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• 1.
Department of Mathematics, University of Haifa.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Some new estimates of the ‘Jensen gap’2016In: Journal of inequalities and applications (Print), ISSN 1025-5834, E-ISSN 1029-242X, Vol. 2016, article id 39Article in journal (Refereed)

Let (μ,Ω) be a probability measure space. We consider the so-called ‘Jensen gap’ J(φ,μ,f)=∫ Ω φ(f(s))dμ(s)−φ(∫ Ω f(s)dμ(s)) for some classes of functions φ. Several new estimates and equalities are derived and compared with other results of this type. Especially the case when φ has a Taylor expansion is treated and the corresponding discrete results are pointed out.

• 2.
Department of Mechanics and Mathematics, L.N. Gumilyov Eurasian National University, Astana .
Department of Mechanics and Mathematics, L.N. Gumilyov Eurasian National University, Astana. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Boundedness and compactness of a class of Hardy type operators2016In: Journal of inequalities and applications (Print), ISSN 1025-5834, E-ISSN 1029-242X, no 1, article id 324Article in journal (Refereed)

We establish characterizations of both boundedness and of compactness of a general class of fractional integral operators involving the Riemann-Liouville, Hadamard, and Erdelyi-Kober operators. In particular, these results imply new results in the theory of Hardy type inequalities. As applications both new and well-known results are pointed out.

Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
On a new class of Hardy-type inequalities2012In: Journal of inequalities and applications (Print), ISSN 1025-5834, E-ISSN 1029-242X, Vol. 2012, no 259Article in journal (Refereed)

In this paper, we generalize a Hardy-type inequality to the class of arbitrary non-negative functions bounded from below and above with a convex function multiplied with positive real constants. This enables us to obtain new generalizations of the classical integral Hardy's, Hardy-Hilbert's, Hardy-Littlewood-P\'{o}lya's and P\'{o}lya-Knopp's inequalities as well as of Godunova's and of some recently obtained inequalities in multidimensional settings. Finally, we apply a similar idea to functions bounded from below and above with a superquadratic function.

• 4.
Department of Mathematics, Faculty of Exact and Natural Sciences, Ivane Javakhishvili Tbilisi State University.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Sharp Hp- Lptype inequalities of weighted maximal operators of Vilenkin-Nörlund means and its applications2016In: Journal of inequalities and applications (Print), ISSN 1025-5834, E-ISSN 1029-242X, no 1, article id 242Article in journal (Refereed)

We prove and discuss some new Hp-Lptype inequalities of weighted maximal operators of Vilenkin-Nörlund means with monotone coefficients. It is also proved that these inequalities are the best possible in a special sense. We also apply these results to prove strong summability for such Vilenkin-Nörlund means. As applications, both some well-known and new results are pointed out

• 5.
Luleå tekniska universitet.
Department of Mathematics and Statistics, McMaster University, Hamilton. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Duality theorem over the cone of monotone functions and sequences in higher dimensions2002In: Journal of inequalities and applications (Print), ISSN 1025-5834, E-ISSN 1029-242X, Vol. 7, no 1, p. 79-108Article in journal (Refereed)

Let f be a non-negative function defined on ℝ+n which is monotone in each variable separately. If 1 < p < ∞, g ≥ 0 and v a product weight function, then equivalent expressions for sup ∫ℝ(+)(n) fg/(ℝ+nfpv)1/p are given, where the supremum is taken over all such functions f. Variants of such duality results involving sequences are also given. Applications involving weight characterizations for which operators defined on such functions (sequences) are bounded in weighted Lebesgue (sequence) spaces are also pointed out.

• 6.
Luleå tekniska universitet.
Faculty of Textile Technology, University of Zagreb. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Carlson type inequalities1998In: Journal of inequalities and applications (Print), ISSN 1025-5834, E-ISSN 1029-242X, Vol. 2, no 2, p. 121-135Article in journal (Refereed)

A scale of Carlson type inequalities are proved and the best constants are found. Some multidimensional versions of these inequalities are also proved and it is pointed out that also a well-known inequality by Beurling-Kjellberg is included as an endpoint case.

• 7.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Lucian Blaga University of Sibiu.
Some multiplicative inequalities for inner products and of the Carlson type2008In: Journal of inequalities and applications (Print), ISSN 1025-5834, E-ISSN 1029-242XArticle in journal (Refereed)

We prove a multiplicative inequality for inner products, which enables us to deduce improvements of inequalities of the Carlson type for complex functions and sequences, and also other known inequalities.

• 8.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Some new sharp limit Hardy-type inequalities via convexity2014In: Journal of inequalities and applications (Print), ISSN 1025-5834, E-ISSN 1029-242X, Vol. 2014, article id 6Article in journal (Refereed)

Some new limit cases of Hardy-type inequalities are proved, discussed and compared. In particular, some new refinements of Bennett's inequalities are proved. Each of these refined inequalities contain two constants, and both constants are in fact sharp. The technique in the proofs is new and based on some convexity arguments of independent interest.

• 9.
Department of Differential Equations, Faculty of Mechanics and Mathematics, Moscow Lomonosov State University.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
On the precise asymptotics of the constant in Friedrich's inequality for functions vanishing on the part of the boundary with microinhomogeneous structure2007In: Journal of inequalities and applications (Print), ISSN 1025-5834, E-ISSN 1029-242XArticle in journal (Refereed)

We construct the asymptotics of the sharp constant in the Friedrich-type inequality for functions, which vanish on the small part of the boundary Γ1ɛ. It is assumed that Γ1ɛ consists of (1/δ)n-1 pieces with diameter of order O(ɛδ). In addition, δ=δ(ɛ) and δ→0 as ɛ→0.

• 10.
Institute of Mathematics, Academy of Sciences of Czech Republic.
Department of Mathematics, Faculty of Science and Arts, Kirikkale University. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Some new iterated Hardy-type inequalities: The case θ=12013In: Journal of inequalities and applications (Print), ISSN 1025-5834, E-ISSN 1029-242X, article id 515Article in journal (Refereed)

In this paper we characterize the validity of the Hardy-type inequality parallel to parallel to integral(infinity)(s)h(z)dz parallel to(p,u,(0,t))parallel to(q,w,(0,infinity)) <= c parallel to h parallel to(1,v,(0,infinity)), where 0 < p < infinity, 0 < q <= +infinity, u, w and v are weight functions on (0, infinity). It is pointed out that this characterization can be used to obtain new characterizations for the boundedness between weighted Lebesgue spaces for Hardy-type operators restricted to the cone of monotone functions and for the generalized Stieltjes operator.

• 11.
Department of Mathematics and Statistics, McMaster University, Hamilton.
Mathematical Institute, Czech Academy of Sciences. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
On some fractional order Hardy inequalities1997In: Journal of inequalities and applications (Print), ISSN 1025-5834, E-ISSN 1029-242X, Vol. 1, no 1, p. 25-46Article in journal (Refereed)

Weighted inequalities for fractional derivatives (= fractional order Hardy-type inequalities) have recently been proved in [4] and [1]. In this paper, new inequalities of this type are proved and applied. In particular, the general mixed norm case and a general twodimensional weight are considered. Moreover, an Orlicz norm version and a multidimensional fractional order Hardy inequality are proved. The connections to related results are pointed out.

• 12.
Eurasian National University, Astana, Institute of Mathematics, Kazakhstan Ministry of Education and Sciences, KIMEP University, Abai Ave. 4, Almaty.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Faculty of Mathematics and Information Technologies, Eurasian National University, Munaitpasov st., 5, Astana.
A New discrete Hardy-type inequality with kernels and monotone functions2015In: Journal of inequalities and applications (Print), ISSN 1025-5834, E-ISSN 1029-242X, article id 321Article in journal (Refereed)

new discrete Hardy-type inequality with kernels and monotone functions is proved for the case $$1< q< p<\infty$$. This result is discussed in a general framework and some applications related to Hölder’s summation method are pointed out.

• 13.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
On Friedrichs-type inequalities in domains rarely perforated along the boundary2011In: Journal of inequalities and applications (Print), ISSN 1025-5834, E-ISSN 1029-242X, Vol. 2011Article in journal (Refereed)

This paper is devoted to the Friedrichs inequality, where the domain isperiodically perforated along the boundary. It is assumed that the functionssatisfy homogeneous Neumann boundary conditions on the outer boundary andthat they vanish on the perforation. In particular, it is proved that thebest constant in the inequality converges to the best constant in aFriedrichs-type inequality as the size of the perforation goes to zero muchfaster than the period of perforation. The limit Friedrichs-type inequalityis valid for functions in the Sobolev space $H^{1}$.

• 14.
Department of Mathematics, University of West Bohemia.
Department of Mathematics, University of West Bohemia. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Some higher order Hardy inequalities2012In: Journal of inequalities and applications (Print), ISSN 1025-5834, E-ISSN 1029-242XArticle in journal (Refereed)

We investigate the k-th order Hardy inequality (1-1) for functions satisfying rather general boundary conditions (1-2), show which of these conditions are admissible and derive sufficient, and necessary and sufficient, conditions (for 0 1) on u, v for (1-1) to hold.

• 15.
Narvik University College and Norut Narvik, UiT, The Arctic University of Norway, Narvik.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Universidade do Algarve, FCT, Campus de Gambelas, Universidade do Algarve, Faro.
Some sharp inequalities for integral operators with homogeneous kernel2016In: Journal of inequalities and applications (Print), ISSN 1025-5834, E-ISSN 1029-242X, Vol. 2016, article id 114Article in journal (Refereed)

One goal of this paper is to show that a big number of inequalities for functions in L p (R + ) Lp(R+), p≥1 p≥1, proved from time to time in journal publications are particular cases of some known general results for integral operators with homogeneous kernels including, in particular, the statements on sharp constants. Some new results are also included, e.g. the similar general equivalence result is proved and applied for 0<p

• 16.
Department of Mathematics and Informatics, Sofia University.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Department of Mathematics, University of Zagreb.
The Beckenbach-Dresher inequality in the psi-direct sums of spaces and related results2012In: Journal of inequalities and applications (Print), ISSN 1025-5834, E-ISSN 1029-242X, Vol. 2012, no 1Article in journal (Refereed)

Let ~ A : [0; 1] ! R be a concave function with ~ A(0) = ~ A(1) = 1. There is a corresponding map k:k ~ A for which the inverse Minkowski inequality holds. Several properties of that map are obtained. Also, we consider the Beckenbach{Dresher type inequality connected with A-direct sums of Banach spaces and of ordered spaces. In the last section we investigate the properties of functions A!;q and k:k!;q , (0 < ! < 1; q < 1) related to the Lorentz sequence space. Other posibilities for parameters ! and q are considered, the inverse HAolder inequalities and more variants of the Beckenbach{Dresher inequalities are obtained.

• 17.
Department of Mathematics, Federal University of Agriculture, Abeokuta.
Department of Mathematics, Obafemi Awolowo University, Ile-Ife, Osun. Department of Mathematics, Obafemi Awolowo University, Ile-Ife, Osun. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Time scale Hardy-type inequalities with ‘broken’ exponent p2015In: Journal of inequalities and applications (Print), ISSN 1025-5834, E-ISSN 1029-242X, Vol. 2015, article id 2015:17Article in journal (Refereed)

In this paper, some new Hardy-type inequalities involving ‘broken’ exponents are derived on arbitrary time scales. Our approach uses both convexity and superquadracity arguments, and the results obtained generalize, complement and provide refinements of some known results in literature.

• 18. Okpoti, Christopher
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Weight characterizations for the discrete Hardy inequality with kernel2006In: Journal of inequalities and applications (Print), ISSN 1025-5834, E-ISSN 1029-242XArticle in journal (Refereed)

A discrete Hardy-type inequality (∑n=1∞(∑k=1ndn,kak)qun)1/q≤C(∑n=1∞anpvn)1/p is considered for a positive "kernel" d={dn,k}, n,k∈ℤ+, and p≤q. For kernels of product type some scales of weight characterizations of the inequality are proved with the corresponding estimates of the best constant C. A sufficient condition for the inequality to hold in the general case is proved and this condition is necessary in special cases. Moreover, some corresponding results for the case when {an}n=1∞ are replaced by the nonincreasing sequences {an*}n=1∞ are proved and discussed in the light of some other recent results of this type.

• 19.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Departamento de Matemtica, Centro CEAF, Instituto Superior Tecnico.
What should have happened if Hardy had discovered this?2012In: Journal of inequalities and applications (Print), ISSN 1025-5834, E-ISSN 1029-242X, Vol. 2012, no 2Article in journal (Refereed)

First we present and discuss an important proof of Hardy's inequality via Jensen's inequality which Hardy and his collaborators did not discover during the 10 years of research until Hardy finally proved his famous inequality in 1925. If Hardy had discovered this proof, it obviously would have changed this prehistory, and in this article the authors argue that this discovery would probably also have changed the dramatic development of Hardy type inequalities in an essential way. In particular, in this article some results concerning powerweight cases in the finite interval case are proved and discussed in this historical perspective. Moreover, a new Hardy type inequality for piecewise constant p = p(x) is proved with this technique, limiting cases are pointed out and put into this frame.

• 20.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
L.N. Gumilyov Eurasian National University.
Some new Hardy-type inequalities for Riemann-Liouville fractional q-integral operator2015In: Journal of inequalities and applications (Print), ISSN 1025-5834, E-ISSN 1029-242X, article id 296Article in journal (Refereed)
• 21.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Computing Centre, Russian Academy of Sciences, Far-Eastern Branch, Khabarovsk.
Weighted integral inequalities with the geometric mean operator2002In: Journal of inequalities and applications (Print), ISSN 1025-5834, E-ISSN 1029-242X, Vol. 7, no 5, p. 727-746Article in journal (Refereed)

The geometric mean operator is defined by Gf(x) = exp(1/x∫0x logf(t)dt). A precise two-sided estimate of the norm ||G|| = supf≠0 ||Gf||Luq/||f||Lvq for 0 < p, q ≤ ∞ is given and some applications and extensions are pointed out

• 22. Wedestig, Anna
Weighted inequalities for the Sawyer two-dimensional Hardy operator and its limiting geometric mean operator2005In: Journal of inequalities and applications (Print), ISSN 1025-5834, E-ISSN 1029-242X, no 4, p. 387-394Article in journal (Refereed)

We consider Tf= 0 x1 0 x2 f ( t1, t2) d t1 d t2 and a corresponding geometric mean operator Gf=exp ( 1/ x1 x2) 0 x1 0 x2 logf ( t1, t2) d t1 d t2 . E. T. Sawyer showed that theHardy-type inequality Tf Luq C f Lvp could be characterized by three independentconditions on the weights. We give a simple proof of the fact thatif the weight v is of product type, then in fact only onecondition is needed. Moreover, by using this information and byperforming a limiting procedure we can derive a weightcharacterization of the corresponding two-dimensional Pólya-Knopp inequality with the geometric mean operator G involved.

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