arXiv subjects

Paweł Pasteczka

Publications and source records attributed to Paweł Pasteczka.

52 records · Page 3Linked to original sources

arXiv2018

Jensen-type geometric shapes

We present both necessary and sufficient conditions to the convex closed shape $X$ such that the inequality $$ \frac{1}{|X|} \int_X f(x)\:dx \le \frac{1}{|\partial X|} \int_{\partial X} f(x)\:dx$$ is valid for every convex function $f \colon X \to \mathbb{R}$ ($\partial X$ stands for the boundary of $X$). It is proved that this inequality holds if $X$ is (i) an $n$-dimensional parallelotope, (ii) an $n$-dimensional ball, (iii) a convex polytope having an inscribed sphere (tangent to all its facets) with center in the center of mass of $\partial X$.

math.CA↗

arXiv2018

On Kedlaya type inequalities for weighted means

In 2016 we proved that for every symmetric, repetition invariant and Jensen concave mean $\mathscr{M}$ the Kedlaya-type inequality $$ \mathscr{A}\big(x_1,\mathscr{M}(x_1,x_2),\ldots,\mathscr{M}(x_1,\ldots,x_n)\big)\le \mathscr{M} \big(x_1, \mathscr{A}(x_1,x_2),\ldots,\mathscr{A}(x_1,\ldots,x_n)\big) $$ holds for an arbitrary $(x_n)$ ($\mathscr{A}$ stands for the arithmetic mean). We are going to prove the weighted counterpart of this inequality. More precisely, if $(x_n)$ is a vector with corresponding (non-normalized) weights $(λ_n)$ and $\mathscr{M}_{i=1}^n(x_i,λ_i)$ denotes the weighted mean then, under analogous conditions on $\mathscr{M}$, the inequality $$ \mathscr{A}_{i=1}^n \big(\mathscr{M}_{j=1}^i (x_j,λ_j),\:λ_i\big) \le \mathscr{M}_{i=1}^n \big(\mathscr{A}_{j=1}^i (x_j,λ_j),\:λ_i\big) $$ holds for every $(x_n)$ and $(λ_n)$ such that the sequence $(\frac{λ_k}{λ_1+\cdots+λ_k})$ is decreasing.

math.CA↗

arXiv2018

On the best Hardy constant for quasi-arithmetic means and homogeneous deviation means

The aim of this paper is to characterize the so-called Hardy means, i.e., those means $M\colon\bigcup_{n=1}^\infty \mathbb{R}_+^n\to\mathbb{R}_+$ that satisfy the inequality $$ \sum_{n=1}^\infty M(x_1,\dots,x_n) \le C\sum_{n=1}^\infty x_n $$ for all positive sequences $(x_n)$ with some finite positive constant $C$. The smallest constant $C$ satisfying this property is called the Hardy constant of the mean $M$. In this paper we determine the Hardy constant in the cases when the mean $M$ is either a concave quasi-arithmetic or a concave and homogeneous deviation mean.

math.CA↗

arXiv2018

On the quasi-arithmetic Gauss-type iteration

For a sequence of continuous, monotone functions $f_1,\dots,f_n \colon I \to \mathbb{R}$ ($I$ is an interval) we define the mapping $M \colon I^n \to I^n$ as a Cartesian product of quasi-arithmetic means generated by $f_j$-s. It is known that, for every initial vector, the iteration sequence of this mapping tends to the diagonal of $I^n$. We will prove that whenever all $f_j$-s are $\mathcal{C}^2$ with nowhere vanishing first derivative, then this convergence is quadratic. Furthermore, the limit $\frac{\text{Var}\, M^{k+1}(v)}{(\text{Var}\, M^{k}(v))^2}$ will be calculated in a nondegenerated case.

math.CA↗

arXiv2017

On Hardy type inequalities for weighted means

The aim of this paper is to establish weighted Hardy type inequality in a broad family of means. In other words, for a fixed vector of weights $(λ_n)_{n=1}^\infty$ and a weighted mean $\mathscr{M}$, we search for the smallest number $C$ such that $$\sum_{n=1}^{\infty} λ_n \mathscr{M} \big((x_1,\dots,x_n),(λ_1,\dots,λ_n)\big) \le C \sum_{n=1}^{\infty} λ_nx_n \text{ for all admissible }x.$$ The main results provide a definite answer in the case when $\mathcal{M}$ is monotone and satisfies the weighted counterpart of the Kedlaya inequality. In particular, if $\mathcal{M}$ is symmetric, Jensen-concave, and the sequence $\big(\tfrac{λ_n}{λ_1+\cdots+λ_n}\big)$ is nonincreasing. In addition, it is proved that if $\mathcal{M}$ is a symmetric and monotone mean, then the biggest possible weighted Hardy constant is achieved if $λ$ is the constant vector.

math.CA↗

arXiv2017

Interval-type theorems concerning quasi-arithmetic means

Family of quasi-arithmetic means has a natural, partial order (point-wise order) $A^{[f]}\le A^{[g]}$ if and only if $A^{[f]}(v)\le A^{[g]}(v)$ for all admissible vectors $v$ ($f,\,g$ and, later, $h$ are continuous and monotone and defined on a common interval). Therefore one can introduce the notion of interval-type sets (sets $\mathcal{I}$ such that whenever $A^{[f]} \le A^{[h]} \le A^{[g]}$ for some $A^{[f]},\,A^{[g]} \in \mathcal{I}$ then $A^{[h]} \in \mathcal{I}$ too). Our aim is to give examples of interval-type sets involving vary smoothness assumptions of generating functions.

math.CA↗

arXiv2017

Interval-type theorems concerning means

Each family $\mathcal{M}$ of means has a natural, partial order (point-wise order), that is $M \le N$ iff $M(x) \le N(x)$ for all admissible $x$. In this setting we can introduce the notion of interval-type set (a subset $\mathcal{I} \subset \mathcal{M}$ such that whenever $M \le P \le N$ for some $M,\,N \in \mathcal{I}$ and $P \in \mathcal{M}$ then $P \in \mathcal{I}$). For example, in the case of power means there exists a natural isomorphism between interval-type sets and intervals contained in real numbers. Nevertheless there appear a number of interesting objects for a families which cannot be linearly ordered. In the present paper we consider this property for Gini means and Hardy means. Moreover some results concerning $L^\infty$ metric among (abstract) means will be obtained.

math.CA↗

arXiv2016

Lower estimation of the difference among quasi-arithmetic means

Quasi-arithmetic means are defined for every continuous, strictly monotone function $f \colon U \rightarrow \mathbb{R}$, ($U$ -- an interval). For an $n$-tuple $a \in U^n$ with corresponding vector of weights $w=(w_1,\dots,w_n)$ ($w_i>0$, $\sum w_i=1$) it equals $f^{-1}\left( \sum_{i=1}^{n} w_i f(a_i)\right)$. In 1960s Cargo and Shisha defined a metric in a family of quasi-arithmetic means defined on a common interval as the maximal possible difference between these means taken over all admissible vectors with corresponding weights. During the years 2013--16 we proved that, having two quasi-arithmetic means, we can majorized distance between them in terms of Arrow-Pratt index $f''/f'$. In this paper we are going to proof that this operator can be also used to establish certain lower boundaries of this distance.

math.CA↗

arXiv2015

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 $M\colon\bigcup_{n=1}^\infty \mathbb{R}_+^n\to\mathbb{R}_+$ that satisfy the inequality $$ \sum_{n=1}^\infty M(x_1,\dots,x_n) \le C\sum_{n=1}^\infty x_n $$ for all positive sequences $(x_n)$ with some finite positive constant $C$. 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 $C$ satisfying the above inequality.

math.CA↗

arXiv2015

Limit properties in a family of quasi-arithmetic means

It is known that the family of power means tends to maximum pointwise if we pass argument to infinity. We will give some necessary and sufficient condition for the family of quasi-arithmetic means generated by a functions satisfying certain smoothness conditions to have analogous property.

math.FA↗

arXiv2014

Iterated Quasi-Arithmetic Mean-Type Mappings

For a family of quasi-arithmetic means satisfying certain smoothness condition we majorize the speed of convergence of the iterative sequence of self-mappings having a mean on each entry, described in the definition of Gaussian product, to relevant mean-type mapping. We apply this result to approximate any continuous function which is invariant with respect to such a self-mappings.

math.CA↗

arXiv2014

Scales of quasi-arithmetic means determined by invariance property

It is well known that if $\mathcal{P}_t$ denotes a set of power means then the mapping $\mathbb{R} \ni t \mapsto \mathcal{P}_t(v) \in (\min v, \max v)$ is both 1-1 and onto for any non-constant sequence $v = (v_1,\dots,\,v_n)$ of positive numbers. Shortly: the family of power means is a scale. If $I$ is an interval and $f \colon I \rightarrow \mathbb{R}$ is a continuous, strictly monotone function then $f^{-1}(\tfrac{1}{n} \sum f(v_i))$ is a natural generalization of power means, so called quasi-arithmetic mean generated by $f$. A famous folk theorem says that the only homogeneous, quasi-a\-rith\-me\-tic means are power means. We prove that, upon replacing the homogeneity requirement by an invariant-type axiom, one gets a family of quasi-arithmetic means building up a scale, too.

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arXiv2013

On negative results concerning Hardy means

In the present paper we are going to prove some necessary condition for a mean to be Hardy. This condition is then applied to completely characterize the Hardy property among the Gini means.

math.CA↗

arXiv2013

A new estimate of the difference among quasi-arithmetic means

In the 1960s Cargo and Shisha proved some majorizations for the distance among quasi-arithmetic means (defined as f^{-1}(\sum_{i=1}^{n} w_i f(a_i) for any continuous, strictly monotone function f:I->R, where I is an interval, and a=(a_1,...,a_n) is a vector with entries in I, w=(w_1,...,w_n) is a sequence of corresponding weights w_i>0, w_1+...+w_n=1). Nearly thirty years later, in 1991, Pàles presented an iff condition for a sequence of quasi-arithmetic means to converge to another QA mean. It was closely related with the three parameters' operator (f(x)-f(z))/(f(x)-f(y)). The author presented recently an estimate for the distance among such quasi-arithmetic means whose underlying functions satisfy some smoothness conditions. Used was the operator f -> f''/f' introduced in the 1940s by Mikusiński and Łojasiewicz. It is natural to look for similar estimate(s) in the case of the underlying functions not being smooth. For instance, by the way of using Pàles' operator. This is done in the present note. Moreover, the result strengthens author's earlier estimates.

math.CA↗

arXiv2013

On some type of Hardy's inequality involving generalized power means

We discuss properties of certain generalization of Power Means proposed in 1971 by Carlson, Meany and Nelson. For any fixed parameter (k,s,q) and vector (v_1,...,v_n) they take the q-th power means of all possible k-tuples (v_{i_1},...,v_{i_k}), and then calculate the s-th power mean of the resulting vector of length C_n^k. We work towards a complete answer to the question when such means satisfy inequalities resembling the classical Hardy inequality. We give a definitive answer in a large part of the parameter space. An unexpected corollary is that this family behaves much differently than most of other families admitting Hardy-type inequalities. Namely, arbitrarily small perturbations of parameters may lead to the breakdown of such inequalities.

math.CA↗

arXiv2011

When is a family of generalized means a scale?

For a family {k_t | t \in I} of real C^2 functions defined on U (I, U -- open intervals) and satisfying some mild regularity conditions, we prove that the mapping I \ni t --> k_t^{-1}(\sum_{i=1}^n w_i k_t(a_i)) is a continuous bijection between I and (min a, max a), for every fixed non-constant sequence a = (a_i)_{i=1}^n with values in U and every set, of the same cardinality, of positive weights w=(w_i)_{i=1}^n. In such a situation one says that the family of functions {k_t} generates a scale on U. The precise assumptions in our result read (all indicated derivatives are with respect to x \in U) (i) k'_t does not vanish anywhere in U for every t \in I, (ii) I \ni t \mapsto \frac{k"_t(x)}{k'_t(x)} is increasing, 1--1 on a dense subset of U and onto the image R for every x \in U. This result makes possible few new things as well as new proofs of classical results.

math.FA↗