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Some new weakly singular integral inequalities of Gronwall–Bellman type are established, which generalized some known weakly singular inequalities and can be used in the analysis of various problems in the theory of certain classes of differential equations, integral equations and evolution equations. Some applications to fractional differential and integral equations are also indicated.

We point out weighted Montgomery's identities for function of two variables and apply them to give a new inequalities of the Ostrowski type for mappings of two independent variables and of the Gr\"{; ; ; ; ; ; u}; ; ; ; ; ; ss type inequalities for double weighted integrals.

The classical Bohr's inequality states that | z + w | 2 ⩽ p | z | 2 + q | w | 2 for all z , w ∈ C and all p , q > 1 with 1 p + 1 q = 1 . In this paper, Bohr's inequality is generalized to the context of Hilbert space operators for all positive conjugate exponents p , q ∈ R . In particular, the parallelogram law is recovered and some other interesting operator inequalities and established.

We generalize the identities of J. Pečarić and A.M. Fink for the Chebyshev functional. The identities are shown to be very useful in establishing various bounds for the Chebyshev functional.

A variant of Jensen–Steffensen's inequality is proved. Necessary and sufficient conditions for the equality in Jensen–Steffensen's inequality are established. Several inequalities involving more than two monotonic functions and generalized quasi-arithmetic means with not only positive weights are proved. It is shown that such generalized quasi-arithmetic means have the same comparability properties as those with positive weights.

The Cauchy type mean-value theorems for the Riemann–Liouville fractional derivative are deduced here from known mean-value theorems of the Lagrange type. A general method for deducing these Cauchy type formulas is extracted. Two Cauchy type formulas are then deduced without a priori knowledge about the Lagrange type mean-value theorems.

In this paper, we give some improvements of the well-known Aczél inequality and the related results.

In this paper we obtain a new class of multivariable integral inequalities of Hilbert type. By specializing the upper estimate functions in the hypothesis and the parameters, we obtain many special cases, which include, in particular, the integral inequalities derived recently by Pachpatte.

In this paper, some new Hardy-type inequalities involving many functions are obtained. These on the one hand generalize and on the other hand improve some existing results by Isumi and Isumi, Levinson, and Pachpatte on this famous type of inequalities. (C) 2000 Academic Press. [References: 19]

In this article, all of the cases of the inequality[formula]being or not being valid are discussed for all real numbersx, y ∈ R andx ≠ 0, wherea, b > 0 anda ≠ b. Some recent developments concerning mean values are simply introduced.

A generalization of Alzer's inequality is proved. It is shown that this inequality is satisfied for a large class of increasing convex sequences.

A generalization is given of the extension of Hadamard's inequality to r -convex functions. A corresponding generalization of the Fink–Mond–Pecaric inequalities for r -convex functions is established.

In this paper we generalize the classical Chung's strong law of large numbers. We also prove some results about almost sure A -summability for sequences of independent random variables, where A is a suitable real infinite matrix.

The aim of this paper is the generalizations of Opial type inequalities for the case of many independent variables.

Versions of the upper Hadamard inequality are developed for r -convex and r -concave functions.

Using some results of Maligranda, we obtain a generalization of the well known Jensen's inequality which is also a common generalization of inequalities of Hölder and Minkowski.

In this paper are given some simple proofs and extensions of some inequalities for concave functions obtained recently by Brenner and Alzer.

Weighted versions of the Favard and Benwald inequalities are proved in the class of monotone and concave (convex) functions. Some necessary majorization estimates and a double-weight characterization for a Favard-type inequality are included. Godkänd; 1995; 20070125 (evan)

ƒ defined on the set T of matrices with values in the set of matrices is said to be semi-convex if X, Y∈T, X≤Y implies that λX+(1−λ)Y∈T, 0

In this paper we provide an account of various results that have been obtained concerning Bohr′s inequality with emphasis on several generalizations.

An inequality of G. Szegő is generalized to Wright-convex functions.

This paper is a complete review and unified treatment of recent results conerning the Neuberg-Pedoe and Oppenheim inequalities. Some new proofs and generalizations of these results are also added.

In this paper we use a generalization of the Cebysev inequality for products of functions defined on cells X in R r as proved recently by R. R. Janic and the present authors to obtain a number of quite general inequalities. As particular cases we obtain multidimensional integral versions of inequalities of Mitra, Stankovic and Milovanovic, Dunkel and Vasic and Milovanovic.

In a paper published in 1969 [ 11 Beesack proved some inequalities involving nth iterated kernels and convolutions. In the present paper we shall extend these results to general mean values of kernels and convolutions and also obtain some reverse or complementary inequalities involving such means. In this introductory section we establish our notation, define generalized iterated kernels, and prove two lemmas. The second lemma gives the strong form of Jensen’s inequality which we will later require. In Section 2 we define the general nth-order means and prove a number of inequalities relating these, including an analogue of the classical inequality ( 4

A complete set of Lr (r≠0) form very general Opial type weighted inequalities is given for a general linear differential operator L. These involve its related initial value problem solution y, Ly, the associated Green's function H, and initial conditions point x0∈R. This work is inspired by work of R. P. Agarwal and P. Y. H. Pang (1995, “Opial Inequalities with Applications in Differential and Difference Equations,” Kluwer Academic, Dordrecht) and G. A. Anastassiou (1998, Math. Inequalities Appl.1, No. 2, 193–200) and generalizes their related results. An application to proving uniqueness in solutions of initial value problems is given at the end.

In this paper we use basic properties of superquadratic functions to obtain new inequalities including Fejer's type and Hermite–Hadamard type inequalities. For superquadratic functions which are also convex, we get refinements of known results.

We consider an inverse Holder type inequality of J. Bergh [ Math Z. 215 (1994), 205-208] yielding for quasi-concave functions. We prove that this inequality holds in a more general class of functions endowed with two quasimonotonicity growth conditions. Some classes of quasi-monotone functions in mean are introduced and some new Bergh-type inequalities in these classes are proved. Our proofs are short and completely different from that of J. Bergh.

On etablit quelques resultats lies a l'integralite de B. Jessen pour les fonctions convexes

Many converses of Jensen's inequality for convex functions can be found in the literature. Here we give matrix versions, with matrix weights, of these inequalities. Some applications to the Hadamard product of matrices are also given.

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, Holder's and Minkowski's, and their converse inequalities.

We establish some inequalities for the Bessel functions of the first kind J ν ( x ) and for the modified Bessel functions I ν ( x ) and K ν ( x ). Some of these results are obtained using the method of Giordano and Laforgia which is based essentially on the arithmetic–geometric mean inequality. Other results are established as a consequence of the classical Holder and Gram inequalities.

In this paper we shall give a very short proof of the Jensen-Boas inequality. In the proof we shall use only Jensen’s inequality for sums, i.e.. (2) where P, = C;=, pi, pi > 0 (i = l,..., n), xi E I (i = l,..., n), and the Jensen- Steffensen inequality. Inequality (2) can be easily obtained from (1). If A(a) < A(v,) < n(y*) < ... < A( y,- ,) < I(b), then from the Jensen- Steffensen inequality we have the inequalities i.e.

In this paper we give a sequence of interpolating inequalities for the well-known Jensen inequality for convex functions.

Steffensen [ 1 ] proved the following result: Assume that two integrable functions f(t) and g(t) are defined on the interval [a, b], that f(t) never increases, and that 0