Abstract
In this paper, we give a generalization of Cauchy-Schwarz inequality in unitary spaces and obtain its integral analogs. As an application, we establish an inequality for covariances.
Journal of Inequalities and Applications volume 2015, Article number: 201 (2015)
In this paper, we give a generalization of Cauchy-Schwarz inequality in unitary spaces and obtain its integral analogs. As an application, we establish an inequality for covariances.
Let u and v be two vectors in a unitary space \(\mathbb{H}\). The Cauchy-Schwarz inequality is well known,
where \(\langle\cdot,\cdot\rangle\) and \(\|\cdot\|\) denote the inner product and norm in \(\mathbb{H}\), respectively. Its integral form in the space of real-valued functions \(L^{2}[a,b]\) is
The Cauchy-Schwarz inequality is one of the most important inequalities in mathematics. To date, a large number of generalizations and refinements of the inequalities (1.1) and (1.2) have been investigated in the literature (see [1] and references therein, also see [2–9]).
In this note, we will present some new generalizations of the Cauchy-Schwarz inequality (1.1).
Suppose that \(\mathbb{H}\) is a unitary space (complex inner product space) with standard inner product \(\langle\cdot,\cdot\rangle\) and norm \(\|\cdot\|\), namely \(\langle x,y\rangle=x^{T}\overline{y}\) and \(\|x\| =\sqrt{\langle x,x\rangle}\) (see [10]). Let \(X=( x_{1},x_{2},\ldots,x_{n})\) denote the n-tuple of vectors \(x_{i} \in \mathbb{H}\), \(i=1,\ldots,n\). For two n-tuples \(X=(x_{1},\ldots,x_{n})\) and \(Y=(y_{1},\ldots,y_{n})\) of \(\mathbb{H}\), we define this A-product of vector \(x_{i}\) and \(y_{i}\) for X and Y by
where \(a=\frac{x_{1}+\cdots+x_{n}}{n}\) and \(b=\frac{y_{1}+\cdots+y_{n}}{n}\).
Our main results are the following theorems.
Let \(X=( x_{1},\ldots,x_{n})\) and \(Y=(y_{1},\ldots ,y_{n})\) be two n-tuples of the unitary space \(\mathbb{H}\), then
Equality holds if \(y_{i}=(x_{i}-2a)\lambda\) (\(i=1,\ldots,n\)) for any \(\{ x_{1},\ldots, x_{n}\}\), where λ is a non-negative constant.
In particular, if \(n=1\), then (1.3) is the Cauchy-Schwarz inequality (1.1).
For complex numbers \(\mathbb{C}\), by Theorem 1, we have the following.
Suppose that \(x_{1},\ldots,x_{n}\) and \(y_{1},\ldots ,y_{n}\) are complex numbers. Set
then
Equality holds if \(y_{i}=(x_{i}-2a)\lambda\) (\(i=1,\ldots,n\)) for any \(\{x_{1},\ldots, x_{n}\}\), where λ is a non-negative constant.
Let \(H\oplus H\oplus\cdots\oplus H \) denote the direct sum of n unitary space \(\mathbb{H}\) with norm \(\|X\|= (\sum_{i=1}^{n}\|x_{i}\|^{2} )^{\frac{1}{2}}\). Set \(f(X,Y)=\sum_{i=1}^{n} x_{i}\otimes_{A}y_{i}\). Since \(f(X,X)\) is not always non-negative, \(f(X,Y)\) is not an inner product in the above direct sum. Hence, (1.3) is different from the Cauchy-Schwarz inequality in the above direct sum.
If we set \(|X\otimes_{A} Y|=\sum_{i=1}^{n}| x_{i}\otimes_{A}y_{i}|\), then (1.3) can be restated as
Furthermore, we obtain the following integral form of (1.3) (only consider real-valued functions).
Let μ be a positive measure such that \(\mu(\Omega)=1\), f and g be real-valued functions in \(L^{2}(\mu)\), and let
then
Equality holds if \(g(x)=(f(x)-2 \int_{\Omega}f \, d\mu)\lambda \), where λ is a non-negative constant.
Using the basic properties of the norm of a unitary space, we get
By (2.1), using the Cauchy-Schwarz inequality (1.1) and the discrete form of the Cauchy-Schwarz inequality, it follows that
Similarly to (2.2), we have
Combining (2.2) and (2.3), we infer that
This is the inequality (1.3), as desired. □
We first prove the following inequality:
In fact, by the Cauchy-Schwarz inequality (1.2), we obtain
This is the inequality (2.4).
Similarly, we have
From (2.4) and (2.5), we find that
The inequality (1.4) follows. □
Let \((a_{1},b_{1}),\ldots,(a_{n},b_{n}) \) be n items of bivariate real data, \(x=\{a_{1},\ldots,a_{n}\}\) and \(y=\{b_{1},\ldots,b_{n}\}\), then their covariance \(\operatorname{Cov}(x,y)\) is defined as [11]
where \(a=\frac{a_{1}+\cdots+a_{n}}{n}\) and \(b=\frac{b_{1}+\cdots+b_{n}}{n}\).
For the covariance \(\operatorname{Cov}(x,y)\), it is well known that Pearson’s product moment inequality is
where \(\operatorname{SD}(x)=\sqrt{\frac{1}{n}\sum_{i=1}^{n}(a_{i}-a)^{2}}\) and \(\operatorname{SD}(y)=\sqrt{\frac{1}{n}\sum_{i=1}^{n}(b_{i}-b)^{2}}\).
Similarly, now we define this covariance of two n-tuples \(X=( x_{1},x_{2},\ldots ,x_{n})\) and \(Y=(y_{1},y_{2},\ldots,y_{n})\) of the unitary space \(\mathbb{H}\) as
where \(a=\frac{x_{1}+\cdots+x_{n}}{n}\) and \(b=\frac{y_{1}+\cdots+y_{n}}{n}\).
Set \(\alpha_{i}=x_{i}-a\) and \(\beta_{i}=y_{i}-b\), \(i=1,\ldots,n\). Note that
Hence, (1.3) can be written in the following form:
where \(\|X\|= (\sum^{n}_{i=1}\|x_{i}\|^{2} )^{1/2}\) and \(\|Y\|= (\sum^{n}_{i=1}\|y_{i}\|^{2} )^{1/2}\).
Using the triangle inequality on the left side of (3.1), we obtain
Finally, we can simply state the above result, as follows.
Let \(X=( x_{1},x_{2},\ldots,x_{n})\) and \(Y=(y_{1},y_{2},\ldots,y_{n})\) be two n-tuples of the unitary space \(\mathbb {H}\), then
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This work was partially supported by the National Natural Science Foundation of China (Grant No. 11171053).
The author declares to have no competing interests.
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Tuo, L. Generalizations of Cauchy-Schwarz inequality in unitary spaces. J Inequal Appl 2015, 201 (2015). https://doi.org/10.1186/s13660-015-0719-z
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DOI: https://doi.org/10.1186/s13660-015-0719-z