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] .

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.

For the Hardy type inequalities the "breaking point" (=the point where the inequality reverses) is p = 1. Recently, J. Oguntoase and L. E. Persson proved a refined Hardy type inequality with a breaking point at p = 2. In this paper we prove a new scale of refined Hardy type inequality which can have a breaking point at any p ≥ 2. The technique is to first make some further investigations for superquadratic and superterzatic functions of independent interest, among which, a new Jensen type inequality is proved

In this manuscript a number of general inequalities for isotonic subadditive functionals on a set of positive mappings are proved and applied. In particular, it is pointed out that these inequalities both unify and generalize some general forms of the Holder, Popoviciu, Minkowski, Bellman and Power mean inequalities. Also some refinements of some of these results are proved.

In this paper we deal with γ -quasiconvex functions when −1γ 0, to derive sometwo-sided Jensen type inequalities. We also discuss some Jensen-Steffensen type inequalitiesfor 1-quasiconvex functions. We compare Jensen type inequalities for 1-quasiconvex functionswith Jensen type inequalities for superquadratic functions and we extend the result obtained forγ -quasiconvex functions to more general classes of functions.

Some scales of refined Jensen and Hardy type inequalities are derived and discussed. The key object in our technique is ? -quasiconvex functions K(x) defined by K(x)x-? =? (x) , where Φ is convex on [0,b) , 0 < b > ∞ and γ > 0.

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 metric projection operator is an important tool in numerical analysis, optimization, variational inequalities and complementarity problems and has been considered from the point of view of isotonicity, with respect to an ordering compatible with the vector structure on Hilbert spaces and Banach spaces. In this paper, the authors study some inequalities related to the isotonicity of the metric projection operator onto a closed convex set in an ordered Banach space. The concept of antiprojection operator onto a compact nonempty subset of a Hilbert space is introduced and the relationship between the new inequality obtained by the authors and the isotonicity of such an operator is also discussed.

In the context of generalized Orlicz spaces, the products X-Phi 1 circle dot X-Phi 2 and X-Phi 1 circle times X-Phi 2 are studied and conditions are obtained under which these spaces are contained in a suitable space X-Phi. These imbedding results (inequalities) are in a sense sharp and for the case X = L-1, the conditions are even necessary and sufficient. Moreover, a new Holder type inequality is proved.

Some new Carleman-Knopp type inequalities are proved as "end point" inequalities of modern forms of Hardy's inequalities. Both finite and infinite intervals are considered and both the cases p q and q < p are investigated. The obtained results are compared with similar results in the literature and the sharpness of the constants is discussed for the power weight case. Moreover, some reversed Carleman-Knopp inequalities are derived and applied.

Characterization of Lvp[0, ∞) - L μq[O, ∞) boundedness of the general Hardy operator (Hsf)(x) =(∫[0,x] fsudλ) 1/s restricted to monotone functions f ≥ 0 for 0 < p.q.s < ∞ with positive Borel σ -finite measures λ, μ and v is obtained.

A recently discovered Hardy-Pólya type inequality described by a convex function is considered and further developed both in weighted and unweighted cases. Also some corresponding multidimensional and reversed inequalities are pointed out. In particular, some new multidimensional Hardy and Pólya-Knopp type inequalities and some new integral inequalities with general integral operators (without additional restrictions on the kernel) are derived

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).

We derive and discuss a new two-dimensional weightedHardy-type inequality in a rectangle for the class of functions fromthe Sobolev space $H^1$ vanishing on small alternating pieces of theboundary

We state and prove some new weighted Hardy type inequalities with an integral operator A(k) defined by A(k)f(x) := 1/K(x) integral(Omega 2) k(x,y)f(y)d mu(2) (y), where k : Omega(1) x Omega(2) --> R is a general nonnegative kernel, (Omega(1), mu(1)) and (Omega(2), mu(2)) are measure spaces and K(x) := integral(Omega 2) k(x,y)d mu(2) (y), x is an element of Omega(1). In particular, the obtained results unify and generalize most of the results of this type (including the classical ones by Hardy, Hilbert and Godunova).

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.

It is well-known that the Hölder-Rogers inequality implies the Minkowski inequality. Infantozzi [6] observed implicitely and Royden [15] proved explicitely that the reverse implication is also true. In this note we discuss and give a new proof of this perhaps surprising fact

This reviewer remembers Leopold Fejér (1880-1959) saying (in Hungarian) that ``the history of mathematics serves to prove that nobody discovered anything: there was always somebody who had known it before'' (quoted in English in the present paper). The author argues convincingly about who had priority to the inequalities of Hölder (Rogers), Cauchy (Lagrange), Jensen (Hölder) and others. Also equivalences and relations between different forms and different inequalities and several proofs are offered. The paper concludes with biographical sketches of Leonard James Rogers and Otto Ludwig Hölder.

For any Banach space X the n-th James constants J(n)(X) and the n-th Khintchine constants K-p,q(n)(X) are investigated and discussed. Some new properties of these constants are presented. The main result is an estimate of the n-th Khintchine constants K-p,q(n)(X) by the n-th James constants Jn (X). In the case of n = 2 and p = q = 2 this estimate is even stronger and improvs an earlier estimate proved by Kato-Maligranda-Takahashi

Relations between the norms of an operator and its complexification as a mapping from Lp to Lq has been recognized as a serious problem in analysis after the publication of Marcel Riesz's work on convexity and bilinear forms in 1926. We summarize here what it is known about these relations in the case of normed Legesgue spaces and investigate the quasinormed case, i. e. we consider all 0 < p,q ≤ 8 . In particular, in the lower triangle, that is, for 0< p≤q≤∞ these norms are the same. In the upper triangle and the normed case, that is, when 1 ≤ q < p ≤ ∞ the norm of the complexification of a real operator is obviously not bigger than 2 times its real norm. In 1977 Krivine proved that the constant 2 can be replaced by √2 . On the other hand, it was suspected that in the case of quasi-normed Lebesgue spaces (0 < q < p ≤ ∞ ) the corresponding constant could be arbitrarily large, but as we will see this is not the case. More precisely, we prove that this constant for quasi-normed Lebesgue spaces is between 1 and 2. Some additional properties and estimates of this constant with some results about the relation between complex and real norms of operators, including those between two-dimensional Orlicz spaces are presented in the first four chapters. Finally, in Chapter 5, we use the results on the estimates of the norms in the proof of the real Riesz-Thorin interpolation theorem valid in the first quadrant

We prove two new reverse Cauchy-Schwarz inequalities of additive and multiplicative types in a space equipped with a positive sesquilinear form with values in a C*-algebra. We apply our results to get some norm and integral inequalities. As a consequence, we improve a celebrated reverse Cauchy-Schwarz inequality due to G. Polya and G. Szego

It is known that the maximal operator ofWalsh-Marcinkiewicz means is bounded from the dyadic martingale Hardy space Hp to the space Lp for p > 2/3 and the condition p > 2/3 is essential. In the case p = 2/3 the boundedness of the maximal operator does not hold. This means that the investigation of the maximal operator at the endpoint case p = 2/3 plays an important role. The main aim of this paper is to prove a strong convergence theorem for the Walsh-Marcinkiewicz means on the Hardy space H2/3.

We prove a new discrete Hardy-type inequality ||A/||q,u ≤ C||f||p,v, where the matrix operator A is defined by (Af) i:= Σj=1i ai,jfj, ai,j ≥ 0. Moreover, we study die problem of compactness of the operator A, and die dual result is stated

Anon-negative triangularmatrix operator is considered in weighted Lebesgue spaces of sequences. Under some additional conditions on the matrix, some new weight characterizations for discrete Hardy type inequalities with matrix operator are proved for the case 1 < q < p < ∞. Some further results are pointed out

The well-known Grisvard-Jakovlev inequality (see Theorems 1 and 1_ ) can be interpretedas a fractional order Hardy inequality or as a weighted difference inequality. Someinequalities of this type have been recently proved and discussed by the authors and H. Heinig,and this paper coincides mostly with a lecture held by the first author at the International workshopon difference and differential inequalities (July 3 - 7, 1996, Marmara Research Center,Turkey) where some historical remarks, ideas and results from the papers of the authors and H.Heinig have been presented. Additionally we present and prove some new difference inequalitieswith weights. Mostly, we omit the proofs which can be found in the papers mentioned and in thereferences there, and for simplicity, we consider functions on the interval

The paper is devoted to the study of weighted Hardy-type inequalities on the cone of quasi-concave functions, which is equivalent to the study of the boundedness of the Hardy operator between the Lorentz Γ-spaces. For described inequalities we obtain necessary and sufficient conditions to hold for parameters q 1, p > 0 and sufficient conditions for the rest of the range of parameters.

A weight function w(x) on (0,l) or (l,infinity), is said to be quasi-monotone if w(x)x(-a0) <= C(0)w(y)y(-a0) either for all x <= y or for all y <= x, for some a(0) is an element of R, C-0 >= 1. In this paper we discuss, complement and unify several results concerning quasi-monotone functions. In particular, some new results concerning the close connection to index numbers and generalized Bary-Stechkin classes are proved and applied. Moreover, some new regularization results are proved and several applications are pointed out, e. g. in interpolation theory, Fourier analysis, Hardy-type inequalities, singular operators and homogenization theory.

We consider the weighted Hardy inequality (∫0∞ (∫0x f(t)dt)q u(x)dx)1/q ≤ C(∫0∞ f p(x)v(x)dx)1/p for the case 0 < q < p < ∝, p > 1. The weights u(x) and v(x) for which this inequality holds for all f (x) ≥ 0 may be characterized by the Mazya-Rosin or by the Persson-Stepanov conditions. In this paper, we show that these conditions are not unique and can be supplemented by some continuous scales of conditions and we prove their equivalence. The results for the dual operator which do not follow by duality when 0 < q < 1 are also given

A generalization of the Lizorkin theorem on Fourier multipliers is proved. The proofs are based on using the so-called net spaces and interpolation theorems. An example is given of a Fourier multiplier which satisfies the assumptions of the generalized theorem but does not satisfy the assumptions of the Lizorkin theorem.