Some Improvements of the Cauchy-Schwarz Inequality Using the Tapia Semi-Inner-Product

Nicuşor Minculete,Hamid Reza Moradi

doi:10.3390/math8122112

Nicuşor Minculete, Hamid Reza Moradi

Open Access

https://doi.org/10.3390/math8122112

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Abstract

The aim of this article is to establish several estimates of the triangle inequality in a normed space over the field of real numbers. We obtain some improvements of the Cauchy–Schwarz inequality, which is improved by using the Tapia semi-inner-product. Finally, we obtain some new inequalities for the numerical radius and norm inequalities for Hilbert space operators.

Highlights

In an inner product space an important inequality is the inequality of Cauchy–Schwarz [1,2], namely:|h x, yi| ≤ k x kkyk, (1)for all x, y ∈ X, where X is a complex inner product space.Aldaz [3] and Dragomir [4] studied the Cauchy–Schwarz inequality in the complex case.Another inequality that plays a central role in a normed space is the triangle inequality, k x + y k ≤ k x k + k y k, (2)for all x, y ∈ X, where X is a complex normed space
We present some results regarding the several estimates of the triangle inequality in a normed space over the field of real numbers R
In the case when the norm k · k is generated by an inner product h·, ·i, ( x, y) T = h x, yi, for all x, y ∈ X, we find the Cauchy–Schwarz inequality

Summary

Introduction

In an inner product space an important inequality is the inequality of Cauchy–Schwarz [1,2], namely:. Aldaz [3] and Dragomir [4] studied the Cauchy–Schwarz inequality in the complex case. Another inequality that plays a central role in a normed space is the triangle inequality, k x + y k ≤ k x k + k y k,. Dehghan [14] presented a new refinement of the triangle inequality and defined the skew angular x y distance between nonzero vectors x and y by β[ x, y] =.

Inequalities Related to the Triangle Inequality

Some Inequalities Related to the Tapia Semi-Inner-Product

Estimates for Numerical Radii via Cauchy–Schwarz and Triangle Inequalities