Inversion of the Cauchy–Bunyakovskii–Schwarz inequality

In the present paper, the inequality inverse to the Cauchy–Bunyakovskii–Schwarz inequality and generalizing other well-known inversions of this inequality is proved.     

Refined Heinz operator inequalities

Motivated by the well-known Heinz norm inequalities, in this article we study the corresponding Heinz operator inequalities. We derive the whole series of refinements of these operator inequalities, first with the help of the well-known Hermite–Hadamard inequality, and then, utilizing the parametrized family of the so-called Heron means. In such a way, we obtain improvements of some recent results, known from the literature.     

On a reverse Heinz–Kato–Furuta inequality

In the set up of Minkowski spaces, the Schwarz inequality holds with the reverse inequality sign. As a consequence, the same occurs with the triangle inequality. In this note, extensions of this indefinite version of the Schwarz inequality are presented. Namely, a reverse Heinz–Kato–Furuta inequality valid for timelike vectors is included and related inequalities that also hold with the reverse sign are investigated.     

A functional generalization of the Cauchy–Schwarz inequality and some subclasses

In this work, a functional generalization of the Cauchy–Schwarz inequality is presented for both discrete and continuous cases and some of its subclasses are then introduced. It is also shown that many well-known inequalities related to the Cauchy–Schwarz inequality are special cases of the inequality presented.     

The converse of the Loewner–Heinz inequality via perspective

The Loewner–Heinz inequality is not only the most essential one in operator theory, but also a fundamental tool for treating operator inequalities. The aim of this paper is to investigate the converse of the Loewner–Heinz inequality in the view point of perspective and generalized perspective of operator monotone and multiplicative functions. Indeed, we give perspective inequalities equivalent to the Loewner–Heinz inequality.     

On Pólya–Szegö reverse of Schwarz's inequality

We present some new results on the Cauchy–Schwarz inequality in inner product spaces, where four vectors are involved. This naturally extends Pólya–Szegö reverse of Schwarz's inequality onto complex inner product spaces. Applications to the famous Hadamard's inequality about determinants and the triangle inequality for norms are given.     

Remarks on a main inequality for several special functions

It is shown that the main inequality for several special functions derived in [Masjed-Jamei M. A main inequality for several special functions. Comput Math Appl. 2010;60:1280–1289] can be put in a concise form, and that the main inequalities of the first kind Bessel function, Laplace and Fourier transforms are not valid as presented in the aforementioned paper. To provide alternative inequalities, we give a generalization, being in some cases an improvement, of the Cauchy–Bunyakovsky–Schwarz inequality which can be applied to real functions not necessarily of constant sign. The corresponding discrete inequality is also obtained, which we use to improve the inequalities of the Riemann zeta and the generalized Hurwitz–Lerch zeta functions. Finally, from the main concise inequality, we derive a Turán-type inequality.     

Some generalizations of the Cauchy–Schwarz and the Cauchy–Bunyakovsky inequalities involving four free parameters and their applications

Some generalizations of the well-known Cauchy–Schwarz inequality and the analogous Cauchy–Bunyakovsky inequality involving four free parameters are given for both discrete and continuous cases. Several particular cases of interest are also analyzed. Some of the applications of our main results include (for example) the Wagner inequality.     

Molnár-dependence in a pre-Hilbert module over an H*-algebra

Molnár-dependence is related to the strong Cauchy-Schwarz inequality in a pre-Hilbert module over an H*-algebra analogously as the linear dependence is related to the Cauchy–Schwarz inequality in a pre-Hilbert space. Necessary and sufficient conditions for two elements of a pre-Hilbert module to be Molnár-dependent are established in this article, what is enabled by proving a stronger inequality than the strong Cauchy–Schwarz one. Furthermore, it is shown that Molnár-dependence is a transitive relation.     

Improved Heinz inequalities via the Jensen functional

By virtue of convexity of Heinz means, in this paper we derive several refinements of Heinz norm inequalities with the help of the Jensen functional and its properties. In addition, we discuss another approach to Heinz operator means which is more convenient for obtaining the corresponding operator inequalities for positive invertible operators.     

Matrix Wielandt inequality via the matrix geometric mean

In this paper, by virtue of the matrix geometric mean and the polar decomposition, we present new Wielandt type inequalities for matrices of any size. To this end, based on results due to J.I. Fujii, we reform a matrix Cauchy–Schwarz inequality, which differs from ones due to Marshall and Olkin. As an application, we show a new block matrix version of Wielandt type inequalities under the block rank additivity condition.     

Refinements of the Cauchy–Bunyakovsky–Schwarz inequality for functions of selfadjoint operators in Hilbert spaces

Some inequalities for continuous functions of selfadjoint operators in Hilbert spaces that improve the Cauchy–Bunyakovsky–Schwarz inequality, are given.     

Interpolating inequalities related to a recent result of Audenaert

Audenaert recently obtained an inequality for unitarily invariant norms that interpolates between the arithmetic–geometric mean inequality and the Cauchy–Schwarz inequality for matrices. A refined version of Audenaert’s inequality for the Hilbert–Schmidt norm is given. Other interpolating inequalities for unitarily invariant norms are also presented.     

Numerous refinements of Pólya and Heinz operator inequalities

In this article, we present an infinite number of refinements of the Heinz inequality for real numbers and operators. Making use of them, infinitely many refinements of the classical Pólya inequality and their operator versions are deduced.     

Equivalence relations among inequalities for some relative operator entropies

Positivity - The relative operator entropy has properties like operator means. In addition, the relative operator entropy has entropy-like properties. In this paper, we prove a Loewner–Heinz...     

The Cauchy–Schwarz inequality and the induced metrics on real vector spaces mainly on the real line

It is very well known that the Cauchy–Schwarz inequality is an important property shared by all inner product spaces and the inner product induces a norm on the space. A proof of the Cauchy–Schwarz inequality for real inner product spaces exists, which does not employ the homogeneous property of the inner product. However, it is shown that a real vector space with a product satisfying properties of an inner product except the homogeneous property induces a metric but not a norm. It is remarkable to see that the metric induced on the real line by such a product has highly contrasting properties relative to the absolute value metric. In particular, such a product on the real line is given so that the induced metric is not complete and the set of rational numbers is not dense in the real line.     

Turán-type inequalities for Struve,modified Struve and Anger–Weber functions

The aim of this paper is to establish the Turán-type inequalities for Struve functions, modified Struve functions, Anger–Weber functions and Hurwitz zeta function, by using a new form of the Cauchy–Bunyakovsky–Schwarz inequality.     

On some integral inequalities related to the Cauchy–Bunyakovsky–Schwarz inequality

Some new results that provide refinements and reverses of the Cauchy–Bunyakovsky–Schwarz (CBS-)inequality in the general setting of measure theory and under some boundedness conditions for the functions involved are given.     

A matrix reverse Cauchy–Schwarz inequality

The matrix geometric mean is concave. We complete this important fact with a reverse result. This follows from an interesting non-commutative extension of a classical reverse Cauchy–Schwarz inequality (Cassel, 1951). Our investigation also leads to state a reverse result to von-Neuman’s trace inequality.     

Algebraic analysis of multigrid algorithms

We study the convergence rate of multilevel algorithms from an algebraic point of view. This requires a detailed analysis of the constant in the strengthened Cauchy–Schwarz inequality between the coarse‐grid space and a so‐called complementary space. This complementary space may be spanned by standard hierarchical basis functions, prewavelets or generalized prewavelets. Using generalized prewavelets, we are able to derive a constant in the strengthened Cauchy–Schwarz inequality which is less than 0.31 for the L² and H¹ bilinear form. This implies a convergence rate less than 0.15. So, we are able to prove fast multilevel convergence. Furthermore, we obtain robust estimations of the convergence rate for a large class of anisotropic ellipic equations, even for some that are not H¹‐elliptic. Copyright © 1999 John Wiley & Sons, Ltd.