Hermite–Hadamard’s type inequalities for convex functions of selfadjoint operators in Hilbert spaces

Some Hermite–Hadamard’s type inequalities for convex functions of selfadjoint operators in Hilbert spaces under suitable assumptions for the involved operators are given. Applications in relation with the celebrated Hölder–McCarthy’s inequality for positive operators and Ky Fan’s inequality for real numbers are given as well.     

Integral representations and holonomic q-difference structure of Askey–Wilson type functions

We derive in a unified way the difference equations for Askey–Wilson polynomials and their Stieltjes transforms, by using basic properties of the de Rham cohomology associated with q-integral representations (Jackson integrals of BC ₁ type) of these functions.     

On Baskakov–Szász–Mirakyan-type operators preserving exponential type functions

The current article deals with the study of Baskakov–Szász–Mirakyan operators which reproduces constant and exponential functions. We discuss a uniform estimate and establish a quantitative result for the modified operators.     

Some Hermite–Hadamard type inequalities for functions of generalized convex derivative

We obtain some new inequalities of Hermite–Hadamard type. We consider functions that have convex or generalized convex derivative. Additional inequalities are proven for functions whose second derivative in absolute values are convex. Applications of the main results are presented.

     

Weighted composition operators and locally convex algebras

The Gleason-Kahane-Zelazko theorem characterizes the continuous homo-morphism of an associative, locally multiplicatively convex, sequentially complete algebra A into the field C among all linear forms on A. This characterization will be applied along two different directions. In the case in which A is a commutative Banach algebra, the theorem yields the representation of some classes of continuous linear maps A:A→A as weighted composition operators, or as composition operators when A is a continuous algebra endomorphism. The theorem will then be applied to explore the behaviour of continuous linear forms on quasi-regular elements, when A is either the algebra of all Hilbert-Schmidt operators or a Hilbert algebra.     

Singular integral operators on tent spaces over spaces of homogeneous type: Fefferman–Stein box maximal functions and pointwise Carleson type estimates

In this article we generalize the singular integral operator theory on weighted tent spaces to spaces of homogeneous type. This generalization of operator theory is in the spirit of C. Fefferman and Stein since we use some auxiliary functionals on tent spaces which play roles similar to the Fefferman–Stein sharp and box maximal functions in the Lebesgue space setting. Our contribution in this operator theory is twofold: for singular integral operators (including maximal regularity operators) on tent spaces pointwise Carleson type estimates are proved and this recovers known results; on the underlying space no extra geometrical conditions are needed and this could be useful for future applications to parabolic problems in rough settings.     

Hermite–Hadamard type inequalities for the m- and (α, m)-geometrically convex functions

In the paper the authors introduce concepts of the m- and (α, m)-geometrically convex functions and establish some inequalities of Hermite–Hadamard type for these classes of functions.     

Daugavet type inequalities for operators on L p–spaces

Let T be a regular operator from L ^p L ^p. Then , where T_r denotes the regular norm of T, i.e., T_r=|T| where |T| denotes the modulus operator of a regular operator T. For p=1 every bounded linear operator is regular and T=T_r, so that the above inequality generalizes the Daugavet equation for operators on L ¹–spaces. The main result of this paper (Theorem 9) is a converse of the above result. Let T be a regular linear operator on L _p and denote by T _A the operator T_A. Then for all A with (A)>0 if and only if .     

Factorization of Hardy spaces and Hankel operators on convex domains in ℂ n

In this article we study the (small) Hankel operator h_b on the Hardy and Bergman spaces on a smoothly bounded convex domain of finite type in ℂⁿ. We completely characterize the Hankel operators h_b that are bounded, compact, and belong to the Schatten ideal S_p, for 0 < p < ∞. In particular, if h_b denotes the Hankel operator on the Hardy space H² (Ω), we prove that h_b is bounded if and only if b ∈ BMOA, compact if and only if b ∈ VMOA, and in the Schatten class if and only if b ∈e B_p, 0 < p < ∞. This last result extends the analog theorem in the case of the unit disc of Peller [19] and Semmes [21]. In order to characterize the bounded Hankel operators, we prove a factorization theorem for functions in H¹ (Ω), a result that is of independent interest.     

Commutators of Calderón-Zygmund operators related to admissible functions on spaces of homogeneous type and applications to Schrödinger operators

Let X be an RD-space. In this paper, the authors establish the boundedness of the commutator Tbf = bTf-T(bf) on Lp , p∈(1,∞), where T is a Calderón-Zygmund operator related to the admissible function ρ and b∈BMOθ(X)BMO(X). Moreover, they prove that Tb is bounded from the Hardy space H1ρ(X) into the weak Lebesgue space L1weak(X). This can be used to deal with the Schrdinger operators and Schrdinger type operators on the Euclidean space Rn and the sub-Laplace Schrdinger operators on the stratified Lie group G.     

Integral means of weighted Hardy–Bloch functions

Let \({\mathcal{B}^\omega(p, q, B_d)}\) denote the \({\omega}\)-weighted Hardy–Bloch space on the unit ball B _d of \({\mathbb{C}^d}\), \({d\ge 1}\). For \({2< p,q < \infty}\) and \({f\in \mathcal{B}^\omega(p, q, B_d)}\), we obtain sharp estimates on the growth of the p-integral means M _p (f, r) as \({r\to 1-}\).     

PC-fractions and Szegö polynomials associated with starlike univalent functions

Each classS , 0<1, functions starlike of order , can be associated with a Carathéodory function mapping the unit disk onto a subset of the right halfplane. This Carathéodory function determines a certain continued fraction (PC-fraction) and a family of polynomials orthogonal on the unit circle (Szegö polynomials). We compute the PC-fraction and Szegö polynomials corresponding to eachS and do some investigations on these PC-fractions and Szegö polynomials.     

Generalized Baskakov–Durrmeyer type operators

In the present paper, we study some approximation properties of the Durrmeyer type modification of generalized Baskakov operators introduced by Erencin (Appl Math Comput 218(3):4384–4390, 2011). First, we establish a Lorentz-type lemma for the derivatives of the kernel of the generalized Baskakov operators and then obtain a recurrence relation for the moments of their Durrmeyer type modification. Next, we discuss some direct results in simultaneous approximation by these operators e.g. pointwise convergence theorem, Voronovskaja-type theorem and an estimate of error in terms of the modulus of continuity. Finally, we estimate the error in the approximation of functions having derivatives of bounded variation.     

Uniformly summing sets of operators on spaces of vector–valued continuous functions

Let X, Y be Banach spaces. We say that a set is uniformly p–summing if the series is uniformly convergent for whenever (x_n) belongs to . We consider uniformly summing sets of operators defined on a -space and prove, in case X does not contain a copy of c₀, that is uniformly summing iff is, where T (φ x) = (T^#φ) x for all and x∈X. We also characterize the sets with the property that is uniformly summing viewed in . Received: 1 July 2005     

Certain subclasses of uniformly convex functions of order α and type β with varying arguments

In this paper, we define a new subclass of k-uniformly convex functions order α type β with varying argument of coefficients and obtain coefficient estimates. Further we investigate extreme points, growth and distortion bounds, radii of starlikeness and convexity and modified Hadamard products.