GENERALISATIONS OF PRECUPANU’S INEQUALITY FOR

ORTHORNORMAL FAMILIES OF VECTORS IN INNER PRODUCT SPACES

S.S. DRAGOMIR

Abstract. Some generalisations of Precupanu’s inequality for orthornormal families of vectors in real or complex inner product spaces and applications related to Buzano’s, Richard’s and Kurepa’s results are given.

1. Introduction

In 1976, T. Precupanu [6] obtained the following result related to the Schwarz inequality in a real inner product space (H; ⟨·, ·⟩) :

Theorem 1. For any a ∈ H, x, y ∈ H\ {0} , we have the inequality:

(1.1)  \frac{− ∥a∥ ∥b∥ + ⟨a, b⟩}{2} ≤ \begin{matrix}⟨x, a⟩ & ⟨x, b⟩ \\ ∥x∥ & 2 \end{matrix} + \begin{matrix}⟨y, a⟩ & ⟨y, b⟩ \\ ∥y∥ & 2 \end{matrix} − 2 · \begin{matrix}⟨x, a⟩ & ⟨y, b⟩ & ⟨x, y⟩ \\ ∥x∥ & 2 & ∥y∥ & 2 \end{matrix}

≤ \frac{∥a∥ ∥b∥ + ⟨a, b⟩}{2}.

In the right-hand side or in the left-hand side of (1.1) we have equality if and only if there are λ, μ ∈ ℝ such that

(1.2)  λ\frac{⟨x, a⟩}{∥x∥^{2}} · x + μ\frac{⟨y, b⟩}{∥y∥^{2}} · y = \frac{1}{2} (λa + μb) .

Note for instance that [6], if y ⊥ b, i.e., ⟨y, b⟩ = 0, then by (1.1) one may deduce:

(1.3)  \frac{− ∥a∥ ∥b∥ + ⟨a, b⟩}{2} ∥x∥^{2} ≤ ⟨x, a⟩ ⟨x, b⟩ ≤ \frac{∥a∥ ∥b∥ + ⟨a, b⟩}{2}

for any a, b, x ∈ H, an inequality that has been obtained previously by U. Richard [7]. The case of equality in the right-hand side or in the left-hand side of (1.3) holds if and only if there are λ, μ ∈ ℝ with

(1.4)  2λ ⟨x, a⟩ x = (λa + μb) ∥x∥^{2} .

For a = b, we may obtain from (1.1) the following inequality [6]

(1.5)  0 ≤ \begin{matrix}⟨x, a⟩ & 2 \\ ∥x∥ & 2 \end{matrix} + \begin{matrix}⟨y, a⟩ & 2 \\ ∥y∥ & 2 \end{matrix} − 2 · \begin{matrix}⟨x, a⟩ & ⟨y, b⟩ & ⟨x, y⟩ \\ ∥x∥ & 2 & ∥y∥ & 2 \end{matrix} ≤ ∥a∥^{2} .

Date: November 10, 2004.

1991 Mathematics Subject Classification. 46C05, 46C09, 26D15.

Key words and phrases. Schwarz’s inequality, Buzano’s inequality, Kurepa’s inequality, Richard’s inequality, Precupanu’s inequality, Inner products.

This inequality implies [6]:

(1.6)  \frac{⟨x, y⟩}{∥x∥ ∥y∥} ≥ \frac{1}{2} [\frac{⟨x, a⟩}{∥x∥ ∥a∥} + \frac{⟨y, a⟩}{∥y∥ ∥a∥}]^{2} − \frac{3}{2}.

In [5], M.H. Moore pointed out the following reverse of the Schwarz inequality

(1.7)  |⟨y, z⟩| ≤ ∥y∥ ∥z∥ ,  y, z ∈ H,

where some information about a third vector x is known:

Theorem 2. Let (H; ⟨·, ·⟩) be an inner product space over the real field ℝ and x, y, z ∈ H such that:

(1.8)  |⟨x, y⟩| ≥ (1 − ε) ∥x∥ ∥y∥ ,  |⟨x, z⟩| ≥ (1 − ε) ∥x∥ ∥z∥ ,

where ε is a positive real number, reasonably small. Then

(1.9)  |⟨y, z⟩| ≥ max \{1 − ε − \sqrt{2ε}, 1 − 4ε, 0\} ∥y∥ ∥z∥ .

Utilising Richard’s inequality (1.3) written in the following equivalent form:

(1.10)  2 · \begin{matrix}⟨x, a⟩ & ⟨x, b⟩ \\ ∥x∥ & 2 \end{matrix} − ∥a∥ ∥b∥ ≤ ⟨a, b⟩ ≤ 2 · \begin{matrix}⟨x, a⟩ & ⟨x, b⟩ \\ ∥x∥ & 2 \end{matrix} + ∥a∥ ∥b∥

for any a, b ∈ H and a ∈ H\ {0} , Precupanu has obtained the following Moore’s type result:

Theorem 3. Let (H; ⟨·, ·⟩) be a real inner product space. If a, b, x ∈ H and 0 < ε_{1} < ε_{2} are such that:

(1.11)  ε_{1} ∥x∥ ∥a∥ ≤ ⟨x, a⟩ ≤ ε_{2} ∥x∥ ∥a∥ ,
ε_{1} ∥x∥ ∥b∥ ≤ ⟨x, b⟩ ≤ ε_{2} ∥x∥ ∥b∥ ,

then

(1.12)  (2ε_{1}^{2} − 1) ∥a∥ ∥b∥ ≤ ⟨a, b⟩ ≤ (2ε_{1}^{2} + 1) ∥a∥ ∥b∥ .

Remark that the right inequality is always satisfied, since by Schwarz’s inequality, we have ⟨a, b⟩ ≤ ∥a∥ ∥b∥. The left inequality may be useful when one assumes that ε_{1} ∈ (0, 1]. In that case, from (1.12), we obtain

(1.13)  − ∥a∥ ∥b∥ ≤ (2ε_{1}^{2} − 1) ∥a∥ ∥b∥ ≤ ⟨a, b⟩

provided ε_{1} ∥x∥ ∥a∥ ≤ ⟨x, a⟩ and ε_{1} ∥x∥ ∥b∥ ≤ ⟨x, b⟩ , which is a refinement of Schwarz’s inequality

− ∥a∥ ∥b∥ ≤ ⟨a, b⟩ .

In the complex case, apparently independent of Richard, M.L. Buzano obtained in [2] the following inequality

(1.14)  |⟨x, a⟩ ⟨x, b⟩| ≤ \frac{∥a∥ ∥b∥ + |⟨a, b⟩|}{2} · ∥x∥^{2} ,

provided x, a, b are vectors in the complex inner product space (H; ⟨·, ·⟩) .

In the same paper [6], Precupanu, without mentioning Buzano’s name in relation to the inequality (1.14), observed that, on utilising (1.14), one may obtain the following result of Moore type:

Theorem 4. Let (H; ⟨·, ·⟩) be a (real or) complex inner product space. If x, a, b ∈ H are such that

(1.15)  |⟨x, a⟩| ≥ (1 − ε) ∥x∥ ∥a∥ ,  |⟨x, b⟩| ≥ (1 − ε) ∥x∥ ∥b∥ ,

then

(1.16)  |⟨a, b⟩| ≥ (1 − 4ε + 2ε^{2}) ∥a∥ ∥b∥ .

Note that the above theorem is useful when, for ε ∈ (0, 1], the quantity 1 − 4ε + 2ε^{2} > 0, i.e., ε ∈ (0, 1 − \frac{\sqrt{2}}{2}] .

Remark 1. When the space is real, the inequality (1.16) provides a better lower bound for |⟨a, b⟩| than the second bound in Moore’s result (1.9). However, it is not known if the first bound in (1.9) remains valid for the case of complex spaces. From Moore’s original proof, apparently, the fact that the space (H; ⟨·, ·⟩) is real plays an essential role.

Before we point out some new results for orthonormal families of vectors in real or complex inner product spaces, we state the following result that complements the Moore type results outlined above for real spaces:

Theorem 5. Let (H; ⟨·, ·⟩) be a real inner product space and a, b, x, y ∈ H\ {0} . (i) If there exist δ_{1}, δ_{2} ∈ (0, 1] such that

\frac{⟨x, a⟩}{∥x∥ ∥a∥} ≥ δ_{1}, \frac{⟨y, a⟩}{∥y∥ ∥a∥} ≥ δ_{2}

and δ_{1} + δ_{2} ≥ 1, then
(1.17)  \frac{⟨x, y⟩}{∥x∥ ∥y∥} ≥ \frac{1}{2} (δ_{1} + δ_{2})^{2} − \frac{3}{2}  (≥ −1) .
(ii) If there exist μ_{1} (μ_{2}) ∈ ℝ such that

μ_{1} ∥a∥ ∥b∥ ≤ \begin{matrix}⟨x, a⟩ & ⟨x, b⟩ \\ ∥x∥ & 2 \end{matrix} (≤ μ_{2} ∥a∥ ∥b∥)

and 1 ≥ μ_{1} ≥ 0 (−1 ≤ μ_{2} ≤ 0) , then
(1.18)  [−1 ≤] 2μ_{1} − 1 ≤ \frac{⟨a, b⟩}{∥a∥ ∥b∥} (≤ 2μ_{2} + 1 [≤ 1]) .

The proof is obvious by the inequalities (1.6) and (1.10). We omit the details.

2. Inequalities for orthonormal Families

We recall that the finite family {e_{i}}_{i∈I} is orthonormal in (H; ⟨·, ·⟩) , a real or complex inner product space, if

⟨e_{i}, e_{j}⟩ = \{ \begin{matrix}0 & if i ≠ j \\ 1 & if i = j \end{matrix}

where i, j ∈ I.

The following result may be stated.

Theorem 6. Let {e_{i}}_{i∈I} and {f_{j}}_{j∈J} be two finite families of orthonormal vectors in (H; ⟨·, ·⟩) . For any x, y ∈ H\ {0} one has the inequality

(2.1) |∑_{i∈I} ⟨x, e_{i}⟩ ⟨e_{i}, y⟩ + ∑_{j∈J} ⟨x, f_{j}⟩ ⟨f_{j}, y⟩

− 2 ∑_{i∈I,j∈J} ⟨x, e_{i}⟩ ⟨f_{j}, y⟩ ⟨e_{i}, f_{j}⟩ − \frac{1}{2} ⟨x, y⟩| ≤ \frac{1}{2} ∥x∥ ∥y∥ .

The case of equality holds in (2.1) if and only if there exists a λ ∈ 𝕂 such that

(2.2)  x − λy = 2 (∑_{i∈I} ⟨x, e_{i}⟩ e_{i} − λ ∑_{j∈J} ⟨y, f_{j}⟩ f_{j}) .

Proof. We know that, if u, v ∈ H, v ≠ 0, then

(2.3)  ∥u − \frac{⟨u, v⟩}{∥v∥^{2}} · v∥^{2} = \frac{∥u∥^{2} ∥v∥^{2} − |⟨u, v⟩|^{2}}{∥v∥^{2}}

showing that, in Schwarz’s inequality

(2.4)  |⟨u, v⟩|^{2} ≤ ∥u∥^{2} ∥v∥^{2} ,

the case of equality, for v ≠ 0, holds if and only if

(2.5)  u = \frac{⟨u, v⟩}{∥v∥^{2}} · v,

i.e. there exists a λ ∈ ℝ such that u = λv.

Now, let u := 2 ∑_{i∈I} ⟨x, e_{i}⟩ e_{i} − x and v := 2 ∑_{j∈J} ⟨y, f_{j}⟩ f_{j} − y.

Observe that

∥u∥^{2} = ∥2 ∑_{i∈I} ⟨x, e_{i}⟩ e_{i}∥^{2} − 4 Re ⟨∑_{i∈I} ⟨x, e_{i}⟩ e_{i}, x⟩ + ∥x∥^{2}

= 4 ∑_{i∈I} |⟨x, e_{i}⟩|^{2} − 4 ∑_{i∈I} |⟨x, e_{i}⟩|^{2} + ∥x∥^{2} = ∥x∥^{2} ,

and, similarly

∥v∥^{2} = ∥y∥^{2} .

Also,

⟨u, v⟩ = 4 ∑_{i∈I,j∈J} ⟨x, e_{i}⟩ ⟨f_{j}, y⟩ ⟨e_{i}, f_{j}⟩ + ⟨x, y⟩

− 2 ∑_{i∈I} ⟨x, e_{i}⟩ ⟨e_{i}, y⟩ − 2 ∑_{j∈J} ⟨x, f_{j}⟩ ⟨f_{j}, y⟩ .

Therefore, by Schwarz’s inequality (2.4) we deduce the desired inequality (2.1). By (2.5), the case of equality holds in (2.1) if and only if there exists a λ ∈ 𝕂 such that

2 ∑_{i∈I} ⟨x, e_{i}⟩ e_{i} − x = λ (2 ∑_{j∈J} ⟨y, f_{j}⟩ f_{j} − y) ,

which is equivalent to (2.2).

Remark 2. If in (2.2) we choose x = y, then we get the inequality:

(2.6) |∑_{i∈I} |⟨x, e_{i}⟩|^{2} + ∑_{j∈J} |⟨x, f_{j}⟩|^{2} − 2 ∑_{i∈I,j∈J} ⟨x, e_{i}⟩ ⟨f_{j}, x⟩ ⟨e_{i}, f_{j}⟩ − \frac{1}{2} ∥x∥^{2}|

≤ \frac{1}{2} ∥x∥^{2}

for any x ∈ H.

If in the above theorem we assume that I = J and f_{i} = e_{i}, i ∈ I, then we get from (2.2) the Schwarz inequality |⟨x, y⟩| ≤ ∥x∥ ∥y∥ .

If I ∩ J = ∅, I ∪ J = K, g_{k} = e_{k}, k ∈ I, g_{k} = f_{k}, k ∈ J and {g_{k}}_{k∈K} is orthonormal, then from (2.1) we get:

(2.7)  |∑_{k∈K} ⟨x, g_{k}⟩ ⟨g_{k}, y⟩ − \frac{1}{2} ⟨x, y⟩| ≤ \frac{1}{2} ∥x∥ ∥y∥ ,  x, y ∈ H

which has been obtained earlier by the author in [3].

If I and J reduce to one element, namely e_{1} = \frac{e}{∥e∥}, f_{1} = \frac{f}{∥f∥} with e, f ≠ 0, then from (2.1) we get

(2.8) |\begin{matrix}⟨x, e⟩ & ⟨e, y⟩ \\ ∥e∥ & 2 \end{matrix} + \begin{matrix}⟨x, f⟩ & ⟨f, y⟩ \\ ∥f∥ & 2 \end{matrix} − 2 · \begin{matrix}⟨x, e⟩ & ⟨f, y⟩ & ⟨e, f⟩ \\ ∥e∥ & 2 & ∥f∥ & 2 \end{matrix} − \frac{1}{2} ⟨x, y⟩|

≤ \frac{1}{2} ∥x∥ ∥y∥ , x, y ∈ H

which is the corresponding complex version of Precupanu’s inequality (1.1). If in (2.8) we assume that x = y, then we get

(2.9)  |\begin{matrix}|⟨x, e⟩| & 2 \\ ∥e∥ & 2 \end{matrix} + \begin{matrix}|⟨x, f⟩| & 2 \\ ∥f∥ & 2 \end{matrix} − 2 · \begin{matrix}⟨x, e⟩ & ⟨f, e⟩ & ⟨e, f⟩ \\ ∥e∥ & 2 & ∥f∥ & 2 \end{matrix} − \frac{1}{2} ∥x∥^{2}| ≤ \frac{1}{2} ∥x∥^{2} .

The following corollary may be stated:

Corollary 1. With the assumptions of Theorem 6, we have:

|∑_{i∈I} ⟨x, e_{i}⟩ ⟨e_{i}, y⟩ + ∑_{j∈J} ⟨x, f_{j}⟩ ⟨f_{j}, y⟩ − 2 ∑_{i∈I,j∈J} ⟨x, e_{i}⟩ ⟨f_{j}, y⟩ ⟨e_{i}, f_{j}⟩| (2.10)

≤ \frac{1}{2} |⟨x, y⟩| + |∑_{i∈I} ⟨x, e_{i}⟩ ⟨e_{i}, y⟩ + ∑_{j∈J} ⟨x, f_{j}⟩ ⟨f_{j}, y⟩

− 2 ∑_{i∈I,j∈J} ⟨x, e_{i}⟩ ⟨f_{j}, y⟩ ⟨e_{i}, f_{j}⟩ − \frac{1}{2} |⟨x, y⟩||

≤ \frac{1}{2} [|⟨x, y⟩| + ∥x∥ ∥y∥] .

Proof. The first inequality follows by the triangle inequality for the modulus. The second inequality follows by (2.1) on adding the quantity \frac{1}{2} |⟨x, y⟩| on both sides.

Remark 3. (1) If we choose in (2.10), x = y, then we get:

|∑_{i∈I} |⟨x, e_{i}⟩|^{2} + ∑_{j∈J} |⟨x, f_{j}⟩|^{2} − 2 ∑_{i∈I,j∈J} ⟨x, e_{i}⟩ ⟨f_{j}, x⟩ ⟨e_{i}, f_{j}⟩| (2.11)

≤ |∑_{i∈I} |⟨x, e_{i}⟩|^{2} + ∑_{j∈J} |⟨x, f_{j}⟩|^{2}

− 2 ∑_{i∈I,j∈J} ⟨x, e_{i}⟩ ⟨f_{j}, x⟩ ⟨e_{i}, f_{j}⟩ − \frac{1}{2} ∥x∥^{2}| + \frac{1}{2} ∥x∥^{2}

≤ ∥x∥^{2} .

We observe that (2.11) will generate Bessel’s inequality if {e_{i}}_{i∈I} , {f_{j}}_{j∈J} are disjoint parts of a larger orthonormal family.

(2) From (2.8) one can obtain:

(2.12) |\begin{matrix}⟨x, e⟩ & ⟨e, y⟩ \\ ∥e∥ & 2 \end{matrix} + \begin{matrix}⟨x, f⟩ & ⟨f, y⟩ \\ ∥f∥ & 2 \end{matrix} − 2 · \begin{matrix}⟨x, e⟩ & ⟨f, y⟩ & ⟨e, f⟩ \\ ∥e∥ & 2 & ∥f∥ & 2 \end{matrix} | ≤ \frac{1}{2} [∥x∥ ∥y∥ + |⟨x, y⟩|]

and in particular

(2.13)  |\begin{matrix}|⟨x, e⟩| & 2 \\ ∥e∥ & 2 \end{matrix} + \begin{matrix}|⟨x, f⟩| & 2 \\ ∥f∥ & 2 \end{matrix} − 2 · \begin{matrix}⟨x, e⟩ & ⟨f, e⟩ & ⟨e, f⟩ \\ ∥e∥ & 2 & ∥f∥ & 2 \end{matrix} | ≤ ∥x∥^{2} ,

for any x, y ∈ H.

The case of real inner products will provide a natural generalisation for Precupanu’s inequality (1.1):

Corollary 2. Let (H; ⟨·, ·⟩) be a real inner product space and {e_{i}}_{i∈I} , {f_{j}}_{j∈J} two finite families of orthonormal vectors in (H; ⟨·, ·⟩) . For any x, y ∈ H\ {0} one has the double inequality:

(2.14) \frac{1}{2} [|⟨x, y⟩| − ∥x∥ ∥y∥]

≤ ∑_{i∈I} ⟨x, e_{i}⟩ ⟨y, e_{i}⟩ + ∑_{j∈J} ⟨x, f_{j}⟩ ⟨y, f_{j}⟩ − 2 ∑_{i∈I,j∈J} ⟨x, e_{i}⟩ ⟨y, f_{j}⟩ ⟨e_{i}, f_{j}⟩

≤ \frac{1}{2} [∥x∥ ∥y∥ + |⟨x, y⟩|] .

In particular, we have

(2.15)  0 ≤ ∑_{i∈I} ⟨x, e_{i}⟩^{2} + ∑_{j∈J} ⟨x, f_{j}⟩^{2} − 2 ∑_{i∈I,j∈J} ⟨x, e_{i}⟩ ⟨x, f_{j}⟩ ⟨e_{i}, f_{j}⟩
≤ ∥x∥^{2} ,

for any x ∈ H.

Remark 4. Similar particular inequalities to those incorporated in (2.7) – (2.13) may be stated, but we omit them.

3. Refinements of Kurepa’s Inequality

Let (H; ⟨·, ·⟩) be a real inner product space generating the norm ∥·∥ . The complexification H_{ℂ} of H is defined as a complex linear space H × H of all ordered pairs (x, y) , x, y ∈ H endowed with the operations:

(x, y) + (x′, y′) := (x + x′, y + y′) ,  x, x′, y, y′ ∈ H;
(σ + iτ) · (x, y) := (σx − τy, τx + σy) ,  x, y ∈ H and σ, τ ∈ ℝ.

On H_{ℂ} := H × H, endowed with the above operations, one can now canonically define the scalar product ⟨·, ·⟩_{ℂ} by:

(3.1)  ⟨z, z′⟩_{ℂ} := ⟨x, x′⟩ + ⟨y, y′⟩ + i [⟨x′, y⟩ − ⟨x, y′⟩]

where z = (x, y) , z′ = (x′, y′) ∈ H_{ℂ}. Obviously,

∥z∥_{ℂ}^{2} = ∥x∥^{2} + ∥y∥^{2} , z = (x, y) ∈ H_{ℂ}.

One can also define the conjugate of a vector z = (x, y) by z̄:= (x, −y) . It is easy to see that, the elements of H_{ℂ}, under defined operations, behave as formal “complex” combinations x+iy with x, y ∈ H. Because of this, we may write z = x+iy instead of z = (x, y) . Thus, z̄ = x − iy.

Under this setting, S. Kurepa [4] proved the following refinement of Schwarz’s inequality:

(3.2)  |⟨a, z⟩_{ℂ}|^{2} ≤ \frac{1}{2} ∥a∥^{2} [∥z∥_{ℂ}^{2} + |⟨z, z̄⟩_{ℂ}|] ≤ ∥a∥^{2} ∥z∥_{ℂ}^{2} ,

for any a ∈ H and z ∈ H_{ℂ}.

This was motivated by generalising the de Bruijn result for sequences of real and complex numbers obtained in [1].

The following result may be stated.

Theorem 7. Let (H; ⟨·, ·⟩) be a real inner product space and {e_{i}}_{i∈I} , {f_{j}}_{j∈J} two finite families in H. If (H_{ℂ}; ⟨·, ·⟩_{ℂ}) is the complexification of (H; ⟨·, ·⟩) , then for any w ∈ H_{ℂ}, we have the inequalities

|∑_{i∈I} ⟨w, e_{i}⟩_{ℂ}^{2} + ∑_{j∈J} ⟨w, f_{j}⟩_{ℂ}^{2} − 2 ∑_{i∈I,j∈J} ⟨w, e_{i}⟩_{ℂ} ⟨w, f_{j}⟩_{ℂ} ⟨e_{i}, f_{j}⟩| (3.3)

≤ \frac{1}{2} |⟨w, w̄⟩_{ℂ}| + |∑_{i∈I} ⟨w, e_{i}⟩_{ℂ}^{2} + ∑_{j∈J} ⟨w, f_{j}⟩_{ℂ}^{2}

− 2 ∑_{i∈I,j∈J} ⟨w, e_{i}⟩_{ℂ} ⟨w, f_{j}⟩_{ℂ} ⟨e_{i}, f_{j}⟩ − \frac{1}{2} ⟨w, w̄⟩_{ℂ}|

≤ \frac{1}{2} [∥w∥_{ℂ}^{2} + |⟨w, w̄⟩_{ℂ}|] ≤ ∥w∥_{ℂ}^{2} .

Proof. Define g_{j} ∈ H_{ℂ}, g_{j} := (e_{j}, 0) , j ∈ I. For any k, j ∈ I we have

⟨g_{k}, g_{j}⟩_{ℂ} = ⟨(e_{k}, 0) , (e_{j}, 0)⟩_{ℂ} = ⟨e_{k}, e_{j}⟩ = δ_{kj},

therefore {g_{j}}_{j∈I} is an orthonormal family in (H_{ℂ}; ⟨·, ·⟩_{ℂ}) .

If we apply Corollary 1 for (H_{ℂ}; ⟨·, ·⟩_{ℂ}) , x = w, y = w̄, we may write:

|∑_{i∈I} ⟨w, e_{i}⟩_{ℂ} ⟨e_{i}, w̄⟩_{ℂ} + ∑_{j∈J} ⟨w, f_{j}⟩ ⟨f_{j}, w̄⟩ (3.4)

− 2 ∑_{i∈I,j∈J} ⟨w, e_{i}⟩_{ℂ} ⟨f_{j}, \overline{w}⟩_{ℂ} ⟨e_{i}, f_{j}⟩|

≤ \frac{1}{2} ∥w∥_{ℂ} \left∥{ \atop w̄}\right∥_{ℂ} + |∑_{i∈I} ⟨w, e_{i}⟩_{ℂ} ⟨e_{i}, w̄⟩_{ℂ} + ∑_{j∈J} ⟨w, f_{j}⟩ ⟨f_{j}, w̄⟩

− 2 ∑_{i∈I,j∈J} ⟨w, e_{i}⟩_{ℂ} ⟨f_{j}, \overline{w}⟩_{ℂ} ⟨e_{i}, f_{j}⟩ − \frac{1}{2} ⟨w, w̄⟩_{ℂ}|

≤ \frac{1}{2} [|⟨w, w̄⟩_{ℂ}| + ∥w∥_{ℂ} \left∥{ \atop w̄}\right∥_{ℂ}] .

However, for w := (x, y) ∈ H_{ℂ}, we have w̄ = (x, −y) and

⟨e_{j}, w̄⟩_{ℂ} = ⟨(e_{j}, 0) , (x, −y)⟩_{ℂ} = ⟨e_{j}, x⟩ + i ⟨e_{j}, y⟩

and

⟨w, e_{j}⟩_{ℂ} = ⟨(x, y) , (e_{j}, 0)⟩_{ℂ} = ⟨x, e_{j}⟩ + i ⟨e_{j}, y⟩

showing that ⟨e_{j}, w̄⟩ = ⟨w, e_{j}⟩ for any j ∈ I. A similar relation is true for f_{j} and since

∥w∥_{ℂ} = \left∥{ \atop w̄}\right∥_{ℂ} = (∥x∥^{2} + ∥y∥^{2})^{\frac{1}{2}} ,

hence from (3.4) we deduce the desired inequality (3.3).

Remark 5. It is obvious that, if one family, say {f_{j}}_{j∈J} is empty, then, on observing that all sums ∑_{j∈J} should be zero, from (3.3) one would get [3]

|∑_{i∈I} ⟨w, e_{i}⟩_{ℂ}^{2}| (3.5)

≤ \frac{1}{2} |⟨w, w̄⟩_{ℂ}| + |∑_{i∈I} ⟨w, e_{i}⟩_{ℂ}^{2} − \frac{1}{2} ⟨w, w̄⟩_{ℂ}|

≤ \frac{1}{2} [∥w∥_{ℂ}^{2} + |⟨w, w̄⟩_{ℂ}|] ≤ ∥w∥_{ℂ}^{2} .

If in (3.5) one assumes that the family {e_{i}}_{i∈I} contains only one element e = \frac{a}{∥a∥}, a ≠ 0, then by selecting w = z, one would deduce

|⟨a, z⟩_{ℂ}|^{2} ≤ |⟨a, z⟩_{ℂ}^{2} − \frac{1}{2} ⟨z, z̄⟩_{ℂ}| + \frac{1}{2} |⟨z, z̄⟩_{ℂ}|
≤ \frac{1}{2} ∥a∥^{2} [∥z∥_{ℂ}^{2} + |⟨z, z̄⟩_{ℂ}|] ,

which is a refinement for Kurepa’s inequality (3.2).

References

[1] N.G. de BRUIJN, Problem 12, Wisk. Opgaven, 21 (1960), 12-14.

[2] M.L. BUZANO, Generalizzazione della disiguaglianza di Cauchy-Schwarz (Italian), Rend. Sem. Mat. Univ. e Politech. Torino, 31 (1971/73), 405-409 (1974).

[3] S.S. DRAGOMIR, Refinements of Buzano’s and Kurepa’s inequalities in inner product spaces, Preprint, RGMIA Res. Rep. Coll., 7 (2004).

[4] S. KUREPA, On the Buniakowsky-Cauchy-Schwarz inequality, Glasnick Mathematički, 1(21)(2) (1966), 147-158.

[5] M.H. MOORE, An inner product inequality, SIAM J. Math. Anal., 4(1973), No. 3, 514-518.

[6] T. PRECUPANU, On a generalisation of Cauchy-Buniakowski-Schwarz inequality, Anal. St. Univ. “Al. I. Cuza” Iaşi, 22(2) (1976), 173-175.

[7] U. RICHARD, Sur des inégalités du type Wirtinger et leurs application aux équationes différentielles ordinaires, Colloquium of Analysis held in Rio de Janeiro, August, 1972, pp. 233-244.

School of Computer Science and Mathematics, Victoria University of Technology, PO Box 14428, MCMC 8001, VIC, Australia.

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