REVERSING THE CBS-INEQUALITY FOR SEQUENCES OF VECTORS IN HILBERT SPACES WITH APPLICATIONS (II)

S.S. DRAGOMIR

Abstract. Several new reverses for the Cauchy-Bunyakovsky-Schwarz (CBS) inequality for sequences of vectors in Hilbert spaces which complement the ones obtained in part one are given. Applications in reversing the generalised triangle inequality and for Fourier coefficients are given as well.

1. Introduction

Let (H; ⟨·, ·⟩) be an inner product space over the real or complex number field 𝕂.

One of the key inequalities in inner product spaces with numerous applications is the Schwarz inequality

(1.1)  |⟨x, y⟩|^{2} ≤ ∥x∥^{2} ∥y∥^{2} ,  x, y ∈ H

which is known as the quadratic version of it, while

(1.2)  |⟨x, y⟩| ≤ ∥x∥ ∥y∥ ,  x, y ∈ H

is the simple version. The case of equality holds in either (1.1) or (1.2) if and only if the vectors x and y are linearly dependent.

By a multiplicative reverse of the Schwarz inequality in either simple or quadratic version, we understand an inequality of the form

(1.3)  (1 ≤) \begin{matrix}∥x∥ & ∥y∥ \\ | & ⟨x, y⟩ & | \end{matrix} ≤ k_{1}  or  (1 ≤) \begin{matrix}∥x∥ & 2 & ∥y∥ & 2 \\ |⟨x, y⟩| & 2 \end{matrix} ≤ k_{2}

with appropriate k_{1} and k_{2} and under various assumptions for the vectors x and y, while by the additive reverse we understand an inequality of the form

(1.4) (0 ≤) ∥x∥ ∥y∥ − |⟨x, y⟩| ≤ h_{1}  or  (0 ≤) ∥x∥^{2} ∥y∥^{2} − |⟨x, y⟩|^{2} ≤ h_{2}.

Similar definitions apply when |⟨x, y⟩| is replaced by Re ⟨x, y⟩ or |Re ⟨x, y⟩| .

The following reverses for the Schwarz inequality hold (see [4], or the monograph on line [5, p. 27]).

Theorem 1. Let (H; ⟨·, ·⟩) be an inner product space over the real or complex number field 𝕂. If x, a ∈ H and r > 0 are such that

(1.5)  x ∈ B (x, r) := {z ∈ H| ∥z − a∥ ≤ r} ,

Date: 21 February, 2005.

2000 Mathematics Subject Classification. 46C05, 26D15.

Key words and phrases. CBS-inequality, Fourier coefficients, Triangle inequality.

then we have the inequalities

(1.6)  (0 ≤) ∥x∥ ∥a∥ − |⟨x, a⟩| ≤ ∥x∥ ∥a∥ − |Re ⟨x, a⟩|

≤ ∥x∥ ∥a∥ − Re ⟨x, a⟩ ≤ \frac{1}{2}r^{2}.

The constant \frac{1}{2} is best possible in (1.5) in the sense that it cannot be replaced by a smaller quantity.

An additive version for the Schwarz inequality that may be more useful in applications is incorporated in [4] (see also [5, p. 28]).

Theorem 2. Let (H; ⟨·, ·⟩) be an inner product space over 𝕂 and x, y ∈ H and γ, Γ ∈ 𝕂 with Γ ≠ −γ and either

(1.7)  Re ⟨Γy − x, x − γy⟩ ≥ 0,

or, equivalently,

(1.8)  ∥x − \frac{γ + Γ}{2}y∥ ≤ \frac{1}{2} |Γ − γ| ∥y∥

holds. Then we have the inequalities

(1.9)  0 ≤ ∥x∥ ∥y∥ − |⟨x, y⟩|

≤ ∥x∥ ∥y∥ − |Re [\frac{Γ̄ + γ̄}{|Γ + γ|} · ⟨x, y⟩]|

≤ ∥x∥ ∥y∥ − Re [\frac{Γ̄ + γ̄}{|Γ + γ|} · ⟨x, y⟩]

≤ \frac{1}{4} · \frac{|Γ − γ|^{2}}{|Γ + γ|} ∥y∥^{2} .

The constant \frac{1}{4} in the last inequality is best possible.

We remark that a simpler version of the above result may be stated if one assumed that the scalars are real:

Corollary 1. If M ≥ m > 0, and either

(1.10)  Re ⟨My − x, x − my⟩ ≥ 0,

or, equivalently,

(1.11)  ∥x − \frac{m + M}{2} y∥ ≤ \frac{1}{2} (M − m) ∥y∥

holds, then

(1.12)  0 ≤ ∥x∥ ∥y∥ − |⟨x, y⟩|

≤ ∥x∥ ∥y∥ − |Re ⟨x, y⟩|

≤ ∥x∥ ∥y∥ − Re ⟨x, y⟩

≤ \frac{1}{4} · \frac{(M − m)^{2}}{M + m} ∥y∥^{2} .

The constant \frac{1}{4} is sharp.

Now, let (K, ⟨·, ·⟩) be a Hilbert space over 𝕂, p_{i} ≥ 0, i ∈ ℕ with ∑_{i=1}^{∞} p_{i} = 1. Consider ℓ_{p}^{2} (K) as the space

ℓ_{p}^{2} (K) := {x = (x_{i}) |x_{i} ∈ K, i ∈ ℕ and ∑_{i=1}^{∞} p_{i} ∥x_{i}∥^{2} < ∞} .

It is well known that ℓ_{p}^{2} (K) endowed with the inner product

⟨x, y⟩_{p} := ∑_{i=1}^{∞} p_{i} ⟨x_{i}, y_{i}⟩

is a Hilbert space over 𝕂. The norm ∥·∥_{p} of ℓ_{p}^{2} (K) is given by

∥x∥_{p} := (∑_{i=1}^{∞} p_{i} ∥x_{i}∥^{2})^{\frac{1}{2}} .

If x, y ∈ ℓ_{p}^{2} (K) , then the following Cauchy-Bunyakovsky-Schwarz (CBS) inequality holds true:

(1.13)  ∑_{i=1}^{∞} p_{i} ∥x_{i}∥^{2} ∑_{i=1}^{∞} p_{i} ∥y_{i}∥^{2} ≥ |∑_{i=1}^{∞} p_{i} ⟨x_{i}, y_{i}⟩|^{2}

with equality iff there exists a λ ∈ 𝕂 such that x_{i} = λy_{i} for each i ∈ ℕ. If

α ∈ ℓ_{p}^{2} (K) := {α = (α_{i})_{i∈ℕ}| α_{i} ∈ 𝕂, i ∈ ℕ and ∑_{i=1}^{∞} p_{i} |α_{i}|^{2} < ∞}

and x ∈ ℓ_{p}^{2} (K) , then the following (CBS)-type inequality is also valid:

(1.14)  ∑_{i=1}^{∞} p_{i} |α_{i}|^{2} ∑_{i=1}^{∞} p_{i} ∥x_{i}∥^{2} ≥ ∥∑_{i=1}^{∞} p_{i}α_{i}x_{i}∥^{2}

with equality if and only if there exists a vector v ∈ K such that x_{i} = \overline{α_{i}}v for each i ∈ ℕ.

Note that the inequality (1.14) follows by the obvious identity

(1.15) ∑_{i=1}^{n} p_{i} |α_{i}|^{2} ∑_{i=1}^{n} p_{i} ∥x_{i}∥^{2} − ∥∑_{i=1}^{n} p_{i}α_{i}x_{i}∥^{2} = \frac{1}{2} ∑_{i=1}^{n} ∑_{j=1}^{n} p_{i}p_{j} ∥\overline{α_{i}}x_{j} − \overline{α_{j}}x_{i}∥^{2}

for each n ∈ ℕ, n ≥ 1.

In [6], by the use of some preliminary results obtained in [3], various reverses for the (CBS)-type inequalities (1.13) and (1.14) for sequences of vectors in Hilbert spaces were obtained. Applications for bounding the distance to a finite-dimensional subspace and in reversing the generalised triangle inequality have also been provided.

The aim of the present paper is to provide different results by employing some inequalities discovered in [4]. Similar applications are pointed out.

4 S.S. DRAGOMIR

2. Reverses of the (CBS)-Inequality for Two Sequences in ℓ_{p}^{2} (K) The following proposition may be stated.

Proposition 1. Let x, y ∈ ℓ_{p}^{2} (K) and r > 0. If

(2.1)  ∥x_{i} − y_{i}∥ ≤ r for each i ∈ ℕ,

then

(2.2)  (0 ≤) (∑_{i=1}^{∞} p_{i} ∥x_{i}∥^{2} ∑_{i=1}^{∞} p_{i} ∥y_{i}∥^{2})^{\frac{1}{2}} − |∑_{i=1}^{∞} p_{i} ⟨x_{i}, y_{i}⟩|

≤ (∑_{i=1}^{∞} p_{i} ∥x_{i}∥^{2} ∑_{i=1}^{∞} p_{i} ∥y_{i}∥^{2})^{\frac{1}{2}} − |∑_{i=1}^{∞} p_{i} Re ⟨x_{i}, y_{i}⟩|

≤ (∑_{i=1}^{∞} p_{i} ∥x_{i}∥^{2} ∑_{i=1}^{∞} p_{i} ∥y_{i}∥^{2})^{\frac{1}{2}} − ∑_{i=1}^{∞} p_{i} Re ⟨x_{i}, y_{i}⟩

≤ \frac{1}{2}r^{2}.

The constant \frac{1}{2} in front of r^{2} is best possible in the sense that it cannot be replaced by a smaller quantity. Proof. If (2.1) holds true, then

∥x − y∥_{p}^{2} = ∑_{i=1}^{∞} p_{i} ∥x_{i} − y_{i}∥^{2} ≤ r^{2} ∑_{i=1}^{∞} p_{i} = r^{2}

and thus ∥x − y∥_{p} ≤ r.

Applying the inequality (1.6) for the inner product (ℓ_{p}^{2} (K) , ⟨·, ·⟩_{p}) , we deduce the desired result (2.2).

The sharpness of the constant follows by Theorem 1 and we omit the details.

The following result may be stated as well. Proposition 2. Let x, y ∈ ℓ_{p}^{2} (K) and γ, Γ ∈ 𝕂 with Γ ≠ −γ. If either

(2.3)  Re ⟨Γy_{i} − x_{i}, x_{i} − γy_{i}⟩ ≥ 0 for each i ∈ ℕ

or, equivalently,

(2.4)  ∥x_{i} − \frac{γ + Γ}{2}y_{i}∥ ≤ \frac{1}{2} |Γ − γ| ∥y_{i}∥ for each i ∈ ℕ

holds, then:

(2.5)  (0 ≤) (∑_{i=1}^{∞} p_{i} ∥x_{i}∥^{2} ∑_{i=1}^{∞} p_{i} ∥y_{i}∥^{2})^{\frac{1}{2}} − |∑_{i=1}^{∞} p_{i} ⟨x_{i}, y_{i}⟩|

≤ (∑_{i=1}^{∞} p_{i} ∥x_{i}∥^{2} ∑_{i=1}^{∞} p_{i} ∥y_{i}∥^{2})^{\frac{1}{2}} − |Re [\frac{Γ̄ + γ̄}{|Γ + γ|} ∑_{i=1}^{∞} p_{i} ⟨x_{i}, y_{i}⟩]|

≤ (∑_{i=1}^{∞} p_{i} ∥x_{i}∥^{2} ∑_{i=1}^{∞} p_{i} ∥y_{i}∥^{2})^{\frac{1}{2}} − Re [\frac{Γ̄ + γ̄}{|Γ + γ|} ∑_{i=1}^{∞} p_{i} ⟨x_{i}, y_{i}⟩]

≤ \frac{1}{4} · \frac{|Γ − γ|^{2}}{|Γ + γ|} ∑_{i=1}^{∞} p_{i} ∥y_{i}∥^{2} .

The constant \frac{1}{4} is best possible in (2.5). Proof. Since, by (2.3),

Re ⟨Γy − x, x − γy⟩_{p} = ∑_{i=1}^{∞} p_{i} Re ⟨Γy_{i} − x_{i}, x_{i} − γy_{i}⟩ ≥ 0,

hence, on applying the inequality (1.9) for the Hilbert space (ℓ_{p}^{2} (K) , ⟨·, ·⟩_{p}) , we deduce the desired inequality (2.5).

The best constant follows by Theorem 2 and we omit the details. Corollary 2. If the conditions (2.3) and (2.4) hold for Γ = M, γ = m with M ≥ m > 0, then

(2.6)  (0 ≤) (∑_{i=1}^{∞} p_{i} ∥x_{i}∥^{2} ∑_{i=1}^{∞} p_{i} ∥y_{i}∥^{2})^{\frac{1}{2}} − |∑_{i=1}^{∞} p_{i} ⟨x_{i}, y_{i}⟩|

≤ (∑_{i=1}^{∞} p_{i} ∥x_{i}∥^{2} ∑_{i=1}^{∞} p_{i} ∥y_{i}∥^{2})^{\frac{1}{2}} − |∑_{i=1}^{∞} p_{i} Re ⟨x_{i}, y_{i}⟩|

≤ (∑_{i=1}^{∞} p_{i} ∥x_{i}∥^{2} ∑_{i=1}^{∞} p_{i} ∥y_{i}∥^{2})^{\frac{1}{2}} − ∑_{i=1}^{∞} p_{i} Re ⟨x_{i}, y_{i}⟩

≤ \frac{1}{4} · \frac{(M − m)^{2}}{M + m} ∑_{i=1}^{∞} p_{i} ∥y_{i}∥^{2} .

The constant \frac{1}{4} is best possible.

3. Reverses of the (CBS)-Inequality for Mixed Sequences The following result holds:

Theorem 3. Let α ∈ ℓ_{p}^{2} (K) , x ∈ ℓ_{p}^{2} (K) and v ∈ K\ {0} , r > 0. If

(3.1)  ∥x_{i} − \overline{α_{i}}v∥ ≤ r |α_{i}| for each i ∈ ℕ

(note that if α_{i} ≠ 0 for any i ∈ ℕ, then the condition (3.1) is equivalent to the simpler one

(3.2)  ∥\frac{x_{i}}{\overline{α_{i}}} − v∥ ≤ r for each i ∈ ℕ),

then

(3.3)  (0 ≤) (∑_{i=1}^{∞} p_{i} |α_{i}|^{2} ∑_{i=1}^{∞} p_{i} ∥x_{i}∥^{2})^{\frac{1}{2}} − ∥∑_{i=1}^{∞} p_{i}α_{i}x_{i}∥

≤ (∑_{i=1}^{∞} p_{i} |α_{i}|^{2} ∑_{i=1}^{∞} p_{i} ∥x_{i}∥^{2})^{\frac{1}{2}} − |⟨∑_{i=1}^{∞} p_{i}α_{i}x_{i}, \frac{v}{∥v∥}⟩|

≤ (∑_{i=1}^{∞} p_{i} |α_{i}|^{2} ∑_{i=1}^{∞} p_{i} ∥x_{i}∥^{2})^{\frac{1}{2}} − |Re ⟨∑_{i=1}^{∞} p_{i}α_{i}x_{i}, \frac{v}{∥v∥}⟩|

≤ (∑_{i=1}^{∞} p_{i} |α_{i}|^{2} ∑_{i=1}^{∞} p_{i} ∥x_{i}∥^{2})^{\frac{1}{2}} − Re ⟨∑_{i=1}^{∞} p_{i}α_{i}x_{i}, \frac{v}{∥v∥}⟩

≤ \frac{1}{2} · \frac{r^{2}}{∥r∥} ∑_{i=1}^{∞} p_{i} |α_{i}|^{2} .

The constant \frac{1}{2} is best possible in (3.3).

Proof. From (3.1) we deduce

∥x_{i}∥^{2} − 2 Re ⟨α_{i}x_{i}, v⟩ + |α_{i}|^{2} ∥v∥^{2} ≤ r^{2} |α_{i}|^{2} ,

which is clearly equivalent to

(3.4)  ∥x_{i}∥^{2} + |α_{i}|^{2} ∥v∥^{2} ≤ 2 Re ⟨α_{i}x_{i}, v⟩ + r^{2} |α_{i}|^{2}

for each i ∈ ℕ.

If we multiply (3.4) by p_{i} ≥ 0, i ∈ ℕ and sum over i ∈ ℕ, then we deduce

(3.5) ∑_{i=1}^{∞} p_{i} ∥x_{i}∥^{2} + ∥v∥^{2} ∑_{i=1}^{∞} p_{i} |α_{i}|^{2} ≤ 2 Re ⟨∑_{i=1}^{∞} p_{i}α_{i}x_{i}, v⟩ + r^{2} ∑_{i=1}^{∞} p_{i} |α_{i}|^{2} .

Since, obviously

(3.6)  2 ∥v∥ (∑_{i=1}^{∞} p_{i} |α_{i}|^{2} ∑_{i=1}^{∞} p_{i} ∥x_{i}∥^{2})^{\frac{1}{2}} ≤ ∑_{i=1}^{∞} p_{i} ∥x_{i}∥^{2} + ∥v∥^{2} ∑_{i=1}^{∞} p_{i} |α_{i}|^{2} ,

hence, by (3.5) and (3.6), we deduce

2 ∥v∥ (∑_{i=1}^{∞} p_{i} |α_{i}|^{2} ∑_{i=1}^{∞} p_{i} ∥x_{i}∥^{2})^{\frac{1}{2}} ≤ 2 Re ⟨∑_{i=1}^{∞} p_{i}α_{i}x_{i}, v⟩ + r^{2} ∑_{i=1}^{∞} p_{i} |α_{i}|^{2} ,

which is clearly equivalent to the last inequality in (3.3).

The other inequalities are obvious.

The best constant follows by Theorem 1.

The following corollary may be stated.

REVERSING THE CBS-INEQUALITY 7

Corollary 3. Let α ∈ ℓ_{p}^{2} (K) , x ∈ ℓ_{p}^{2} (K) , e ∈ H, ∥e∥ = 1 and γ, Γ ∈ 𝕂 with Γ ≠ −γ. If

(3.7) ∥x_{i} − \overline{α_{i}} \frac{γ + Γ}{2} · e∥ ≤ \frac{1}{2} |Γ − γ| |α_{i}|

for each i ∈ ℕ, or, equivalently,

(3.8)  Re ⟨Γ\overline{α_{i}}e − x_{i}, x_{i} − γ\overline{α_{i}}e⟩

for each i ∈ ℕ (note that, if α_{i} ≠ 0 for any i ∈ ℕ, then (3.7) is equivalent to

(3.9) ∥\frac{x_{i}}{\overline{α_{i}}} − \frac{γ + Γ}{2}e∥ ≤ \frac{1}{2} |Γ − γ|

for each i ∈ ℕ and (3.8) is equivalent to

(3.10)  Re ⟨Γe − \frac{x_{i}}{\overline{α_{i}}} , \frac{x_{i}}{\overline{α_{i}}} − γe⟩ ≥ 0

for each i ∈ ℕ), then the following reverse of the (CBS)-inequality is valid:

(3.11)  (0 ≤) (∑_{i=1}^{∞} p_{i} |α_{i}|^{2} ∑_{i=1}^{∞} p_{i} ∥x_{i}∥^{2})^{\frac{1}{2}} − ∥∑_{i=1}^{∞} p_{i}α_{i}x_{i}∥

≤ (∑_{i=1}^{∞} p_{i} |α_{i}|^{2} ∑_{i=1}^{∞} p_{i} ∥x_{i}∥^{2})^{\frac{1}{2}} − |⟨∑_{i=1}^{∞} p_{i}α_{i}x_{i}, e⟩|

≤ (∑_{i=1}^{∞} p_{i} |α_{i}|^{2} ∑_{i=1}^{∞} p_{i} ∥x_{i}∥^{2})^{\frac{1}{2}} − |Re [\frac{Γ̄ + γ̄}{|Γ + γ|} ⟨∑_{i=1}^{∞} p_{i}α_{i}x_{i}, e⟩]|

≤ (∑_{i=1}^{∞} p_{i} |α_{i}|^{2} ∑_{i=1}^{∞} p_{i} ∥x_{i}∥^{2})^{\frac{1}{2}} − Re [\frac{Γ̄ + γ̄}{|Γ + γ|} ⟨∑_{i=1}^{∞} p_{i}α_{i}x_{i}, e⟩]

≤ \frac{1}{4} · \frac{|Γ − γ|^{2}}{|Γ + γ|} ∑_{i=1}^{∞} p_{i} |α_{i}|^{2} .

The constant \frac{1}{4} is best possible. Remark 1. If M ≥ m > 0, α_{i} ≠ 0 and for e as above, either

(3.12) ∥\frac{x_{i}}{\overline{α_{i}}} − \frac{M + m}{2}e∥ ≤ \frac{1}{2} (M − m) for each i ∈ ℕ

or, equivalently,

Re ⟨Me − \frac{x_{i}}{\overline{α_{i}}} , \frac{x_{i}}{\overline{α_{i}}} − me⟩ ≥ 0 for each i ∈ ℕ

holds, then

(0 ≤) (∑_{i=1}^{∞} p_{i} |α_{i}|^{2} ∑_{i=1}^{∞} p_{i} ∥x_{i}∥^{2})^{\frac{1}{2}} − ∥∑_{i=1}^{∞} p_{i}α_{i}x_{i}∥

≤ (∑_{i=1}^{∞} p_{i} |α_{i}|^{2} ∑_{i=1}^{∞} p_{i} ∥x_{i}∥^{2})^{\frac{1}{2}} − |⟨∑_{i=1}^{∞} p_{i}α_{i}x_{i}, e⟩|

≤ (∑_{i=1}^{∞} p_{i} |α_{i}|^{2} ∑_{i=1}^{∞} p_{i} ∥x_{i}∥^{2})^{\frac{1}{2}} − |Re ⟨∑_{i=1}^{∞} p_{i}α_{i}x_{i}, e⟩|

≤ (∑_{i=1}^{∞} p_{i} |α_{i}|^{2} ∑_{i=1}^{∞} p_{i} ∥x_{i}∥^{2})^{\frac{1}{2}} − Re ⟨∑_{i=1}^{∞} p_{i}α_{i}x_{i}, e⟩

≤ \frac{1}{4} · \frac{(M − m)^{2}}{M + m} ∑_{i=1}^{∞} p_{i} |α_{i}|^{2} .

The constant \frac{1}{4} is best possible.

4. Reverses for the Generalised Triangle Inequality

In 1966, Diaz and Metcalf [2] proved the following interesting reverse of the generalised triangle inequality:

(4.1)  r ∑_{i=1}^{∞} ∥x_{i}∥ ≤ ∥∑_{i=1}^{∞} x_{i}∥ ,

provided the vectors x_{1}, …, x_{n} ∈ H\ {0} satisfy the assumption

(4.2)  0 ≤ r ≤ \frac{Re ⟨x_{i}, a⟩}{∥x_{i}∥},  i ∈ \{1, …, n\} ,

where a ∈ H, ∥a∥ = 1 and (H; ⟨·, ·⟩) is a real or complex inner product space.

In an attempt to provide other sufficient conditions for (4.1) to hold, the author pointed out in [7] that

(4.3)  \sqrt{1 − ρ^{2}} ∑_{i=1}^{∞} ∥x_{i}∥ ≤ ∥∑_{i=1}^{∞} x_{i}∥

where the vectors x_{i}, i ∈ {1, …, n} satisfy the condition

(4.4)  ∥x_{i} − a∥ ≤ ρ,  i ∈ {1, …, n} ,

where r ∈ H, ∥a∥ = 1 and ρ ∈ (0, 1) .

Following [7], if M ≥ m > 0 and the vectors x_{i} ∈ H, i ∈ {1, …, n} verify either

(4.5)  Re ⟨Ma − x_{i}, x_{i} − ma⟩ ≥ 0,  i ∈ {1, …, n} ,

or, equivalently,

(4.6)  ∥x_{i} − \frac{M + m}{2} · a∥ ≤ \frac{1}{2} (M − m) ,  i ∈ \{1, …, n\} ,

where a ∈ H, ∥a∥ = 1, then

(4.7)  \frac{2\sqrt{mM}}{M + m} ∑_{i=1}^{n} ∥x_{i}∥ ≤ ∥∑_{i=1}^{n} x_{i}∥ .

It is obvious from Theorem 3, that, if

(4.8)  ∥x_{i} − v∥ ≤ r,  for i ∈ {1, …, n} ,

where x_{i} ∈ H, i ∈ {1, …, n} , v ∈ H\ {0} and r > 0, then we can state the inequality

(4.9)  (0 ≤) (\frac{1}{n} ∑_{i=1}^{n} ∥x_{i}∥^{2})^{\frac{1}{2}} − ∥\frac{1}{n} ∑_{i=1}^{n} x_{i}∥

≤ (\frac{1}{n} ∑_{i=1}^{n} ∥x_{i}∥^{2})^{\frac{1}{2}} − |⟨\frac{1}{n} ∑_{i=1}^{n} x_{i}, \frac{v}{∥v∥}⟩|

≤ (\frac{1}{n} ∑_{i=1}^{n} ∥x_{i}∥^{2})^{\frac{1}{2}} − |Re ⟨\frac{1}{n} ∑_{i=1}^{n} x_{i}, \frac{v}{∥v∥}⟩|

≤ (\frac{1}{n} ∑_{i=1}^{n} ∥x_{i}∥^{2})^{\frac{1}{2}} − Re ⟨\frac{1}{n} ∑_{i=1}^{n} x_{i}, \frac{v}{∥v∥}⟩

≤ \frac{1}{2} · \frac{r^{2}}{∥v∥}.

Since, by the (CBS)-inequality we have

(4.10)  \frac{1}{n} ∑_{i=1}^{n} ∥x_{i}∥ ≤ (\frac{1}{n} ∑_{i=1}^{n} ∥x_{i}∥^{2})^{\frac{1}{2}} ,

hence, by (4.9) and (4.5) we have:

(4.11)  (0 ≤) ∑_{i=1}^{n} ∥x_{i}∥ − ∥∑_{i=1}^{n} x_{i}∥ ≤ \frac{1}{2}n · \frac{r^{2}}{∥v∥}

provided that (4.8) holds true.

Utilising Corollary 3, we may state that, if

(4.12)  ∥x_{i} − \frac{γ + Γ}{2} · e∥ ≤ \frac{1}{2} |Γ − γ| ,  i ∈ \{1, …, n\} ,

or, equivalently,

(4.13)  Re ⟨Γe − x_{i}, x_{i} − γe⟩ ≥ 0,  i ∈ {1, …, n} ,

where e ∈ H, ∥e∥ = 1, γ, Γ ∈ 𝕂, Γ ≠ −γ and x_{i} ∈ H, i ∈ {1, …, n} , then

(4.14)  (0 ≤) (\frac{1}{n} ∑_{i=1}^{n} ∥x_{i}∥^{2})^{\frac{1}{2}} − ∥\frac{1}{n} ∑_{i=1}^{n} x_{i}∥

≤ (\frac{1}{n} ∑_{i=1}^{n} ∥x_{i}∥^{2})^{\frac{1}{2}} − |⟨\frac{1}{n} ∑_{i=1}^{n} x_{i}, e⟩|

≤ (\frac{1}{n} ∑_{i=1}^{n} ∥x_{i}∥^{2})^{\frac{1}{2}} − |Re [\frac{Γ̄ + γ̄}{|Γ + γ|} ⟨\frac{1}{n} ∑_{i=1}^{n} x_{i}, e⟩]|

≤ (\frac{1}{n} ∑_{i=1}^{n} ∥x_{i}∥^{2})^{\frac{1}{2}} − Re [\frac{Γ̄ + γ̄}{|Γ + γ|} ⟨\frac{1}{n} ∑_{i=1}^{n} x_{i}, e⟩]

≤ \frac{1}{4} · \frac{|Γ − γ|^{2}}{|Γ + γ|} .

Now, making use of (4.10) and (4.14) we can establish the following additive reverse of the generalised triangle inequality

(4.15)  (0 ≤) ∑_{i=1}^{n} ∥x_{i}∥ − ∥∑_{i=1}^{n} x_{i}∥ ≤ \frac{1}{4}n · \frac{|Γ − γ|^{2}}{|Γ + γ|} ,

provided either (4.12) or (4.13) hold true.

5. Applications for Fourier Coefficients

Let (H; ⟨·, ·⟩) be a Hilbert space over the real or complex number field 𝕂 and {e_{i}}_{i∈I} an orthonormal basis for H. Then (see for instance [1, p. 54 – 61]):

(i) Every element x ∈ H can be expanded in a Fourier series, i.e.,

x = ∑_{i∈I} ⟨x, e_{i}⟩ e_{i},

where ⟨x, e_{i}⟩ , i ∈ I are the Fourier coefficients of x; (ii) (Parseval identity)

∥x∥^{2} = ∑_{i∈I} ⟨x, e_{i}⟩ e_{i}, x ∈ H;

(iii) (Extended Parseval’s identity)

⟨x, y⟩ = ∑_{i∈I} ⟨x, e_{i}⟩ ⟨e_{i}, y⟩ , x, y ∈ H;

(iv) (Elements are uniquely determined by their Fourier coefficients)

⟨x, e_{i}⟩ = ⟨y, e_{i}⟩ for every i ∈ I implies that x = y.

We must remark that all the results from the second and third sections may be stated for K = 𝕂 where 𝕂 is the Hilbert space of complex (real) numbers endowed with the usual norm and inner product.

Therefore we can state the following reverses of the Schwarz inequality:

Proposition 3. Let (H; ⟨·, ·⟩) be a Hilbert space over 𝕂 and {e_{i}}_{i∈I} an orthonormal base for H. If x, y ∈ H, y ≠ 0, a ∈ 𝕂 (ℂ, ℝ) with r > 0 such that

(5.1)  |\frac{⟨x, e_{i}⟩}{⟨y, e_{i}⟩} − a| ≤ r  for each i ∈ I,

then we have the following reverse of the Schwarz inequality:

(5.2)  (0 ≤) ∥x∥ ∥y∥ − |⟨x, y⟩|

≤ ∥x∥ ∥y∥ − |Re [⟨x, y⟩ · \frac{ā}{|a|}]|

≤ ∥x∥ ∥y∥ − Re [⟨x, y⟩ · \frac{ā}{|a|}]

≤ \frac{1}{2} · \frac{r^{2}}{|a|} ∥y∥^{2} .

The constant \frac{1}{2} is best possible in (5.2).

The proof is similar to the one in Theorem 3, where instead of x_{i} we take ⟨x, e_{i}⟩, instead of α_{i} we take ⟨e_{i}, y⟩ , ∥·∥ = |·| , p_{i} = 1 and use the Parseval identities mentioned above in (ii) and (iii). We omit the details.

The following result may be stated as well.

Proposition 4. Let (H; ⟨·, ·⟩) be a Hilbert space over 𝕂 and {e_{i}}_{i∈I} an orthonormal base for H. If x, y ∈ H, y ≠ 0, e, γ, Γ ∈ 𝕂 with |e| = 1, Γ ≠ −γ and

(5.3)  |\frac{⟨x, e_{i}⟩}{⟨y, e_{i}⟩} − \frac{γ + Γ}{2} · e| ≤ \frac{1}{2} |Γ − γ|

or equivalently,

(5.4)  Re [(Γe − \frac{⟨x, e_{i}⟩}{⟨y, e_{i}⟩}) (\frac{⟨e_{i}, x⟩}{⟨e_{i}, y⟩} − γ̄ē)] ≥ 0

for each i ∈ I, then

(5.5)  (0 ≤) ∥x∥ ∥y∥ − |⟨x, y⟩|

≤ ∥x∥ ∥y∥ − |Re [\frac{Γ̄ + γ̄}{|Γ + γ|} ⟨x, y⟩ · ē]|

≤ ∥x∥ ∥y∥ − Re [\frac{Γ̄ + γ̄}{|Γ + γ|} ⟨x, y⟩ · ē]

≤ \frac{1}{4} · \frac{|Γ − γ|^{2}}{|Γ + γ|} ∥y∥^{2} .

The constant \frac{1}{4} is best possible.

Remark 2. If Γ = M ≥ m = γ > 0, then one may state simpler inequalities from (5.5). We omit the details.

References

[1] F. DEUTSCH, Best Approximation in Inner Product Spaces, CMS Books in Mathematics, Springer-Verlag, New York, Berlin, Heidelberg, 2002.

[2] J.B. DIAZ and F.T. METCALF, A complementary triangle inequality in Hilbert and Banach spaces, Proceedings Amer. Math. Soc., 17(1) (1966), 88-97.

[3] S.S. DRAGOMIR, Reverses of Schwarz, triangle and Bessel inequalities in inner product spaces, J. Ineq. Pure & Appl. Math., 5(3) (2004), Art. 74. [ONLINE

http://jipam.vu.edu.au/article.php?sid=432].

[4] S.S. DRAGOMIR, New reverses of Schwarz, triangle and Bessel inequalities in inner product spaces, The Australian J. Math. Anal. & Appl., 1(1) (2004), Art. 1. [ONLINE

http://ajmaa.org/].

[5] S.S. DRAGOMIR, Advances in Inequalities of the Schwarz, Grüss and Bessel Type in Inner Product Spaces, RGMIA Monographs, Victoria University, 2004. [ONLINE

http://rgmia.vu.edu.au/monographs/advancees.htm].

[6] S.S. DRAGOMIR, Reversing the CBS-inequality for sequences of vectors in Hilbert spaces with applications (I), RGMIA Res. Rep. Coll., 8(2005), Supplement, Article 2. [ONLINE

http://rgmia.vu.edu.au/v8(E).html].

[7] S.S. DRAGOMIR, Reverses of the triangle inequality in inner product spaces, RGMIA Res. Rep. Coll., 7 (2004), Supplement, Article 7. [ONLINE http://rgmia.vu.edu.au/v7(E).html].

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

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