A New Generalization on Cauchy-Schwarz Inequality

Abstract

We extend the well-known Cauchy-Schwarz inequality involving any number of real or complex functions and also give a necessary and sufficient condition for the equality. This is another generalized version of the Cauchy-Schwarz inequality.

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Songting Yin 1

Academic Editor:Yoshihiro Sawano

Department of Mathematics and Computer Science, Tongling University, Tongling, Anhui 244000, China

Received 7 April 2017; Accepted 23 May 2017; 13 June 2017

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Let a=(_a1 ,_a2 ,...,_an ) and b=(_b1 ,_b2 ,...,_bn ) be two vectors in ^Rn . Then the discrete version of Cauchy-Schwarz inequality is (see [1, 2]) [figure omitted; refer to PDF] and its integral representation in the space of the continuous real-valued functions C((a,b),R) reads (see [2, 3]) [figure omitted; refer to PDF]

In undergraduate teaching material, the above two inequalities are presented. Some other forms, such as matrix form and determinant form, are shown in many literatures. It is well known that the Cauchy-Schwarz inequality plays an important role in different branches of modern mathematics such as Hilbert space theory, probability and statistics, classical real and complex analysis, numerical analysis, qualitative theory of differential equations, and their applications. Up to now, a large number of generalizations and refinements of the Cauchy-Schwarz inequality have been investigated in the literatures (see [4, 5]). In [6], Harvey generalized it to an inequality involving four vectors. Namely, for any a,b,c,d∈^Rn , it holds that [figure omitted; refer to PDF] It is a new generalized version of the Cauchy-Schwarz inequality. Recently, the result was refined by Choi [7] to a stronger one: [figure omitted; refer to PDF] for any _a(k) ,_b(k) ∈^Rn , k=1,...,m, and m≥2. Furthermore, he also gained the complex version of the inequality. Since the articles are not so difficult to understand, they are more important for undergraduate students to study.

To meet the need for the teaching, we will consider in this paper the counterparts of (4) and extend them into the following: [figure omitted; refer to PDF] for 2m real-valued functions {_fk ,_hk ∈^C0 ((a,b),R)|"k=1,...,m, m≥2} (Theorem 2). Instead of the Euclidean norm on ^Rn or ^Cn used in [6, 7], here the norm and inner product of functions are defined by [figure omitted; refer to PDF] In this work, we further give a necessary and sufficient condition if the equality holds in (5). The complex version of inequality (5) is also obtained (Theorem 4). The whole proof is direct as in [6, 7], but the calculations have to be made with some adjustments in the integral case.

2. The Real Case of the Cauchy-Schwarz Inequality

Theorem 1.

Let e(x), f(x), g(x), and h(x) be four continuous real-valued functions on [a,b]. Then [figure omitted; refer to PDF] where the equality holds if and only if e(x)f(y)-e(y)f(x)=±(g(x)h(y)-g(y)h(x)) for all x,y∈(a,b).

Proof.

Notice that ^(e)2 =^∫ab e^(x)2 dx[...],^(f)2 =^∫ab f^(x)2 dx[...] and (e,f)=^∫ab e(x)f(x)dx[...]. Then we have [figure omitted; refer to PDF] where D={(x,y)∈^R2 |"a<=x,y<=b}. Using the formula above, we obtain [figure omitted; refer to PDF] which gives the desired inequality. The condition for the equality is obvious. This finishes the proof.

Theorem 2.

Let _f1 ,...,_fm , _h1 ,...,_hm be 2m continuous real-valued functions. Then, for m≥2, one has [figure omitted; refer to PDF] where the equality holds if and only if _fk (x)_hk (y)-_fk (y)_hk (x)=±(_fl (x)_hl (y)-_fl (y)_hl (x)) for all k,l=1,...,m and all x,y∈(a,b).

Proof.

It follows from Theorem 1 that [figure omitted; refer to PDF] for 1<=k<l<=m. Thus, we have [figure omitted; refer to PDF]

3. The Complex Case of the Cauchy-Schwarz Inequality

In this section, we consider the complex case. The norm and the inner product of complex-valued functions are defined by [figure omitted; refer to PDF] First, we give the following.

Theorem 3.

Let e, f, g, and h be four continuous complex-valued functions on [a,b]. Then [figure omitted; refer to PDF] where the equality holds if and only if e(x)f(y)-e(y)f(x)=±(g(x)h(y)-g(y)h(x)) for all x,y∈(a,b).

Proof.

Note that e and f are complex-valued functions. By using the above definition on ^(e)2 ,^(f)2 , and (e,f), we have [figure omitted; refer to PDF] where D={(x,y)∈^R2 |"a<=x,y<=b}. Therefore, [figure omitted; refer to PDF] Then the inequality follows, as required.

By a similar argument, we further obtain the following.

Theorem 4.

Let _f1 ,...,_fm ,_h1 ,...,_hm be 2m continuous complex-valued functions on [a,b]. Then, for m≥2, one has [figure omitted; refer to PDF] where the equality holds if and only if _fk (x)_hk (y)-_fk (y)_hk (x)=±(_fl (x)_hl (y)-_fl (y)_hl (x)) for all k,l=1,...,m and all x,y∈(a,b).

Remark 5.

(i) In this paper, we study the Cauchy-Schwarz inequality by using the real or complex valued functions defined on an interval (a,b). Indeed, it holds for any other continuous functions defined on a measurable set.

(ii) To prove the results, we follow the arguments from [6, 7]. We remark that, by using the Gramian matrix G defined by [figure omitted; refer to PDF] we can also prove Theorem 1. In fact, since G=(ABBC)≥0, it holds that (det Adet Bdet Bdet C)≥0, from which Theorem 1 follows. Further, using 2n×2n Gramian matrix, we generalized the result to Theorem 2. Complex cases are also derived in the same way.

Acknowledgments

This work is supported by AHNSF (1608085MA03) and TLXYRC (2015tlxyrc09).

References

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[3] V. Y. Buniakowski, "Sur quelques inegalites concernant les integrales aux differences finies,", Mémoires de l'Académie impériale des sciences de St. Pétersbourg , vol. 1, no. 9, pp. 1-18, 1859.

[4] S. S. Dragomir, "A survey on Cauchy-Bunyakovsky-Schwarz type discrete inequalities,", JIPAM. Journal of Inequalities in Pure and Applied Mathematics , vol. 4, no. 3, article 63, 2003.

[5] J. C. Kuang, Applied Inequalities , ShanDong Science and Technology Press, Jinan, China, 2004.

[6] N. J. Harvey, "A generalization of the Cauchy-Schwarz inequality involving four vectors,", Journal of Mathematical Inequalities , vol. 9, no. 2, pp. 489-491, 2015.

[7] D. Choi, "A generalization of the Cauchy-Schwarz inequality,", Journal of Mathematical Inequalities , vol. 10, no. 4, pp. 1009-1012, 2016.