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.
1. Introduction
Let and be two vectors in . Then the discrete version of Cauchy-Schwarz inequality is (see [1, 2]) and its integral representation in the space of the continuous real-valued functions reads (see [2, 3])
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 , it holds that It is a new generalized version of the Cauchy-Schwarz inequality. Recently, the result was refined by Choi [7] to a stronger one:for any , and . 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:for real-valued functions (Theorem 2). Instead of the Euclidean norm on or used in [6, 7], here the norm and inner product of functions are defined by 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 where the equality holds if and only if for all .
Proof. Notice that and . Then we have where . Using the formula above, we obtainwhich gives the desired inequality. The condition for the equality is obvious. This finishes the proof.
Theorem 2. Let , be 2m continuous real-valued functions. Then, for , one has where the equality holds if and only if for all and all .
Proof. It follows from Theorem 1 that for . Thus, we have
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 First, we give the following.
Theorem 3. Let , , , and be four continuous complex-valued functions on []. Then where the equality holds if and only if for all .
Proof. Note that and are complex-valued functions. By using the above definition on , and , we havewhere . Therefore,Then the inequality follows, as required.
By a similar argument, we further obtain the following.
Theorem 4. Let be 2m continuous complex-valued functions on []. Then, for , one has where the equality holds if and only if for all and all .
Remark 5. (i) In this paper, we study the Cauchy-Schwarz inequality by using the real or complex valued functions defined on an interval . 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 defined by we can also prove Theorem 1. In fact, since , it holds that , from which Theorem 1 follows. Further, using Gramian matrix, we generalized the result to Theorem 2. Complex cases are also derived in the same way.
Conflicts of Interest
The author declares that they have no conflicts of interest.
Acknowledgments
This work is supported by AHNSF (1608085MA03) and TLXYRC (2015tlxyrc09).