252 research outputs found
Characterization of the Hardy property of means and the best Hardy constants
The aim of this paper is to characterize in broad classes of means the
so-called Hardy means, i.e., those means that satisfy the inequality for all
positive sequences with some finite positive constant . One of the
main results offers a characterization of Hardy means in the class of
symmetric, increasing, Jensen concave and repetition invariant means and also a
formula for the best constant satisfying the above inequality
The Nagaev-Guivarc'h method via the Keller-Liverani theorem
The Nagaev-Guivarc'h method, via the perturbation operator theorem of Keller
and Liverani, has been exploited in recent papers to establish local limit and
Berry-Essen type theorems for unbounded functionals of strongly ergodic Markov
chains. The main difficulty of this approach is to prove Taylor expansions for
the dominating eigenvalue of the Fourier kernels. This paper outlines this
method and extends it by proving a multi-dimensional local limit theorem, a
first-order Edgeworth expansion, and a multi-dimensional Berry-Esseen type
theorem in the sense of Prohorov metric. When applied to uniformly or
geometrically ergodic chains and to iterative Lipschitz models, the above cited
limit theorems hold under moment conditions similar, or close, to those of the
i.i.d. case
Continued fraction digit averages an Maclaurin's inequalities
A classical result of Khinchin says that for almost all real numbers
, the geometric mean of the first digits in the
continued fraction expansion of converges to a number (Khinchin's constant) as . On the other hand,
for almost all , the arithmetic mean of the first continued
fraction digits approaches infinity as . There is a
sequence of refinements of the AM-GM inequality, Maclaurin's inequalities,
relating the -th powers of the -th elementary symmetric means of
numbers for . On the left end (when ) we have the
geometric mean, and on the right end () we have the arithmetic mean.
We analyze what happens to the means of continued fraction digits of a
typical real number in the limit as one moves steps away from either
extreme. We prove sufficient conditions on to ensure to ensure
divergence when one moves steps away from the arithmetic mean and
convergence when one moves steps away from the geometric mean. For
typical we conjecture the behavior for , .
We also study the limiting behavior of such means for quadratic irrational
, providing rigorous results, as well as numerically supported
conjectures.Comment: 32 pages, 7 figures. Substantial additions were made to previous
version, including Theorem 1.3, Section 6, and Appendix
ON THE HOMOGENIZATION OF MEANS
The aim of this paper is to introduce several notions of homogenization in various classes of weighted means, which include quasiarithmetic and semideviation means. In general, the homogenization is an operator which attaches a homogeneous mean to a given one. Our results show that, under some regularity or convexity assumptions, the homogenization of quasiarithmetic means are power means, and homogenization of semideviation means are homogeneous semideviation means. In other results, we characterize the comparison inequality, the Jensen concavity, and Minkowski- and Holder-type inequalities related to semideviation means
A note on the proofs of generalized Radon inequality
In this paper, we introduce and prove several generalizations of the Radon inequality. The proofs in the current paper unify and also are simpler than those in early published work. Meanwhile, we find and show the mathematical equivalences among the Bernoulli inequality, the weighted AM-GM inequality, the Hölder inequality, the weighted power mean inequality and the Minkowski inequality. Finally, some applications involving the results proposed in this work are shown
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