Measures for the width of a periodic function are discussed, based on five different mappings of a (periodic) function on a circle to a (non–periodic) function on a line or line segment. The uncertainty relations corresponding to these measures are also obtained by using a generalization of the Cauchy–Schwarz inequality.

We survey the basic theory of the strengthened Cauchy-Buniakowskii-Schwarz inequality and its applications in multilevel methods for the solution of linear systems arising from nite element or nite diierence discretisation of elliptic partial diierential equations. Proofs are given both in a nite element context, and in purely algebraic form.

We present some identities related to the Cauchy-Schwarz inequality in complex inner product spaces. A new proof of the basic result on the subject of strengthened CauchySchwarz inequalities is derived using these identities. Also, an analogous version of this result is given for strengthened Hölder inequalities.

We give some ratio and difference reverses of the Cauchy–Schwarz inequality for complex functions of self-adjoint operators in Hilbert spaces, under suitable assumptions for the involved operators. Several examples for particular functions of interest are provided as well. Mathematics subject classification (2010): 47A63, 47A99.

In 1964, Gähler 1 introduced the concept of 2-norm and 2-inner product spaces as generalization of norm and inner product spaces. A systematic presentation of the results related to the theory of 2-inner product spaces can be found in the book in 2, 3 and in list of references in it. Generalization of 2-inner product space for n ≥ 2 was developed by Misiak 4 in 1989. Gunawan and Mashadi 5 in 20...

In a 1918 paper [Sch18], Schur proved a remarkable inequality that related group representations, Hermitian forms and determinants. He also gave concise necessary and sufficient conditions for equality. In [Mar64], Marcus gave a beautiful short proof of Schur’s inequality by applying the Cauchy-Schwarz inequality to symmetric tensors, but he did not discuss the case of equality. In [Wil69], Wil...

This paper aims to present some inequalities for unitarily invariant norms. In section 2, we give a refinement of the Cauchy-Schwarz inequality for matrices. In section 3, we obtain an improvement for the result of Bhatia and Kittaneh [Linear Algebra Appl. 308 (2000) 203-211]. In section 4, we establish an improved Heinz inequality for the Hilbert-Schmidt norm. Finally, we present an inequality...