Josip Pečarić

Inequalities involving functions and their integrals and derivatives

I. Landau-Kolmogorov and related inequalities.- II. An inequality ascribed to Wirtinger and related results.- III. Opials inequality.- IV. Hardys, Carlemans and related inequalities.- V. Hilberts and related inequalities.- VI. Inequalities of Lyapunov and of De la Vallee Poussin.- VII. Zmorovi?s and related inequalities.- VIII. Carlsons and related inequalities.- IX. Inequalities involving kernels.- X. Convolution, rearrangement and related inequalities.- XI. Inequalities of Caplygin type.- XII. Inequalities of Gronwall type of a single variable.- XIII. Gronwall inequalities in higher dimension.- XIV. Gronwall inequalities on other spaces: discrete, functional and abstract.- XV. Integral inequalities involving functions with bounded derivatives.- XVI. Inequalities of Bernstein-Mordell type.- XVII. Methods of proofs for integral inequalities.- XVIII. Particular inequalities.- Name Index.

Recent advances in geometric inequalities

The Existence of a Triangle.- Duality between Geometric Inequalities and Inequalities for Positive Numbers.- Homogeneous Symmetric Polynomial Geometric Inequalities.- Duality between Different Triangle Inequalities and Triangle Inequalities with (R, r, s).- Transformations for the Angles of a Triangle.- Some Trigonometric Inequalities.- Some Other Transformations.- Convex Functions and Geometric Inequalities.- Miscellaneous Inequalities with Elements of a Triangle.- Special Triangles.- Triangle and Point.- Inequalities with Several Triangles.- The Mobius-Neuberg and the Mobius-Pompeiu Theorems.- Inequalities for Quadrilaterals.- Inequalities for Polygons.- Inequalities for a Circle.- Particular Inequalities in Plane Geometry.- Inequalities for Simplexes in En (n ? 2).- Inequalities for Tetrahedra.- Other Inequalities in En (n ? 2).

Inequalities for differentiable mappings with application to special means and quadrature formulæ

Abstract Improvements are obtained to some recent error estimates of Dragomir and Agarwal, based on convexity, for the trapezoidal formula. Corresponding estimates are established for the midpoint formula. A parallel development is made based on concavity.

Hadamard-type inequalities for s-convex functions

In this paper we establish some new inequalities for differentiable functions based on concavity and s-convexity. We also prove several Hadamard-type inequalities for products of two convex and s-convex functions.

Improvement and further generalization of inequalities of Ostrowski-Grüss type

Abstract We improve some inequalities of Ostrowski-Gruss type and further generalize them. We apply the obtained results to the estimation of error bounds for some numerical quadrature rules. Also, some bounds for the differences of some special means are discussed.

Improved arithmetic-geometric and Heinz means inequalities for Hilbert space operators

The main objective of this paper is an improvement of the original weighted operator arithmetic-geometric mean inequality in Hilbert space. We define the difference operator between the arithmetic and geometric means and investigate its properties. Due to the derived properties, we obtain a refinement and a converse of the observed operator mean inequality. As an application, we establish one significant operator mean, which interpolates the arithmetic and geometric means, that is, the Heinz operator mean. We also obtain an improvement of this interpolation.

A matrix version of the Ky Fan generalization of the kantorovich inequality

Classical difference and ratio inequalities for means are extended to means of matrices with matrix weights.

On strengthened Hardy and Pólya-Knopp's inequalities

In this paper we prove a strengthened general inequality of the Hardy-Knopp type and also derive its dual inequality. Furthermore, we apply the obtained results to unify the strengthened classical Hardy and Polya-Knopps inequalities deriving them as special cases of the obtained general relations. We discuss Polya-Knopps inequality, compare it with Levin-Cochran-Lees inequalities and point out that these results are mutually equivalent. Finally, we also point out a reversed Polya-Knopp type inequality.

Unified treatment of Gautschi-Kershaw type inequalities for the gamma function

Abstract Gautschi, Kershaw, Lorch, Laforgia and other authors gave several inequalities for the ratio Γ(x + 1) Γ(x + s) where, as usual, Γ denotes the gamma function. In this paper we give a unified treatment of all their results and prove, among other things, new inequalities for the above ratio, which involve the psi function. Inequalities for the ratio of two gamma functions are useful, for example, to deduce Bernstein-type inequalities for ultraspherical polynomials. We give an example of this type.

Generalization of the power means and their inequalities

Abstract In this paper we gave a generalization of power means which include positive nonlinear functionals. For these means we obtained generalizations of fundamental mean inequality, Holders and Minkowskis, and their converse inequalities.