Inequality In Inner Product Spaces Research Articles

Triangle inequalities in inner-product spaces

Tereshin's and Panaitopol's are known inequalities involving the median, circumradius and sides of the triangle. In this short note we generalize the inequalities to inner-product spaces. As an application we derive inequality for the median and the radius of the circumscribed sphere of an $n$-dimensional simplex.

Some Bessel type additive inequalities in inner product spaces

"In this paper we obtain some additive inequalities related to the cele- brated Bessel's inequality in inner product spaces. They complement the results obtained by Boas-Bellman, Bombieri, Selberg and Heilbronn, which have been applied for almost orthogonal series and in Number Theory."

Ulam Stability in Real Inner-Product Spaces

Roughly speaking an equation is called Ulam stable if near each approximate solution of the equation there exists an exact solution. In this paper we prove that Cauchy-Schwarz equation, Ortogonality equation and Gram equation are Ulam stable.This paper is concerned with the Ulam stability of some classical equations arising in thecontext of inner-product spaces. For the general notion of Ulam stability see, e.q., [1]. Roughlyspeaking an equation is called Ulam stable if near every approximate solution there exists anexact solution; the precise meaning in each case presented in this paper is described in threetheorems. Related results can be found in [2, 3, 4]. See also [5] for some inequalities in innerproduct spaces.

On Pólya–Szegö reverse of Schwarz's inequality

AbstractWe 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.

Inequalities for angles between subspaces with applications to Cauchy-Schwarz inequality in inner product spaces

Inequalities for angles between subspaces with applications to Cauchy-Schwarz inequality in inner product spaces

Richard’s inequality, Cauchy–Schwarz’s inequality, and approximate solutions of Sincov’s equation

We observe a connection between Cauchy–Schwarz’s and Richard’s inequalities in inner product spaces and a Ulam-type stability problem for the multiplicative Sincov functional equation. We prove that this equation is superstable for unbounded mappings, i.e., every unbounded approximate solution is an exact solution.

Some results related to Bessel's inequality in inner product spaces

Some results related to Bessel's inequality in inner product spaces

Equality conditions of reverse Schwarz and Grüss inequalities in inner productspaces

Equality conditions of reverse Schwarz and Grüss inequalities in inner productspaces

On reverse Schwarz, Bessel, Grüss and Hadamard inequalities in inner product spaces

ABSTRACTWe present some new results on the Cauchy-Schwarz inequality in inner product spaces. Applications to reverse Bessel and Grüss type inequalities are given and a refinement of the Hadamard inequality is obtained. Sharpness of each introduced inequality is illustrated by suitable examples.

Operator refinements of Schwarz inequality in inner product spaces

Some improvements of the celebrated Schwarz inequality in complex inner product spaces in terms of selfadjoint operators are given. Applications for orthonormal families of vectors are also provided.

An extension of the schwarz inequality in inner product spaces

We extend the improved Schwarz inequality of Dragomir [1, Theorem 2] to any power p ? 2, ||x||p||y||p-|?x,y?|p ? |det [|?x,e?| |?y,e?|(||y||p - |?x,e?|p)1/p (||y||p- |?y,e?|p)1/p]|p for any vectors x, y, e ? Cn with ||e|| = 1. Applications to n-tuples of complex numbers are also included.

Improving Schwarz inequality in inner product spaces

Some improvements of the celebrated Schwarz inequality in complex inner product spaces are given. Applications for n-tuples of complex numbers are provided.

Reverse Schwarz inequalities and some consequences in inner product spaces

Necessary and sufficient conditions for a general reverse of Schwarz's inequality in inner product spaces are given. Results are applied to the triangle inequality and Grus̈s' type inequality.

Reverses of Schwarz inequality in inner product spaces with applications

New reverses of the Schwarz inequality in complex inner products spaces with applications for bounded linear operators are given. Some Gruss' type inequalities and their applications for numerical radius and the operator norm are provided as well.

Another look at some new Cauchy–Schwarz type inner product inequalities

In this paper we take a refreshing look at a Cauchy–Schwarz type inner product inequalities. We also provide a Cauchy–Schwarz based proof and a refinement of Buzano’s inequality in inner product spaces. Some elementary applications such as trace inequalities for unitary matrices and discrete or integral inequalities are given.

Reverses of the triangle inequality in inner product spaces

Reverses of the triangle inequality in inner product spaces

Power series inequalities via a refinement of the Schwarz inequality

In this paper, we obtain some inequalities for functions defined by a power series with nonnegative coefficients. In order to obtain these inequalities, a refinement of the Schwarz inequality in inner product spaces is utilized. Natural applications for some elementary functions of interest are also provided.

Grüss and Ostrowski type inequalities

In this paper we use a method originated in [S. S. Dragomir, Some Grüss type inequalities in inner product spaces, J. Inequal. Pure Appl. Math. 4 (2) (2003) Article 42] to establish some Grüss and Ostrowski type inequalities.

A note on “Some inequalities in inner product spaces related to the generalized triangle inequality” by S.S. Dragomir et al.

In this note, we present an affirmative answer to a question presented in the paper “Some inequalities in inner product spaces related to the generalized triangle inequality” by S.S. Dragomir et al. [Appl. Math. Comput. 217 (18) (2011) 7462–7468].

Some inequalities in inner product spaces related to the generalized triangle inequality

In this paper we obtain some inequalities related to the generalized triangle and quadratic triangle inequalities for vectors in inner product spaces. Some results that employ the Ostrowski discrete inequality for vectors in normed linear spaces are also obtained.