrivial) cancelling mean. A simple method is given, which helps to determine cancelling means for the well-known classes of the Hölder and Stolarsky means. MSC: 39B22, 26D20. [ABSTRACT FROM AUTHOR]

In this paper, the concavity with respect to Hölder means and monotonicity of Jacobian elliptic functions are investigated, and some functional inequalities and elementary estimates for Jacobian elliptic functions are obtained. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, the authors prove some inequalities for the generalized elliptic integrals with respect to Hölder means. [Copyright &y& Elsevier]

Abstract: The generalized Grötzsch function has numerous applications in geometric function theory and analytic number theory and its properties have been investigated by many authors. In this paper we study the concavity of the generalized Grötzsch function with respect to Hölder means. [Copyright &y& Elsevier]

Abstract: In this note we investigate the convexity of zero-balanced hypergeometric functions with respect to Hölder mean. [Copyright &y& Elsevier]

We introduce a variant of the Rényi entropy definition that aligns it with the well-known Hölder mean: in the new formulation, the r-th order Rényi Entropy is the logarithm of the inverse of the r-th order Hölder mean. This brings about new insights into the relationship of the Rényi entropy to quantities close to it, like the information potential and the partition function of statistical mechanics. We also provide expressions that allow us to calculate the Rényi entropies from the Shannon cross-entropy and the escort probabilities. Finally, we discuss why shifting the Rényi entropy is fruitful in some applications. [ABSTRACT FROM AUTHOR]