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Teresa Rajba

Publications and source records attributed to Teresa Rajba.

11 recordsLinked to original sources
arXiv2025

On Popoviciu's concept of convexity for functions of $d$ variables

We establish an integral representation for Popoviciu's convex functions of $d$ variables. This representation serves as a~foundation for deriving several functional inequalities, analogous to those well-known for usual convex functions. Our results generalize and extend the results obtained by S.~Gal, C.~Niculescu, B.~Gavrea, T.~Popoviciu, and others, who considered only differentiable functions of two variables. In contrast to other authors, we do not impose any additional regularity assumptions on the studied functions.

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arXiv2018

Convex order for convolution polynomials of Borel measures

We give necessary and sufficient conditions for Borel measures to satisfy the inequality introduced by Komisarski, Rajba (2018). This inequality is a generalization of the convex order inequality for binomial distributions, which was proved by Mrowiec, Rajba, Wąsowicz (2017), as a probabilistic version of the inequality for convex functions, that was conjectured as an old open problem by I.~Raşa. We present also further generalizations using convex order inequalities between convolution polynomials of finite Borel measures. We generalize recent results obtained by B.~Gavrea (2018) in the discrete case to general case. We give solutions to his open problems and also formulate new problems.

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arXiv2018

A sharpening of a problem on Bernstein polynomials and convex function and related results

We present a short proof of a conjecture proposed by I. Raşa (2017), which is an inequality involving basic Bernstein polynomials and convex functions. This proof was given in the letter to I. Raşa (2017). The methods of our proof allow us to obtain some extended versions of this inequality as well as other inequalities given by I. Raşa. As a tool we use stochastic convex ordering relations. We propose also some generalizations of the binomial convex concentration inequality. We use it to insert some additional expressions between left and right sides of the Raşa inequalities.

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arXiv2017

Muirhead inequality for convex orders and a problem of I. Raşa on Bernstein polynomials

We present a new, very short proof of a conjecture by I. Raşa, which is an inequality involving basic Bernstein polynomials and convex functions. It was affirmed positively very recently by J. Mrowiec, T. Rajba and S. Wąsowicz (2017) by the use of stochastic convex orders, as well as by Abel (2017) who simplified their proof. We give a useful sufficient condition for the verification of some stochastic convex order relations, which in the case of binomial distributions are equivalent to the I. Raşa inequality. We give also the corresponding inequalities for other distributions. Our methods allow us to give some extended versions of stochastic convex orderings as well as the I. Raşa type inequalities. In particular, we prove the Muirhead type inequality for convex orders for convolution polynomials of probability distributions.

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arXiv2016

On the classes of higher-order Jensen-convex functions and Wright-convex functions, II

Recently Nikodem, Rajba and Wąsowicz compared the classes of n-Wright-convex functions and n-Jensen-convex functions by showing that the first one is a proper subclass of the latter one, whenever n is an odd natural number. Till now the case of even n was an open problem. In this paper the complete solution is given: it is shown that the inclusion is proper for any natural n. The classes of strongly n-Wright-convex and strongly n-Jensen-convex functions are also compared (with the same assertion).

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arXiv2016

A solution to the problem of Rasa connected with Bernstein polynomials

During the Conference on Ulam's Type Stability (Rytro, Poland, 2014), Ioan Rasa recalled his 25-years-old problem concerning some inequality involving the Bernstein polynomials. We offer the complete solution (in positive). As a~tool we use stochastic orderings (which we prove for binomial distributions) as well as so-called concentration inequality. Our methods allow us to pose (and solve) the extended version of the problem in question.

math.AP↗
arXiv2014

Extremal measures with prescribed moments

In the approximate integration some inequalities between the quadratures and the integrals approximated by them are called \emph{extremalities}. On the other hand, the set of all quadratures is convex. We are trying to find possible connections between extremalities and extremal quadratures (in the sense of extreme points of a~convex set). Of course, the quadratures are the integrals \wrt~discrete measures and, moreover, a~quadrature is extremal if and only if the associated measure is extremal. Hence the natural problem arises to give some description of extremal measures with prescribed moments in the general (not only discrete) case. In this paper we deal with symmetric measures with prescribed first four moments. The full description (with no symmetry assumptions, and/or not only four moments are prescribed and so on) is far to be done.

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arXiv2010

New integral representations of n-th order convex functions

In this paper we give an integral representation of an $n$-convex function $f$ in general case without additional assumptions on function $f$. We prove that any $n$-convex function can be represented as a sum of two $(n+1)$-times monotone functions and a polynomial of degree at most $n$. We obtain a decomposition of $n$-Wright-convex functions which generalizes and complements results of Maksa and Pales (2009). We define and study relative $n$-convexity of $n$-convex functions. We introduce a measure of $n$-convexity of $f$. We give a characterization of relative $n$-convexity in terms of this measure, as well as in terms of $n$th order distributional derivatives and Radon-Nikodym derivatives. We define, study and give a characterization of strong $n$-convexity of an $n$-convex function $f$ in terms of its derivative $f^{(n+1)}(x)$ (which exists a.e.) without additional assumptions on differentiability of $f$. We prove that for any two $n$-convex functions $f$ and $g$, such that $f$ is $n$-convex with respect to $g$, the function $g$ is the support for the function $f$ in the sense introduced by Wasowicz (2007), up to polynomial of degree at most $n$.

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