# A New Generalization on Cauchy-Schwarz Inequality

Authors: Songting Yin
Journal: Journal of Function Spaces
 (2017)

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## 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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## Body

## 1. Introduction

Leta=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]) (1)∑i=1nai2∑i=1nbi2≥∑i=1naibi2,and its integral representation in the space of the continuous real-valued functions Ca,b,R reads (see [2, 3]) (2)∫abfx2dx∫abgx2dx≥∫abfxgxdx2.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 (3)a2b2+c2d2≥2aTcbTd+aTb2+cTd2-aTd2-bTc2. It is a new generalized version of the Cauchy-Schwarz inequality. Recently, the result was refined by Choi [7] to a stronger one:(4)∑k=1mak2bk2≥∑k=1makTbk2+2m-1∑1≤k<l≤makTalbkTbl-akTblbkTalfor any ak,bk∈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:(5)∑k=1mfk2hk2≥∑k=1mfk,hk2+2m-1∑1≤k<l≤mfk,flhk,hl-fk,hlhk,flfor 2m real-valued functions {fk,hk∈C0a,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 (6)f2≕∫abfx2dx,f,h≕∫abfxhxdx. 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(7)e2f2+g2h2≥e,f2+g,h2+2e,gf,h-e,hf,g, where the equality holds if and only if exfy-eyfx=±gxhy-gyhx for all x,y∈a,b.Proof.
Notice thate2=∫abex2dx‍,f2=∫abfx2dx‍ and e,f=∫abexfxdx‍. Then we have(8)e2f2-e,f2=∫abex2dx×∫abfx2dx-∫abexfxdx2=∫abex2dx∫abfy2dy-∫abexfxdx∫abeyfydy=∬Dex2fy2-exfxeyfydxdy=12∬Dexfy-eyfx2dxdy, where D={x,y∈R2∣a≤x,y≤b}. Using the formula above, we obtain(9)e2f2+g2h2-e,f2-g,h2=12∬Dexfy-eyfx2+gxhy-gyhx2dxdy=12∬Dexfy-eyfx±gxhy-gyhx2∓2exfy-eyfxgxhy-gyhxdxdy≥∓∬Dexfy-eyfxgxhy-gyhxdxdy=∓2∫abexgxdx∫abfyhydy-∫abexhxdx∫abfygydy=∓2e,gf,h-e,hf,g,which gives the desired inequality. The condition for the equality is obvious. This finishes the proof.Theorem 2.
Letf1,…,fm,  h1,…,hm be 2m continuous real-valued functions. Then, for m≥2, one has(10)∑k=1mfk2hk2≥∑k=1mfk,hk2+2m-1∑1≤k<l≤mfk,flhk,hl-fk,hlhk,fl, where the equality holds if and only if fkxhk(y)-fkyhk(x)=±flxhl(y)-flyhl(x) for all k,l=1,…,m and all x,y∈a,b.Proof.
It follows from Theorem1 that(11)fk2hk2+fl2hl2-fk,hk2-fl,hl2≥2fk,flhk,hl-fk,hlhk,fl for 1≤k<l≤m. Thus, we have(12)∑k=1mfk2hk2-fk,hk2=1m-1∑1≤k<l≤mfk2hk2+fl2hl2-fk,hk2-fl,hl2≥2m-1∑1≤k<l≤mfk,flhk,hl-fk,hlhk,fl.


## 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(13)f2≕∫abfxfx¯dx,f,h≕∫abfxhx¯dx,f,h∈C0a,b,C. First, we give the following.Theorem 3.
Lete, f, g, and h be four continuous complex-valued functions on [a,b]. Then (14)e2f2+g2h2≥e,f2+g,h2+2Ree,gf,h-e,hf,g, where the equality holds if and only if exfy-eyfx=±gxhy-gyhx for all x,y∈a,b.Proof.
Note thate and f are complex-valued functions. By using the above definition on e2,f2, and e,f, we have(15)e2f2-e,f2=∫abexex¯dx×∫abfxfx¯dx-∫abexfx¯dx2=∬Dexex¯fyfy¯-exfx¯eyfy¯dxdy=12∬Dexfy-eyfx2dxdy,where D={x,y∈R2∣a≤x,y≤b}. Therefore,(16)e2f2+g2h2-e,f2-g,h2=12∬Dexfy-eyfx2+gxhy-gyhx2dxdy=12∬Dexfy-eyfx±gxhy-gyhx2∓2Reexfy-eyfxgxhy-gyhx¯dxdy≥∓∬DReexfy-eyfxgxhy-gyhx¯dxdy=∓2Re∫abexgx¯dx∫abfyhy¯dy-∫abexhx¯dx∫abfygy¯dy=∓2Ree,gf,h-e,hf,g.Then the inequality follows, as required.By a similar argument, we further obtain the following.Theorem 4.
Letf1,…,fm,h1,…,hm be 2m continuous complex-valued functions on [a,b]. Then, for m≥2, one has(17)∑k=1mfk2hk2≥∑k=1mfk,hk2+2m-1∑1≤k<l≤mRefk,flhk,hl-fk,hlhk,fl, where the equality holds if and only if fkxhk(y)-fkyhk(x)=±flxhl(y)-flyhl(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 intervala,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(18)G=e,ee,fe,ge,hf,ef,ff,gf,hg,eg,fg,gg,hh,eh,fh,gh,h, we can also prove Theorem 1. In fact, since G=ABBC≥0, it holds that detAdetBdetBdetC≥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.



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*Source: 9576375-2017-06-13.xml*