The restricted maximal operators of partial sums with respect to bounded Vilenkin systems are investigated. We derive the maximal subspace of positive numbers, for which this operator is bounded from the Hardy space H p {H-{p}} to the Lebesgue space L p {L-{p}} for all 0 < p ≤ 1 {0<p\leq 1}. We also prove that the result is sharp in a particular sense. 

Let Lambda(q)(omega), q > 0, denote the Lorentz space equipped with the (quasi) norm parallel to f parallel to(Lambda q(omega)) := (integral(1)(0) (f*(t)omega(t))(q)dt/t)(1/q) for a function integral on [0,1] and with omega positive and equipped with some additional growth properties. A generalization of Boas theorem in the form of a two-sided inequality is obtained in the case of both general regular system Phi = {phi(k)}(k=1)(infinity) and generalized Lorentz Lambda(q) (omega) spaces.

This second edition provides a thorough introduction to contemporary convex function theory with many new results. A large variety of subjects are covered, from the one real variable case to some of the most advanced topics.  The new edition includes considerably more material emphasizing the rich applicability of convex analysis to concrete examples.  Chapters 4, 5, and 6 are entirely new, covering important topics such as the Hardy-Littlewood-Pólya-Schur theory of majorization, matrix convexity, and the Legendre-Fenchel-Moreau duality theory.

This book can serve as a reference and source of inspiration to researchers in several branches of mathematics and engineering, and it can also be used as a reference text for graduate courses on convex functions and applications.

Convex functions and their relatives are ubiquitous in a large variety of applications such as optimization theory, mass transportation, mathematical economics, and geometric inequalities related to isoperimetric problems. This chapter is devoted to a succinct presentation of their theory in the context of real normed linear spaces, but most of the illustrations will refer to the Euclidean space RN,">RN,RN, the matrix space MN(R)">MN(R)MN(R) of all N&#x00D7;N">N×NN×N-dimensional real matrices (endowed with the Hilbert–Schmidt norm or with the operator norm), and the Lebesgue spaces Lp(RN)">Lp(RN)Lp(RN) with p&#x2208;[1,&#x221E;]">p∈[1,∞]p∈[1,∞].

The study of convex functions of one real variable offers an excellent glimpse of the beauty and fascination of advanced mathematics. The reader will find here a large variety of results based on simple and intuitive arguments that have remarkable applications. At the same time they provide the starting point of deep generalizations in the setting of several variables, that will be discussed in the next chapters.

The natural domain for a convex function is a convex set. In this chapter we review some basic facts, necessary for a deep understanding of the concept of convexity in real linear spaces. For reader’s convenience, all results concerning the separation of convex sets in Banach spaces are stated in Section 2.2 with proofs covering only the particular (but important) case of Euclidean spaces. Full details in the general case are to be found in Appendix  B

This chapter is aimed to offer a glimpse on the majorization theory and the beautiful inequalities associated to it. Introduced by G. H. Hardy, J. E. Littlewood, and G. Pólya (Messenger Math. 58:145–152, (1929), [208]) in 1929, and popularized by their celebrated book on Inequalities (Hardy et al., Inequalities, Cambridge University Press, 1952, [209]), the relation of majorization has attracted along the time a big deal of attention not only from the mathematicians, but also from people working in various other fields such as statistics, economics, physics, signal processing, data mining, etc. Part of this research activity is summarized in the 900 pages of the recent book by A. W. Marshall, I. Olkin, and B. Arnold (Inequalities: theory of majorization and its applications. Springer, New York (2011), [305]).

In this chapter we investigate three subjects concerning the convexity of functions defined on a space of matrices (or just on a convex subset of it). The first one is devoted to the convex spectral functions, that is, to the convex functions F:Sym(n,R)&#x2192;R">F:Sym(n,R)→RF:Sym(n,R)→R whose values F(A) depend only on the spectrum of A. The main result concerns their description as superpositions f&#x2218;&#x039B;">f∘Λf∘Λ between convex functions f:Rn&#x2192;R">f:Rn→Rf:Rn→R invariant under permutations, and the eigenvalues map &#x039B;">ΛΛ.

Convex optimization is one of the main applications of the theory of convexity and Legendre–Fenchel duality is a basic tool, making more flexible the approach of many concrete problems. The diet problem, the transportation problem, and the optimal assignment problem are among the many problems that during the Second World War and immediately after led L. Kantorovich, T. C. Koopmans, F. L. Hitchcock, and G. B. Danzig to develop the mathematical theory of linear programming. Soon it was realized that most results extend to the framework of convex functions, which marked the birth of convex programming. Later on, W. Fenchel, R. T. Rockafellar, and J. J. Moreau laid the foundations of convex analysis.

In this paper extensions and refinements of Hermite-Hadamard and Fejer type inequalities are derived including monotonicity of some functions related to the Fejer inequality and extensions for functions, which are 1-quasiconvex and for function with bounded second derivative. We deal also with Fejer inequalities in cases that p, the weight function in Fejer inequality, is not symmetric but monotone on [a, b] .

We establish criteria for both boundedness and compactness for some classes of integraloperators with logarithmic singularities in weighted Lebesgue spaces for cases 1 < p 6 q <¥ and 1 < q < p < ¥. As corollaries some corresponding new Hardy inequalities are pointedout.1

The first power weighted version of Hardy’s inequality can be rewritten as∫∞0(xα−1∫x01tαf(t)dt)pdx≤[pp−α−1]p∫∞0fp(x)dx,f≥0,p≥1,α<p−1,where the constant C=[pp−α−1]p is sharp. This inequality holds in the reversed direction when 0≤p<1. In this paper we prove and discuss some discrete analogues of Hardy-type inequalities in fractional h-discrete calculus. Moreover, we prove that the corresponding constants are sharp.

Most Hardy type inequalities concern boundedness of the Hardy type operators in Lebesgue spaces. In this paper we prove some new multi-dimensional Hardy type inequalities in Hölder spaces.

We study one‐ and multi‐dimensional weighted Hardy operators on functions with Hölder‐type behavior. As a main result, we obtain necessary and sufficient conditions on the power weight under which both the left and right hand sided Hardy operators map, roughly speaking, functions with the Hölder behavior only at the singular point to functions differentiable for and bounded after multiplication by a power weight. As a consequence, this implies necessary and sufficient conditions for the boundedness in Hölder spaces due to the corresponding imbeddings. In the multi‐dimensional case we provide, in fact, stronger Hardy inequalities via spherical means. We also separately consider the case of functions with Hölder‐type behavior at infinity (Hölder spaces on the compactified ).

In this paper, we investigate convergence and divergence of partial sums with respect to the 2-dimensional Walsh system on the martingale Hardy spaces. In particular, we find some conditions for the modulus of continuity which provide convergence of partial sums of Walsh-Fourier series. We also show that these conditions are in a sense necessary and suffcient. 

The main aim of this paper is to further develop the recently initiatedresearch concerning geometric construction of some power means wherethe variables are appearing as line segments. It will be demonstratedthat the arithmetic mean, the harmonic mean and the quadratic meancan be constructed for any number of variables and that all power meanswhere the number of variables are n = 2m, m 1 2 N for all powersk = 2􀀀q and k = 2q; q 2 N can be geometrically constructed.

We prove and discuss a new divergence result of Nörlund logarithmic means with respect to Vilenkin system in Hardy space H1.

The primary aim of this chapter is to discuss the connection between the Hermite–Hadamard double inequality and Choquet’s theory. Noticed first by Niculescu (Math Inequal Appl 5(3):479–489, 2002, [356]), Niculescu (Math Inequal Appl 5(4):619–623, 2002, [357]) (during the conference Inequalities 2001, in Timişoara), this connection led him to a partial extension of the majorization theory beyond the classical case of probability measures, using the so-called Steffensen–Popoviciu measures. Their main feature is to offer a large framework under which the Jensen–Steffensen inequality remains available. As a consequence, one obtains the extension of the left-hand side of Hermite–Hadamard double inequality to a context involving signed Borel measures on arbitrary compact convex sets. A similar extension of the right-hand side of this inequality is known only in dimension 1, the higher dimensional case being still open.

In this paper, we derive two-sided estimates of the Lebesgue constants for bounded Vilenkin systems, we also present some applications of importance, e.g., we obtain a characterization for the boundedness of a subsequence of partial sums with respect to Vilenkin–Fourier series of H1 martingales in terms of n's variation. The conditions given in this paper are in a sense necessary and sufficient.

The aim of this paper is to establish some new weighted Hardy-type inequalities involving convex and monotone convex functions using Hilfer fractional derivative and fractional integral operator with generalized Mittag-Leffler function in its kernel. We also discuss one dimensional cases of our related results. As a special case of our general results we obtain the results of Iqbal et al. (2017). Moreover, the refinement of Hardy-type inequalities for Hilfer fractional derivative is also included.

Some classical inequalities are known also in a more general form of Banach lattice norms and/or in continuous forms (i.e., for ‘continuous’ many functions are involved instead of finite many as in the classical situation). The main aim of this paper is to initiate a more consequent study of classical inequalities in this more general frame. We already here contribute by discussing some results of this type and also by deriving some new results related to classical Popoviciu’s, Bellman’s and Beckenbach-Dresher’s inequalities.

Inequalities of the formkuK f kq 6C(kr f kp +kvH f kp) , f > 0,are considered, where K is an integral operator of Volterra type and H is the Hardy operator.Under some assumptions on the kernel K we give necessary and sufficient conditions for suchan inequality to hold.1

Some new extensions and refinements of Hermite–Hadamard and Fejer type inequali-ties for functions which are N-quasiconvex are derived and discussed.

We consider the general Hardy type operator inline image where inline image is a positive and measurable kernel. To characterize the weights u and v so that inline image is still an open problem for any parameters p and q. However, for special cases the solution is known for some parameters p and q. In this paper the current status of this problem is described and discussed mainly for the case inline image In particular, some new scales of characterizations in classical situations are described, some new proofs and results are given and open questions are raised.

In this paper we give an historical synopsis of various Taylor remainders and theirdierent proofs (without being exhaustive). We overview the formulas and the proofs given bysuch names as Bernoulli, Taylor, MacLaurin, Lagrange, Lacroix, Cauchy, Schlomilch, Roche,Cox, Turquan, Bourget, Koenig, Darboux, Amigues, Teixeira, Peano, Blumenthal, Wolfe andGoncalves. We end the paper with a new Taylor remainder which generalizes the well-knownLagrange remainder.

In this note we derive a new Taylor remainder, which extends the well known Lagrange remainder as well as the obscure Goncalves remainder.

Представлено новое неравенство типа Бекенбах-Дрешера в терминах идеальных банаховых структур и неравенство типа Поповичу для бесконечного интерполяционного семейства

We prove the boundedness of Potential operator in weighted generalized Morrey space in terms of Matuszewska-Orlicz indices of weights and apply this result to the Hemholtz equation in r³ with a free term in such a space. We also give a short overview of some typical situations when Potential type operators arise when solving PDEs.

Refinements of some limit Hardy-type inequalities are derived anddiscussed using the concept of superquadracity. We also proved that all threeconstants appearing in the refined inequalities obtained are sharp. The naturalturning point of our refined Hardy inequality is 𝑝 = 2 and for this case we haveeven equality.

The classical Hausdorff-Young and Hardy-Littlewood-Stein inequalities do not hold for p > 2. In this paper we prove that if we restrict to net spaces we can even derive a two-sided estimate for all p > 1. In particular, this result generalizes a recent result by Liflyand E. and Tikhonov S. [7] (MR 2464253).

In this second edition, all chapters in the first edition have been updated with new information. Moreover, a new chapter contains new and complementary information concerning: (a) a convexity approach to prove and explain Hardy-type inequalities; (b) sharp constants; (c) scales of inequalities to characterize Hardy-type inequalities; (d) Hardy-type inequalities in other function spaces; and (e) a number of new open questions.

In [14] we investigated some Vilenkin—Nörlund means with non-increasing coefficients. In particular, it was proved that under some special conditions the maximal operators of such summabily methods are bounded from the Hardy space H_1/(1+α) to the space weak-L_1/(1+α), (0 < α ≦ 1). In this paper we construct a martingale in the space H_1/(1+α), which satisfies the conditions considered in [14], and so that the maximal operators of these Vilenkin—Nörlund means with non-increasing coefficients are not bounded from the Hardy space H_1/(1+α) to the space L_1/(1+α). In particular, this shows that the conditions under which the result in [14] is proved are in a sense sharp. Moreover, as further applications, some well-known and new results are pointed out.

In this paper, we derive the maximal subspace of positive numbers, for which the restricted maximal operator of Fejér means in this subspace is bounded from the Hardy space Hp to the space Lp for all 0 < p ≤ 1/2. Moreover, we prove that the result is in a sense sharp

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.

We find conditions for the weighted boundedness of a general class of multidimensional singular integral operators in generalized Morrey spaces L p,φ (R n ,w), defined by a function φ(x,r) and radial type weight w(|x−x 0 |),x 0 ∈R n . These conditions are given in terms of inclusion into L p,φ (R n ,w), of a certain integral constructions defined by φ and w. In the case of φ=φ(r) we also provide easy to check sufficient conditions for that in terms of indices of φ and w.

The main aim of this paper is to focus on the question to develop classical inequalities to hold in a more general “continuous” form (involving infinitely many functions and/or spaces). First, we discuss such developments concerning Hölder’s and Minkowski’s inequalities. After that we present such new general developments of Popoviciu’s and Bellman’s inequalities. Finally, we present some applications, possible extensions and questions for further research.

For n ε ℤ+ we consider the difference Bn-1 (f)-Bn(f):= 1/an n-1/ηi=0 f(ai/an-1)-1/an+1 nηi=0f(ai/an) where the sequences{ai} and {ai-ai-1} are increasing. Some lower bounds are derived when f is 1-quasiconvex and when f is a closely related superquadratic function. In particular, by using some fairly new results concerning the so called "Jensen gap", these bounds can be compared. Some applications and related results about An-1 (f)-An(f):= 1/an n-1/ηi=0 f(ai/an-1)-1/an+1 nηi=0f(ai/an) are also included.

In this paper we consider the spaces Bw(ℓp), 1 ≤ p ≤ ∞, of infinite matrices A defined by (Equation presented) and prove that Bw(ℓp) is not closed under this product. Moreover, we prove that linear and bounded operators on ℓp are Schur multipliers on Bw(ℓp), a result which is not obvious, since Bw(ℓp) is not a Schur algebra. Most of the results are sharp in the sense that they are given via necessary and sufficient conditions.

Some new refinements of Hardy-type integral inequalities are derived, proved and discussed using the concept of superquadratic and subquadratic functions. The results obtained are generalizations and improvements of inequalities of this type in the literature.

We prove and discuss some new H_p-L_ptype 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

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.

We derive necessary and sufficient conditions (of Muckenhoupt-Bradley type) for the validity of q-analogs of (r, p)-weighted Hardy-type inequalities for all possible positive values of the parameters r and p. We also point out some possibilities to further develop the theory of Hardy-type inequalities in this new direction

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

One goal of this paper is to point out the fact that a big number of inequalities provedfrom time to time in journal publications, both one-dimensional and multi-dimensional, are particularcases of some general results for integral operators with homogeneous kernels, includingin particular, the statements on sharp constants.Some new multidimensional Hardy-Hilbert type inequalities are derived. Moreover, anew multidimensional P´olya-Knopp inequality is proved and some examples of applications arederived from this result. The constants in all inequalities are sharp.

The complete characterization of the weighted L^p− L^r inequalities of supremum operators on the cones of monotone functions for all 0 < p, r≤ ∞ is given.

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.

The sharp constants in Hardy type inequalities are known only in a few cases. In this paper we discuss some situations when such sharp constants are known, but also some new sharp constants are derived both in one-dimensional and multi-dimensional cases.

We obtain two-weighted estimates for the Hardy type operators fromlocal generalized Morrey spaces Lp,ϕloc (X,w1) defined on an arbitrary underlyingquasi-metric measure space (X, μ, ) with the growth condition, toLq,ψloc (X,w2), where w1 = w1[(x, x0)], x0 ∈ X is a weight of radial type,while w2 = w2(x) may be an arbitrary weight. The proof allows to simultaneouslytreat a similar boundedness V Lp,ϕloc (X,w1) → V Lq,ψloc (X,w2) forvanishing Morrey spaces. We obtain sufficient conditions for such estimatesin terms of some integral inequalities imposed on ϕ, ψ and w1.w2. We alsospecially treat the one weight case where w2(x) is also of radial type. Wedo not impose doubling condition on the measure μ, but base our result onthe growth condition.The obtained results show the explicit dependence of the mapping propertiesof the Hardy type operators on the fractional dimension of the set(X, μ, ). An application to spherical Hardy type operators is also given.

The complete characterization of the Hardy-type Lp-Lq inequalities on the weighted cones of quasi-concave functions for all 0 < p, q < ∞ is given.