REVERSES OF THE CAUCHY-BUNYAKOVSKY-SCHWARZ AND HEISENBERG INTEGRAL INEQUALITIES FOR

VECTOR-VALUED FUNCTIONS IN HILBERT SPACES

S.S. DRAGOMIR

Abstract. Some reverses of the Cauchy-Bunyakovsky-Schwarz integral inequality for vector-valued functions in Hilbert spaces and applications for the Heisenberg inequality are provided.

1. Introduction

Assume that (K; ⟨·, ·⟩) is a Hilbert space over the real or complex number field 𝕂. If ρ : [a, b] ⊂ ℝ → [0, ∞) is a Lebesgue integrable function with ∫_{a}^{b} ρ (t) dt = 1, then we may consider the space L_{ρ}^{2} ([a, b] ; K) of all functionsf : [a, b] → K, that are Bochner measurable and ∫_{a}^{b} ρ (t) ∥f (t)∥^{2} dt < ∞. It is well known that L_{ρ}^{2} ([a, b] ; K) endowed with the inner product ⟨·, ·⟩_{ρ} defined by

⟨f, g⟩_{ρ} := ∫_{a}^{b} ρ (t) ⟨f (t) , g (t)⟩ dt

and generating the norm

∥f∥_{ρ} := (∫_{a}^{b} ρ (t) ∥f (t)∥^{2} dt)^{\frac{1}{2}} ,

is a Hilbert space over 𝕂.

Date: 26 July, 2004.

2000 Mathematics Subject Classification. 46C05, 26D15, 26D10.

Key words and phrases. Cauchy-Bunyakovsky-Schwarz inequality, Inner products, Heisenberg inequality.

The following integral inequality is known in the literature as the Cauchy-Bunyakovsky-Schwarz (CBS) inequality

(1.1) ∫_{a}^{b} ρ (t) ∥f (t)∥^{2} dt ∫_{a}^{b} ρ (t) ∥g (t)∥^{2} dt

≥ |∫_{a}^{b} ρ (t) ⟨f (t) , g (t)⟩ dt|^{2} ,

provided f, g ∈ L_{ρ}^{2} ([a, b] ; K) .

The case of equality holds in (1.1) iff there exists a constant λ ∈ 𝕂 such that f (t) = λg (t) for a.e. t ∈ [a, b] .

Another version of the (CBS) inequality for one vector-valued and one scalar function is incorporated in:

(1.2) ∫_{a}^{b} ρ (t) |α (t)|^{2} dt ∫_{a}^{b} ρ (t) ∥f (t)∥^{2} dt

≥ ∥∫_{a}^{b} ρ (t) α (t) f (t) dt∥^{2} ,

provided α ∈ L_{ρ}^{2} ([a, b]) and f ∈ L_{ρ}^{2} ([a, b] ; K) , where L_{ρ}^{2} ([a, b]) denotes the Hilbert space of scalar functions α for which ∫_{a}^{b} ρ (t) |α (t)|^{2} dt < ∞. The equality holds in (1.2) iff there exists a vector e ∈ K such that f (t) = \overline{α (t)}e for a.e. t ∈ [a, b] .

In this paper some reverses of the inequalities (1.1) and (1.2) are given under various assumptions for the functions involved. Natural applications for the Heisenberg inequality for vector-valued functions in Hilbert spaces are also provided.

2. Some Reverse Inequalities, The General Case

The following result holds.

Theorem 1. Let f, g ∈ L_{ρ}^{2} ([a, b] ; K) and r > 0 be such that

(2.1)  ∥f (t) − g (t)∥ ≤ r ≤ ∥g (t)∥

REVERSES OF THE CBS AND HEISENBERG INTEGRAL INEQUALITIES 3

for a.e. t ∈ [a, b] . Then we have the inequalities:

0 ≤ ∫_{a}^{b} ρ (t) ∥f (t)∥^{2} dt ∫_{a}^{b} ρ (t) ∥g (t)∥^{2} dt (2.2)

− |∫_{a}^{b} ρ (t) ⟨f (t) , g (t)⟩ dt|^{2}
≤ ∫_{a}^{b} ρ (t) ∥f (t)∥^{2} dt ∫_{a}^{b} ρ (t) ∥g (t)∥^{2} dt

− [∫_{a}^{b} ρ (t) Re ⟨f (t) , g (t)⟩ dt]^{2}
≤ r^{2} ∫_{a}^{b} ρ (t) ∥f (t)∥^{2} dt.

The constant C = 1 in front of r^{2} is best possible in the sense that it cannot be replaced by a smaller quantity.

Proof. We will use the following result obtained in [1]:

In the inner product space (H; ⟨·, ·⟩) , if x, y ∈ H and r > 0 are such that ∥x − y∥ ≤ r ≤ ∥y∥ , then

(2.3)  0 ≤ ∥x∥^{2} ∥y∥^{2} − |⟨x, y⟩|^{2}
≤ ∥x∥^{2} ∥y∥^{2} − [Re ⟨x, y⟩]^{2} ≤ r^{2} ∥x∥^{2} .

The constant c = 1 in front of r^{2} is best possible in the sense that it cannot be replaced by a smaller quantity.

If (2.1) holds, true, then

∥f − g∥_{ρ}^{2} = ∫_{a}^{b} ρ (t) ∥f (t) − g (t)∥^{2} dt ≤ r^{2} ∫_{a}^{b} ρ (t) dt = r^{2}

and

∥g∥_{ρ}^{2} = ∫_{a}^{b} ρ (t) ∥g (t)∥^{2} dt ≥ r^{2} ∫_{a}^{b} ρ (t) dt = r^{2}

and thus ∥f − g∥_{ρ} ≤ r ≤ ∥g∥_{ρ} . Applying the inequality (2.3) for

(L_{ρ}^{2} ([a, b] ; K) , ⟨·, ·⟩_{p}) , we deduce the desired inequality (2.2).

If we choose ρ (t) = \frac{1}{b−a}, f (t) = x, g (t) = y, x, y ∈ K, t ∈ [a, b] , then from (2.2) we recapture (2.3) for which the constant c = 1 in front of r^{2} is best possible.

We next point out some general reverse inequalities for the second (CBS) inequality (1.2).

4 S.S. DRAGOMIR

Theorem 2. Let α ∈ L_{ρ}^{2} ([a, b]) , g ∈ L_{ρ}^{2} ([a, b] ; K) and a ∈ K, r > 0 such that ∥a∥ > r. If the following condition holds

(2.4)  ∥g (t) − ᾱ (t) a∥ ≤ r |α (t)|

for a.e. t ∈ [a, b] , (note that, if α (t) ≠ 0 for a.e. t ∈ [a, b] , then the condition (2.4) is equivalent to

(2.5)  ∥\frac{g (t)}{ᾱ (t)} − a∥ ≤ r

for a.e. t ∈ [a, b]), then we have the following inequality

(∫_{a}^{b} ρ (t) |α (t)|^{2} dt ∫_{a}^{b} ρ (t) ∥g (t)∥^{2} dt)^{\frac{1}{2}} (2.6)

≤ \frac{1}{\sqrt{∥a∥^{2} − r^{2}}} Re ⟨∫_{a}^{b} ρ (t) α (t) g (t) dt, a⟩

≤ \frac{∥a∥}{\sqrt{∥a∥^{2} − r^{2}}} ∥∫_{a}^{b} ρ (t) α (t) g (t) dt∥;

0 ≤ (∫_{a}^{b} ρ (t) |α (t)|^{2} dt ∫_{a}^{b} ρ (t) ∥g (t)∥^{2} dt)^{\frac{1}{2}} (2.7)

− ∥∫_{a}^{b} ρ (t) α (t) g (t) dt∥
≤ (∫_{a}^{b} ρ (t) |α (t)|^{2} dt ∫_{a}^{b} ρ (t) ∥g (t)∥^{2} dt)^{\frac{1}{2}}

− Re ⟨∫_{a}^{b} ρ (t) α (t) g (t) dt, \frac{a}{∥a∥}⟩

≤ \frac{r^{2}}{\sqrt{∥a∥^{2} − r^{2}} (∥a∥ + \sqrt{∥a∥^{2} − r^{2}})} Re ⟨∫_{a}^{b} ρ (t) α (t) g (t) dt, \frac{a}{∥a∥}⟩

≤ \frac{r^{2}}{\sqrt{∥a∥^{2} − r^{2}} (∥a∥ + \sqrt{∥a∥^{2} − r^{2}})} ∥∫_{a}^{b} ρ (t) α (t) g (t) dt∥;

REVERSES OF THE CBS AND HEISENBERG INTEGRAL INEQUALITIES 5

∫_{a}^{b} ρ (t) |α (t)|^{2} dt ∫_{a}^{b} ρ (t) ∥g (t)∥^{2} dt (2.8)

≤ \frac{1}{∥a∥^{2} − r^{2}} [Re ⟨∫_{a}^{b} ρ (t) α (t) g (t) dt, a⟩]^{2}

≤ \frac{∥a∥^{2}}{∥a∥^{2} − r^{2}} ∥∫_{a}^{b} ρ (t) α (t) g (t) dt∥^{2} ,

and 0 ≤ ∫_{a}^{b} ρ (t) |α (t)|^{2} dt ∫_{a}^{b} ρ (t) ∥g (t)∥^{2} dt (2.9)

− ∥∫_{a}^{b} ρ (t) α (t) g (t) dt∥^{2}

≤ ∫_{a}^{b} ρ (t) |α (t)|^{2} dt ∫_{a}^{b} ρ (t) ∥g (t)∥^{2} dt

− [Re ⟨∫_{a}^{b} ρ (t) α (t) g (t) dt, \frac{a}{∥a∥}⟩]^{2}

≤ \frac{r^{2}}{∥a∥^{2} (∥a∥^{2} − r^{2})} [Re ⟨∫_{a}^{b} ρ (t) α (t) g (t) dt, a⟩]^{2}

≤ \frac{r^{2}}{∥a∥^{2} − r^{2}} ∥∫_{a}^{b} ρ (t) α (t) g (t) dt∥^{2} .

All the inequalities (2.6) – (2.9) are sharp. Proof. From (2.4) we deduce

∥g (t)∥^{2} − 2 Re ⟨g (t) , ᾱ (t) a⟩ + |α (t)|^{2} ∥a∥^{2} ≤ |α (t)|^{2} r^{2} for a.e. t ∈ [a, b] , which is clearly equivalent to:

(2.10)  ∥g (t)∥^{2} + (∥a∥^{2} − r^{2}) |α (t)|^{2} ≤ 2 Re ⟨α (t) g (t) , a⟩

for a.e. t ∈ [a, b] .

If we multiply (2.10) by ρ (t) ≥ 0 and integrate over t ∈ [a, b] , then we deduce ∫_{a}^{b} ρ (t) ∥g (t)∥^{2} dt + (∥a∥^{2} − r^{2}) ∫_{a}^{b} ρ (t) |α (t)|^{2} dt (2.11)

≤ 2 Re ⟨∫_{a}^{b} ρ (t) α (t) g (t) dt, a⟩ .

Now, dividing (2.11) by \sqrt{∥a∥^{2} − r^{2}} > 0, we get

\frac{1}{\sqrt{∥a∥^{2} − r^{2}}} ∫_{a}^{b} ρ (t) ∥g (t)∥^{2} dt (2.12)

+\sqrt{∥a∥^{2} − r^{2}} ∫_{a}^{b} ρ (t) |α (t)|^{2} dt

≤ \frac{2}{\sqrt{∥a∥^{2} − r^{2}}} Re ⟨∫_{a}^{b} ρ (t) α (t) g (t) dt, a⟩ .

On the other hand, by the elementary inequality

\frac{1}{α}p + αq ≥ 2\sqrt{pq}, α > 0, p, q ≥ 0,

we can state that

2\sqrt{∫_{a}^{b} ρ (t) |α (t)|^{2} dt} · \sqrt{∫_{a}^{b} ρ (t) ∥g (t)∥^{2} dt} (2.13)

≤ \frac{1}{\sqrt{∥a∥^{2} − r^{2}}} ∫_{a}^{b} ρ (t) ∥g (t)∥^{2} dt + \sqrt{∥a∥^{2} − r^{2}} ∫_{a}^{b} ρ (t) |α (t)|^{2} dt.

Making use of (2.12) and (2.13), we deduce the first part of (2.6). The second part of (2.6) is obvious by Schwarz’s inequality

Re ⟨∫_{a}^{b} ρ (t) α (t) g (t) dt, a⟩ ≤ ∥∫_{a}^{b} ρ (t) α (t) g (t) dt∥ ∥a∥ .

If ρ (t) = \frac{1}{b−a}, α (t) = 1, g (t) = x ∈ K, then, from (2.6) we get

∥x∥ ≤ \frac{1}{\sqrt{∥a∥^{2} − r^{2}}} Re ⟨x, a⟩ ≤ \begin{matrix}∥x∥ & ∥a∥ \\ √ & \overline{∥a∥^{2} − r^{2}} \end{matrix} ,

provided ∥x − a∥ ≤ r < ∥a∥ , x, a ∈ K. The sharpness of this inequality has been shown in [1], and we omit the details.

The other inequalities are obvious consequences of (2.6) and we omit the details.

3. Some Particular Cases

It has been shown in [1] that, for A, a ∈ 𝕂 (𝕂 = ℂ, ℝ) and x, y ∈ H, where (H; ⟨·, ·⟩) is an inner product over the real or complex number

REVERSES OF THE CBS AND HEISENBERG INTEGRAL INEQUALITIES 7

field 𝕂, the following inequality holds

(3.1)  ∥x∥ ∥y∥ ≤ \frac{1}{2} · \begin{matrix}Re & [(Ā + ā) ⟨x, y⟩] \\ [Re (Aā)] & \frac{1}{2} \end{matrix}

≤ \frac{1}{2} · \frac{|A + a|}{[Re (Aā)]^{\frac{1}{2}}} |⟨x, y⟩|

provided Re (Aā) > 0 and

(3.2)  Re ⟨Ay − x, x − ay⟩ ≥ 0,

or, equivalently,

(3.3)  ∥x − \frac{a + A}{2} · y∥ ≤ \frac{1}{2} |A − a| ∥y∥ ,

holds. The constant \frac{1}{2} is best possible in (3.1).

From (3.1), we can deduce the following results 0 ≤ ∥x∥ ∥y∥ − Re ⟨x, y⟩ (3.4)

≤ \frac{1}{2} · \begin{matrix}Re & [(Ā + ā − 2 [Re (Aā)]^{\frac{1}{2}} ) ⟨x, y⟩] \\ [Re (Aā)] & \frac{1}{2} \end{matrix}

≤ \frac{1}{2} · \frac{|Ā + ā − 2 [Re (Aā)]^{\frac{1}{2}} |}{[Re (Aā)]^{\frac{1}{2}}} |⟨x, y⟩|

and 0 ≤ ∥x∥ ∥y∥ − |⟨x, y⟩| (3.5)

≤ \frac{1}{2} · \frac{|A + a| − 2 [Re (Aā)]^{\frac{1}{2}}}{[Re (Aā)]^{\frac{1}{2}}} |⟨x, y⟩| .

If one assumes that A = M, a = m, M ≥ m > 0, then, from (3.1), (3.4) and (3.5) we deduce the much simpler and more useful results:

(3.6)  ∥x∥ ∥y∥ ≤ \frac{1}{2} · \frac{M + m}{\sqrt{Mm}} Re ⟨x, y⟩ ,

(3.7)  0 ≤ ∥x∥ ∥y∥ − Re ⟨x, y⟩ ≤ \frac{1}{2} · \begin{matrix}(\sqrt{M} − \sqrt{m}) & 2 \\ √ & \overline{Mm} \end{matrix} Re ⟨x, y⟩

and

(3.8)  0 ≤ ∥x∥ ∥y∥ − |⟨x, y⟩| ≤ \frac{1}{2} · \begin{matrix}(\sqrt{M} − \sqrt{m}) & 2 \\ √ & \overline{Mm} \end{matrix} |⟨x, y⟩| ,

provided

Re ⟨My − x, x − my⟩ ≥ 0

or, equivalently

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

Squaring the second inequality in (3.1), we can get the following results as well:

(3.10)  0 ≤ ∥x∥^{2} ∥y∥^{2} − |⟨x, y⟩|^{2} ≤ \frac{1}{4} · \begin{matrix}|A − a| & 2 \\ Re & (Aā) \end{matrix} |⟨x, y⟩|^{2} ,

provided (3.2) or (3.1) holds. Here the constant \frac{1}{4} is also best possible.

Using the above inequalities for vectors in inner product spaces, we are able to state the following theorem concerning reverses of the (CBS) integral inequality for vector-valued functions in Hilbert spaces.

Theorem 3. Let f, g ∈ L_{ρ}^{2} ([a, b] ; K) and γ, Γ ∈ 𝕂 with Re (Γγ̄) > 0. If

(3.11)  Re ⟨Γg (t) − f (t) , f (t) − γg (t)⟩ ≥ 0

for a.e. t ∈ [a, b] , or, equivalently,

(3.12)  ∥f (t) − \frac{γ + Γ}{2} · g (t)∥ ≤ \frac{1}{2} |Γ − γ| ∥g (t)∥

for a.e. t ∈ [a, b] , then we have the inequalities

(∫_{a}^{b} ρ (t) ∥f (t)∥^{2} dt ∫_{a}^{b} ρ (t) ∥g (t)∥^{2} dt)^{\frac{1}{2}} (3.13)

≤ \frac{1}{2} · \begin{matrix}Re & [(Γ̄ + γ̄) ∫_{a}^{b} ρ (t) ⟨f (t) , g (t)⟩ dt] \\ [Re (Γγ̄)] & \frac{1}{2} \end{matrix}

≤ \frac{1}{2} · \frac{|Γ + γ|}{[Re (Γγ̄)]^{\frac{1}{2}}} |∫_{a}^{b} ρ (t) ⟨f (t) , g (t)⟩ dt| ,

0 ≤ (∫_{a}^{b} ρ (t) ∥f (t)∥^{2} dt)^{\frac{1}{2}} (∫_{a}^{b} ρ (t) ∥g (t)∥^{2} dt)^{\frac{1}{2}} (3.14)

− ∫_{a}^{b} ρ (t) Re ⟨f (t) , g (t)⟩ dt

≤ \frac{1}{2} · Re [\{Γ̄ + γ̄ − 2 [Re (Γγ̄)]^{\frac{1}{2}} \} ∫_{a}^{b} ρ (t) ⟨f (t) , g (t)⟩ dt] ; [Re (Γγ̄)]^{\frac{1}{2}}

≤ \frac{1}{2} · \frac{|Γ̄ + γ̄ − 2 [Re (Γγ̄)]^{\frac{1}{2}} |}{[Re (Γγ̄)]^{\frac{1}{2}}} |∫_{a}^{b} ρ (t) ⟨f (t) , g (t)⟩ dt| ,

0 ≤ (∫_{a}^{b} ρ (t) ∥f (t)∥^{2} dt)^{\frac{1}{2}} (∫_{a}^{b} ρ (t) ∥g (t)∥^{2} dt)^{\frac{1}{2}} (3.15)

− |∫_{a}^{b} ρ (t) ⟨f (t) , g (t)⟩ dt|

≤ \frac{1}{2} · \frac{|Γ + γ| − 2 [Re (Γγ̄)]^{\frac{1}{2}}}{[Re (Γγ̄)]^{\frac{1}{2}}} |∫_{a}^{b} ρ (t) ⟨f (t) , g (t)⟩ dt| ,

and 0 ≤ ∫_{a}^{b} ρ (t) ∥f (t)∥^{2} dt ∫_{a}^{b} ρ (t) ∥g (t)∥^{2} dt (3.16)

− |∫_{a}^{b} ρ (t) ⟨f (t) , g (t)⟩ dt|^{2}

≤ \frac{1}{4} · \begin{matrix}|Γ − γ| & 2 \\ Re & (Γγ̄) \end{matrix} |∫_{a}^{b} ρ (t) ⟨f (t) , g (t)⟩ dt|^{2} .

The constants \frac{1}{2} and \frac{1}{4} above are sharp.

In the case where Γ, γ are positive real numbers, the following corollary incorporating more convenient reverses for the (CBS) integral inequality, may be stated. Corollary 1. Let f, g ∈ L_{ρ}^{2} ([a, b] ; K) and M ≥ m > 0. If

(3.17)  Re ⟨Mg (t) − f (t) , f (t) − mg (t)⟩ ≥ 0

for a.e. t ∈ [a, b] , or, equivalently,

(3.18)  ∥f (t) − \frac{m + M}{2} · g (t)∥ ≤ \frac{1}{2} (M − m) ∥g (t)∥

for a.e. t ∈ [a, b] , then we have the inequalities

(∫_{a}^{b} ρ (t) ∥f (t)∥^{2} dt ∫_{a}^{b} ρ (t) ∥g (t)∥^{2} dt)^{\frac{1}{2}} (3.19)

≤ \frac{1}{2} · \frac{M + m}{\sqrt{mM}} ∫_{a}^{b} ρ (t) Re ⟨f (t) , g (t)⟩ dt,

0 ≤ (∫_{a}^{b} ρ (t) ∥f (t)∥^{2} dt)^{\frac{1}{2}} (∫_{a}^{b} ρ (t) ∥g (t)∥^{2} dt)^{\frac{1}{2}} (3.20)

− ∫_{a}^{b} ρ (t) Re ⟨f (t) , g (t)⟩ dt

≤ \frac{1}{2} · \begin{matrix}(\sqrt{M} − \sqrt{m}) & 2 \\ √ & \overline{mM} \end{matrix} ∫_{a}^{b} ρ (t) Re ⟨f (t) , g (t)⟩ dt,

0 ≤ (∫_{a}^{b} ρ (t) ∥f (t)∥^{2} dt)^{\frac{1}{2}} (∫_{a}^{b} ρ (t) ∥g (t)∥^{2} dt)^{\frac{1}{2}} (3.21)

− |∫_{a}^{b} ρ (t) ⟨f (t) , g (t)⟩ dt|

≤ \frac{1}{2} · \begin{matrix}(\sqrt{M} − \sqrt{m}) & 2 \\ √ & \overline{mM} \end{matrix} |∫_{a}^{b} ρ (t) ⟨f (t) , g (t)⟩ dt| ,

and

0 ≤ ∫_{a}^{b} ρ (t) ∥f (t)∥^{2} dt ∫_{a}^{b} ρ (t) ∥g (t)∥^{2} dt (3.22)

− |∫_{a}^{b} ρ (t) ⟨f (t) , g (t)⟩ dt|^{2}

≤ \frac{1}{4} · \frac{(M − m)^{2}}{mM} |∫_{a}^{b} ρ (t) ⟨f (t) , g (t)⟩ dt|^{2} .

The constants \frac{1}{2} and \frac{1}{4} above are best possible.

On utilising the general result of Theorem 2, we are able to state a number of interesting reverses for the (CBS) inequality in the case when one function takes vector-values while the other is a scalar function.

REVERSES OF THE CBS AND HEISENBERG INTEGRAL INEQUALITIES 11

Theorem 4. Let α ∈ L_{ρ}^{2} ([a, b]) , g ∈ L_{ρ}^{2} ([a, b] ; K) , e ∈ K, ∥e∥ = 1, γ, Γ ∈ 𝕂 with Re (Γγ̄) > 0. If

(3.23)  ∥g (t) − ᾱ (t) · \frac{Γ + γ}{2} e∥ ≤ \frac{1}{2} |Γ − γ| |α (t)|

for a.e. t ∈ [a, b] , or, equivalently

(3.24)  Re ⟨Γᾱ (t) e − g (t) , g (t) − γᾱ (t) e⟩ ≥ 0

for a.e. t ∈ [a, b] , (note that, if α (t) ≠ 0 for a.e. t ∈ [a, b] , then (3.23) is equivalent to

(3.25)  ∥\frac{g (t)}{\overline{α (t)}} − \frac{Γ + γ}{2} e∥ ≤ \frac{1}{2} |Γ − γ|

for a.e. t ∈ [a, b] , and (3.24) is equivalent to

(3.26)  Re ⟨Γe − \frac{g (t)}{\overline{α (t)}}, \frac{g (t)}{\overline{α (t)}} − γe⟩ ≥ 0

for a.e. t ∈ [a, b]), then the following reverse inequalities are valid:

(∫_{a}^{b} ρ (t) |α (t)|^{2} dt ∫_{a}^{b} ρ (t) ∥g (t)∥^{2} dt)^{\frac{1}{2}} (3.27)

≤ \frac{Re [(Γ̄ + γ̄) ⟨∫_{a}^{b} ρ (t) α (t) g (t) dt, e⟩]}{2 [Re (Γγ̄)]^{\frac{1}{2}}}

≤ \frac{1}{2} · \frac{|Γ + γ|}{[Re (Γγ̄)]^{\frac{1}{2}}} ∥∫_{a}^{b} ρ (t) α (t) g (t) dt∥;

12 S.S. DRAGOMIR

0 ≤ (∫_{a}^{b} ρ (t) |α (t)|^{2} dt ∫_{a}^{b} ρ (t) ∥g (t)∥^{2} dt)^{\frac{1}{2}} (3.28)

− ∥∫_{a}^{b} ρ (t) α (t) g (t) dt∥

≤ (∫_{a}^{b} ρ (t) |α (t)|^{2} dt ∫_{a}^{b} ρ (t) ∥g (t)∥^{2} dt)^{\frac{1}{2}}

− Re [\frac{Γ̄ + γ̄}{|Γ + γ|} ⟨∫_{a}^{b} ρ (t) α (t) g (t) dt, e⟩]

≤ \frac{|Γ − γ|^{2}}{2\sqrt{Re (Γγ̄)} (|Γ + γ| + 2\sqrt{Re (Γγ̄)})}

× Re [\frac{Γ̄ + γ̄}{|Γ + γ|} ⟨∫_{a}^{b} ρ (t) α (t) g (t) dt, e⟩]

≤ \frac{|Γ − γ|^{2}}{2\sqrt{Re (Γγ̄)} (|Γ + γ| + 2\sqrt{Re (Γγ̄)})} ∥∫_{a}^{b} ρ (t) α (t) g (t) dt∥;

∫_{a}^{b} ρ (t) |α (t)|^{2} dt ∫_{a}^{b} ρ (t) ∥g (t)∥^{2} dt (3.29)

≤ \frac{1}{4} · \frac{1}{Re (Γγ̄)} [Re ((\overline{Γ} + \overline{γ}) ⟨∫_{a}^{b} ρ (t) α (t) g (t) dt, e⟩)]^{2}

≤ \frac{1}{4} · \begin{matrix}|Γ + γ| & 2 \\ Re & (Γγ̄) \end{matrix} ∥∫_{a}^{b} ρ (t) α (t) g (t) dt∥^{2}

and 0 ≤ ∫_{a}^{b} ρ (t) |α (t)|^{2} dt ∫_{a}^{b} ρ (t) ∥g (t)∥^{2} dt (3.30)

− ∥∫_{a}^{b} ρ (t) α (t) g (t) dt∥^{2}

≤ ∫_{a}^{b} ρ (t) |α (t)|^{2} dt ∫_{a}^{b} ρ (t) ∥g (t)∥^{2} dt

− [Re (\frac{\overline{Γ} + \overline{γ}}{|Γ + γ|} ⟨∫_{a}^{b} ρ (t) α (t) g (t) dt, e⟩)]^{2}

REVERSES OF THE CBS AND HEISENBERG INTEGRAL INEQUALITIES 13

≤ \frac{1}{4} · \begin{matrix}|Γ − γ| & 2 \\ |Γ + γ| & 2 & Re & (Γγ̄) \end{matrix} [(Re (\overline{Γ} + \overline{γ}) ⟨∫_{a}^{b} ρ (t) α (t) g (t) dt, e⟩)]^{2}

≤ \frac{1}{4} · \begin{matrix}|Γ − γ| & 2 \\ Re & (Γγ̄) \end{matrix} ∥∫_{a}^{b} ρ (t) α (t) g (t) dt∥^{2} .

The constants \frac{1}{2} and \frac{1}{4} above are sharp.

In the particular case of positive constants, the following simpler version of the above inequalities may be stated. Corollary 2. Let α ∈ L_{ρ}^{2} ([a, b]) \ {0} , g ∈ L_{ρ}^{2} ([a, b] ; K) , e ∈ K, ∥e∥ = 1 and M, m ∈ ℝ with M ≥ m > 0. If

(3.31)  ∥\frac{g (t)}{ᾱ (t)} − \frac{M + m}{2} · e∥ ≤ \frac{1}{2} (M − m)

for a.e. t ∈ [a, b] , or, equivalently,

(3.32)  Re ⟨Me − \frac{g (t)}{ᾱ (t)}, \frac{g (t)}{ᾱ (t)} − me⟩ ≥ 0

for a.e. t ∈ [a, b] , then we have

(∫_{a}^{b} ρ (t) |α (t)|^{2} dt ∫_{a}^{b} ρ (t) ∥g (t)∥^{2} dt)^{\frac{1}{2}} (3.33)

≤ \frac{1}{2} · \frac{M + m}{\sqrt{Mm}} Re ⟨∫_{a}^{b} ρ (t) α (t) g (t) dt, e⟩

≤ \frac{1}{2} · \frac{M + m}{\sqrt{Mm}} ∥∫_{a}^{b} ρ (t) α (t) g (t) dt∥;

14 S.S. DRAGOMIR

0 ≤ (∫_{a}^{b} ρ (t) |α (t)|^{2} dt ∫_{a}^{b} ρ (t) ∥g (t)∥^{2} dt)^{\frac{1}{2}} (3.34)

− ∥∫_{a}^{b} ρ (t) α (t) g (t) dt∥

≤ (∫_{a}^{b} ρ (t) |α (t)|^{2} dt ∫_{a}^{b} ρ (t) ∥g (t)∥^{2} dt)^{\frac{1}{2}}

− Re ⟨∫_{a}^{b} ρ (t) α (t) g (t) dt, e⟩

≤ \frac{(\sqrt{M} − \sqrt{m})^{2}}{2\sqrt{Mm}} Re ⟨∫_{a}^{b} ρ (t) α (t) g (t) dt, e⟩

≤ \frac{(\sqrt{M} − \sqrt{m})^{2}}{2\sqrt{Mm}} ∥∫_{a}^{b} ρ (t) α (t) g (t) dt∥

0 ≤ ∫_{a}^{b} ρ (t) |α (t)|^{2} dt ∫_{a}^{b} ρ (t) ∥g (t)∥^{2} dt (3.35)

≤ \frac{1}{4} · \frac{(M + m)^{2}}{Mm} [Re ⟨∫_{a}^{b} ρ (t) α (t) g (t) dt, e⟩]^{2}

≤ \frac{1}{4} · \frac{(M + m)^{2}}{Mm} ∥∫_{a}^{b} ρ (t) α (t) g (t) dt∥^{2}

and 0 ≤ ∫_{a}^{b} ρ (t) |α (t)|^{2} dt ∫_{a}^{b} ρ (t) ∥g (t)∥^{2} dt (3.36)

− ∥∫_{a}^{b} ρ (t) α (t) g (t) dt∥^{2}
≤ ∫_{a}^{b} ρ (t) |α (t)|^{2} dt ∫_{a}^{b} ρ (t) ∥g (t)∥^{2} dt
− [Re ⟨∫_{a}^{b} ρ (t) α (t) g (t) dt, e⟩]^{2}

≤ \frac{1}{4} · \frac{(M − m)^{2}}{Mm} [Re ⟨∫_{a}^{b} ρ (t) α (t) g (t) dt, e⟩]^{2}

≤ \frac{1}{4} · \frac{(M − m)^{2}}{Mm} ∥∫_{a}^{b} ρ (t) α (t) g (t) dt∥^{2} .

The constants \frac{1}{2} and \frac{1}{4} above are sharp.

4. Reverses of the Heisenberg Inequality

It is well known that if (H; ⟨·, ·⟩) is a real or complex Hilbert space and f : [a, b] ⊂ ℝ →H is an absolutely continuous vector-valued function, then f is differentiable almost everywhere on [a, b] , the derivative f′ : [a, b] → H is Bochner integrable on [a, b] and

(4.1)  f (t) = ∫_{a}^{t} f′ (s) ds for any t ∈ [a, b] .

The following theorem provides a version of the Heisenberg inequalities in the general setting of Hilbert spaces.

Theorem 5. Let φ : [a, b] → H be an absolutely continuous function with the property that b ∥φ (b)∥^{2} = a ∥φ (a)∥^{2} . Then we have the inequality:

(4.2)  (∫_{a}^{b} ∥φ (t)∥^{2} dt)^{2} ≤ 4 ∫_{a}^{b} t^{2} ∥φ (t)∥^{2} dt · ∫_{a}^{b} ∥φ′ (t)∥^{2} dt.

The constant 4 is best possible in the sense that it cannot be replaced by a smaller quantity.

Proof. Integrating by parts, we have successively

∫_{a}^{b} ∥φ (t)∥^{2} dt (4.3)

= t ∥φ (t)∥^{2} |_{a}^{b} − ∫_{a}^{b} t\frac{d}{dt} (∥φ (t)∥^{2}) dt

= b ∥φ (b)∥^{2} − a ∥φ (a)∥^{2} − ∫_{a}^{b} t\frac{d}{dt} ⟨φ (t) , φ (t)⟩ dt

= − ∫_{a}^{b} t [⟨φ′ (t) , φ (t)⟩ + ⟨φ (t) , φ′ (t)⟩] dt

= −2 ∫_{a}^{b} t Re ⟨φ′ (t) , φ (t)⟩ dt

= 2 ∫_{a}^{b} Re ⟨φ′ (t) , (−t) φ (t)⟩ dt.

If we apply the (CBS) integral inequality

∫_{a}^{b} Re ⟨g (t) , h (t)⟩ dt ≤ (∫_{a}^{b} ∥g (t)∥^{2} dt ∫_{a}^{b} ∥h (t)∥^{2} dt)^{\frac{1}{2}}

for g (t) = φ′ (t) , h (t) = −tφ (t) , t ∈ [a, b] , then we deduce the desired inequality (4.2).

The fact that 4 is the best possible constant in (4.2) follows from the fact that in the (CBS) inequality, the case of equality holds iff g (t) = λh (t) for a.e. t ∈ [a, b] and λ a given scalar in 𝕂. We omit the details.

For details on the classical Heisenberg inequality, see, for instance, [2].

The following reverse of the Heisenberg type inequality (4.2) holds.

Theorem 6. Assume that φ : [a, b] → H is as in the hypothesis of Theorem 5. In addition, if there exists a r > 0 such that

(4.4)  ∥φ′ (t) − tφ (t)∥ ≤ r ≤ ∥φ′ (t)∥

for a.e. t ∈ [a, b] , then we have the inequalities

0 ≤ ∫_{a}^{b} t^{2} ∥φ (t)∥^{2} dt ∫_{a}^{b} ∥φ′ (t)∥^{2} dt − \frac{1}{4} (∫_{a}^{b} ∥φ (t)∥^{2} dt)^{2} (4.5)

≤ r^{2} ∫_{a}^{b} t^{2} ∥φ (t)∥^{2} dt.

Proof. We observe, by the identity (4.3), that

(4.6)  \frac{1}{4} (∫_{a}^{b} ∥φ (t)∥^{2} dt)^{2} = (∫_{a}^{b} Re ⟨φ′ (t) , tφ (t)⟩ dt)^{2} .

Now, if we apply Theorem 1 for the choices f (t) = tφ (t) , g (t) = φ′ (t) , and ρ (t) = \frac{1}{b−a}, then by (2.2) and (4.6) we deduce the desired inequality (4.5).

Remark 1. Interchanging the place of tφ (t) with φ′ (t) in Theorem 6, we also have

0 ≤ ∫_{a}^{b} t^{2} ∥φ (t)∥^{2} dt ∫_{a}^{b} ∥φ′ (t)∥^{2} dt − \frac{1}{4} (∫_{a}^{b} ∥φ (t)∥^{2} dt)^{2} (4.7)

≤ ρ^{2} ∫_{a}^{b} ∥φ′ (t)∥^{2} dt,

provided

∥φ′ (t) − tφ (t)∥ ≤ ρ ≤ |t| ∥φ (t)∥

for a.e. t ∈ [a, b] , where ρ > 0 is a given positive number.

The following result also holds.

Theorem 7. Assume that φ : [a, b] → H is as in the hypothesis of Theorem 5. In addition, if there exists M ≥ m > 0 such that

(4.8)  Re ⟨Mtφ (t) − φ′ (t) , φ′ (t) − mtφ (t)⟩ ≥ 0

for a.e. t ∈ [a, b] , or, equivalently,

(4.9)  ∥φ′ (t) − \frac{M + m}{2}tφ (t)∥ ≤ \frac{1}{2} (M − m) |t| ∥φ (t)∥

for a.e. t ∈ [a, b] , then we have the inequalities

∫_{a}^{b} t^{2} ∥φ (t)∥^{2} dt ∫_{a}^{b} ∥φ′ (t)∥^{2} dt (4.10)

≤ \frac{1}{16} · \frac{(M + m)^{2}}{Mm} (∫_{a}^{b} ∥φ (t)∥^{2} dt)^{2}

and

∫_{a}^{b} t^{2} ∥φ (t)∥^{2} dt ∫_{a}^{b} ∥φ′ (t)∥^{2} dt − \frac{1}{4} (∫_{a}^{b} ∥φ (t)∥^{2} dt)^{2} (4.11)

≤ \frac{1}{16} · \frac{(M − m)^{2}}{Mm} (∫_{a}^{b} ∥φ (t)∥^{2} dt)^{2}

respectively.

Proof. We use Corollary 1 for the choices f (t) = φ′ (t) , g (t) = tφ (t) , ρ (t) = \frac{1}{b−a}, to get

∫_{a}^{b} ∥φ′ (t)∥^{2} dt ∫_{a}^{b} t^{2} ∥φ (t)∥^{2} dt

≤ \frac{(M + m)^{2}}{4Mm} (∫_{a}^{b} Re ⟨φ′ (t) , tφ (t)⟩ dt)^{2} .

Since, by (4.6)

(∫_{a}^{b} Re ⟨φ′ (t) , tφ (t)⟩ dt)^{2} = \frac{1}{4} (∫_{a}^{b} ∥φ (t)∥^{2} dt)^{2} ,

hence we deduce the desired result (4.10).

The inequality (4.11) follows from (4.10), and we omit the details.

Remark 2. If one is interested in reverses for the Heisenberg inequality for scalar valued functions, then all the other inequalities obtained above for one scalar function may be applied as well. For the sake of brevity, we do not list them here.

References

[1] S.S. DRAGOMIR, Reverses of Schwarz, triangle and Bessel inequalities in inner product spaces, RGMIA Res. Rep. Coll., 6(2003), Supplement, Art. 19

   [ONLINE http://rgmia.vu.edu.au/v6(E).html].

[2] G.H. HARDY, J.E. LITTLEWOOD and G. POLYA, Inequalities, Cambridge University Press, Cambridge, United Kingdom, 1952.

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