A GENERALISATION OF WAGNER’S INEQUALITY FOR SEQUENCES OF VECTORS IN INNER PRODUCT SPACES

SEVER S. DRAGOMIR

Abstract. An extension of the Cauchy-Buniakowski-Schwartz inequality due to Wagner for sequences of vectors in inner product spaces is given.

1. Introduction

In 1965, S.S. Wagner [3] (see also [1] or [2, p. 85]) pointed out the following generalisation of Cauchy-Buniakowski-Schwartz’s inequality for real numbers.

Theorem 1. Let a = (a_{1}, ..., a_{n}) and b = (b_{1}, ..., b_{n}) be two n-tuples of real numbers. Then for any x ∈ [0, 1] , one has the inequality

(∑_{k=1}^{n} a_{k}b_{k} + x · ∑_{1≤i≠j≤n} a_{i}b_{j})^{2} (1.1)

≤ (∑_{k=1}^{n} a_{k}^{2} + 2x · ∑_{1≤i<j≤n} a_{i}a_{j}) · (∑_{k=1}^{n} b_{k}^{2} + 2x · ∑_{1≤i<j≤n} b_{i}b_{j}) .

For x = 0, we recapture the Cauchy-Buniakowski-Schwartz’s inequality, i.e., (see for example [2, p. 84])

(∑_{k=1}^{n} a_{k}b_{k})^{2} ≤ ∑_{k=1}^{n} a_{k}^{2} ∑_{k=1}^{n} b_{k}^{2},

with equality if and only if there exists a real number r so that a_{k} = rb_{k} for each k ∈ {1, ..., n}.

In this note we extend the above result for sequences of vectors in real or complex inner product spaces.

2. The Results

Let (H; ⟨., .⟩) be an inner product space over K, where K=R or K=C. The following result holds.

Date: January 21, 2003.

1991 Mathematics Subject Classification. Primary 26D15; Secondary 46C10.

Key words and phrases. Cauchy-Buniakowski-Schwartz’s Inequality, Wagner’s Inequality, Inner products spaces.

2 SEVER S. DRAGOMIR

Theorem 2. Let x = (x_{1}, ..., x_{n}) and y = (y_{1}, ..., y_{n}) be two n-tuples of vectors in H. Then for any α ∈ [0, 1] one has the inequality

[∑_{k=1}^{n} Re ⟨x_{k}, y_{k}⟩ + α · ∑_{1≤i≠j≤n} Re ⟨x_{i}, y_{j}⟩]^{2} (2.1)

≤ [∑_{k=1}^{n} ∥x_{k}∥^{2} + 2α · ∑_{1≤i<j≤n} Re ⟨x_{i}, x_{j}⟩] · [∑_{k=1}^{n} ∥y_{k}∥^{2} + 2α · ∑_{1≤i<j≤n} Re ⟨y_{i}, y_{j}⟩] .

Proof. Consider the function f : R → R, given by

(2.2)  f (t) = (1 − α) · ∑_{k=1}^{n} ∥tx_{k} − y_{k}∥^{2} + α · ∥∑_{k=1}^{n} (tx_{k} − y_{k})∥^{2} .

Then

(2.3) f (t) = [(1 − α) · ∑_{k=1}^{n} ∥x_{k}∥^{2} + α · ∥∑_{k=1}^{n} x_{k}∥^{2}] t^{2}

+2 [(1 − α) · ∑_{k=1}^{n} Re ⟨x_{k}, y_{k}⟩ + α · Re ⟨∑_{k=1}^{n} x_{k}, ∑_{k=1}^{n} y_{k}⟩] t

+ [(1 − α) · ∑_{k=1}^{n} ∥y_{k}∥^{2} + α · ∥∑_{k=1}^{n} y_{k}∥^{2}] .

Observe that

(2.4) ∥∑_{k=1}^{n} x_{k}∥^{2} = ∑_{k=1}^{n} ∥x_{k}∥^{2} + 2 · ∑_{1≤i<j≤n} Re ⟨x_{i}, x_{j}⟩

and

(2.5) ∥∑_{k=1}^{n} y_{k}∥^{2} = ∑_{k=1}^{n} ∥y_{k}∥^{2} + 2 · ∑_{1≤i<j≤n} Re ⟨y_{i}, y_{j}⟩ .

Also

(2.6)  Re ⟨∑_{k=1}^{n} x_{k}, ∑_{k=1}^{n} y_{k}⟩ = ∑_{k=1}^{n} Re ⟨x_{k}, y_{k}⟩ + ∑_{1≤i≠j≤n} Re ⟨x_{i}, y_{j}⟩ .

Using (2.3) − (2.6) , we deduce that

(2.7)  f (t) = [∑_{k=1}^{n} ∥x_{k}∥^{2} + 2α · ∑_{1≤i<j≤n} Re ⟨x_{i}, x_{j}⟩] t^{2}

+2 [∑_{k=1}^{n} Re ⟨x_{k}, y_{k}⟩ + α · ∑_{1≤i≠j≤n} Re ⟨x_{i}, y_{j}⟩] t

+ [∑_{k=1}^{n} ∥y_{k}∥^{2} + 2α · ∑_{1≤i<j≤n} Re ⟨y_{i}, y_{j}⟩] .

Since, by (2.2) , f (t) ≥ 0 for any t ∈ R, it follows that the discriminant of the quadratic function given by (2.7) is negative, which is clearly equivalent with the desired inequality (2.1) .

One may obtain an interesting inequality if x and y are assumed to incorporate orthogonal vectors. Corollary 1. Assume that {x_{i}}_{i=1,...,n} are orthogonal, i.e., x_{i} ⊥ x_{j} for any i, j ∈ {1, ..., n}, i ≠ j; and {y_{i}}_{i=1,...,n} are orthogonal as well in the real inner product space (H; ⟨., .⟩) . Then

(2.8)  sup_{α∈[0,1]} [∑_{k=1}^{n} ⟨x_{k}, y_{k}⟩ + α · ∑_{1≤i≠j≤n} ⟨x_{i}, y_{j}⟩]^{2} ≤ ∑_{k=1}^{n} ∥x_{k}∥^{2} ∑_{k=1}^{n} ∥y_{k}∥^{2} .

3. Applications

1. If we assume that H = C, with the inner product ⟨x, y⟩ = x · ȳ, then by (2.1) we may deduce the following Wagner’s type inequality for complex numbers

[∑_{k=1}^{n} Re (a_{k}b̄_{k}) + α · ∑_{1≤i≠j≤n} Re (a_{i}b̄_{j})]^{2} (3.1)

≤ [∑_{k=1}^{n} |a_{k}|^{2} + 2α · ∑_{1≤i<j≤n} Re (a_{i}ā_{j})] · [∑_{k=1}^{n} |b_{k}|^{2} + 2α · ∑_{1≤i<j≤n} Re (b_{i}b̄_{j})] ,

where α ∈ [0, 1] and a = (a_{1}, ..., a_{n}), b = (b_{1}, ..., b_{n}) ∈ C^{n}.

2. Consider the Hilbert space L_{2} (Ω, μ) := {f : Ω → C, ∫_{Ω} |f (x)|^{2} dμ (x) < ∞} where Ω is a μ-measurable space and μ : Ω → [0, ∞] is a positive measure on Ω. Then for H = L_{2} (Ω, μ) and since the inner product generating the norm is given by

⟨f, g⟩ = ∫_{Ω} f (x) ḡ (x) dμ (x) ,

we get the inequality

[∑_{k=1}^{n} ∫_{Ω} Re (f_{k} (x) ḡ_{k} (x)) dμ (x) + α · ∑_{1≤i≠j≤n} ∫_{Ω} Re (f_{i} (x) ḡ_{j} (x)) dμ (x)]^{2} (3.2)

≤ [∑_{k=1}^{n} ∫_{Ω} |f_{k} (x)|^{2} dμ (x) + 2α · ∑_{1≤i<j≤n} ∫_{Ω} Re (f_{i} (x) f̄_{j} (x)) dμ (x)]

× [∑_{k=1}^{n} ∫_{Ω} |g_{k} (x)|^{2} dμ (x) + 2α · ∑_{1≤i<j≤n} ∫_{Ω} Re (g_{i} (x) ḡ_{j} (x)) dμ (x)]

where f_{i}, g_{i} ∈ L_{2} (Ω, μ) , i ∈ {1, ..., n} and α ∈ [0, 1] .

References

[1] P. Flor, Über eine Ungleichung von S.S. Wagner, Elemente Math., 20(1965), 136.

[2] D.S. Mitrinović, J.E. Pečarić and A.M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht/Boston/London, 1993.

[3] S.S. Wagner, Amer. Math. Soc., Notices, 12(1965), 220.

School of Computer Science & Mathematics, Victoria University, PO Box 14428, Melbourne City MC, Victoria 8001, Australia

E-mail address: sever@matilda.vu.edu.au
URL: http://rgmia.vu.edu.au/SSDragomirWeb.html