Integral Gr

INTEGRAL GR ¨ USS INEQUALITY FOR MAPPINGS WITH VALUES IN HILBERT SPACES AND APPLICATIONS

S. S. Dragomir

Abstract. In this paper we prove a version of Gr¨ uss’ integral in- equality for mappings with values in Hilbert spaces. Some applica- tions for convex functions defined on Hilbert spaces are also given.

1. Introduction

In 1935, G. Gr¨ uss proved the following integral inequality [12]

¯ ¯

¯ ¯ 1 b − a

Z _b

a

f (x) g (x) dx − 1 b − a

Z _b

a

f (x) dx · 1 b − a

Z _b

a

g (x) dx

¯ ¯

¯ ¯ (1.1)

≤ 1

4 (Φ − φ) (Γ − γ) ,

provided that f and g are two integrable functions on [a, b] and satisfy the condition

(1.2) φ ≤ f (x) ≤ Φ and γ ≤ g (x) ≤ Γ for a.e. x ∈ [a, b] . The constant ¹ ₄ is the best possible and is achieved for

f (x) = g (x) = sgn µ

x − a + b 2

¶ . The discrete version of (1.1) states that:

If a ≤ a _i ≤ A, b ≤ b _i ≤ B (i = 1, ..., n) where a, A, a _i , b, B, b _i are real numbers, then

(1.3)

¯ ¯

¯ ¯

¯ 1 n

X n i=1

a _i b _i − 1 n

X n i=1

a _i · 1 n

X n i=1

b _i

¯ ¯

¯ ¯

¯ ≤ 1

4 (A − a) (B − b) , where the constant ¹ ₄ is the best possible.

Received September 17, 2001.

2000 Mathematics Subject Classification: 26D15, 46A99.

Key words and phrases: Gr¨ uss inequality, Bochner integral.

For an entire chapter devoted to the history of this inequality see the book [14] where further references are given.

New results in the domain can be found in the papers [1]-[9] and [13].

In the recent paper [2], the author proved the following generalization in inner product spaces.

Theorem 1. Let (X, h·, ·i) be an inner product space over K, K = C, R and e ∈ X, kek = 1. If φ, Φ, γ, Γ ∈ K and x, y ∈ X are such that (1.4) Re hΦe − x, x − φei ≥ 0 and Re hΓe − y, y − γei ≥ 0, holds, then we have the inequality

(1.5) |hx, yi − hx, ei he, yi| ≤ 1

4 |Φ − φ| |Γ − γ| . The constant ¹ ₄ is the best possible.

It has been shown in [1] that the above theorem, for real cases, con- tains the usual integral and discrete Gr¨ uss inequality and also some Gr¨ uss type inequalities for mappings defined on infinite intervals.

Namely, if ρ : (−∞, ∞) → (−∞, ∞) is a probabilistic density func- tion, i.e., R _∞

−∞ ρ (t) dt = 1, then ρ

∈ L ² (−∞, ∞) and obviously

° °

°ρ

° °

° 2 = 1. Consequently, if we assume that f, g ∈ L ² (−∞, ∞) and

(1.6) α · ρ

≤ f ≤ ψ · ρ

, β · ρ

≤ g ≤ θ · ρ

a.e. on (−∞, ∞) , then we have the inequality

¯ ¯

¯ ¯ Z _∞

−∞

f (t) g (t) dt − Z _∞

−∞

f (t) ρ

(t) dt · Z _∞

−∞

g (t) ρ

(t) dt

¯ ¯

¯ ¯ (1.7)

≤ 1

4 (ψ − α) (θ − β) .

Similarly, if l = (l _i ) _i∈N ∈ l ² (R) with P

i∈N |l _i | ² = 1 and x = (x _i ) _i∈N , y = (y _i ) _i∈N ∈ l ² (R) are such that

(1.8) α · l _i ≤ x _i ≤ ψ · l _i , β · l _i ≤ y _i ≤ θ · l _i for all i ∈ N, then we have

(1.9)

¯ ¯

¯ ¯

¯ X

i∈N

x _i y _i − X

i∈N

x _i l _i · X

i∈N

y _i l _i

¯ ¯

¯ ¯

¯ ≤ 1

4 (ψ − α) (θ − β) .

In this paper we point out a Gr¨ uss type inequality for Bochner mea-

surable functions with values in Hilbert spaces.

2. Preliminary results The following lemma holds.

Lemma 1. Let (H; h·, ·i) be a Hilbert space over the real or complex number field K, f : Ω → H a Bochner measurable function on Ω ⊂ R ⁿ such that f ∈ L ₂ (Ω, H) and ρ : Ω → [0, ∞) a Lebesgue integrable mapping on Ω so that R

Ω ρ (t) dµ (t) = 1. If there exists the vectors x, X ∈ H such that

(2.1) Re hX − f (t) , f (t) − xi ≥ 0 for a.e. t ∈ Ω, then we have the inequality

0 ≤ Z

Ω

ρ (t) kf (t)k ² dµ (t) −

° °

° ° Z

Ω

ρ (t) f (t) dµ (t)

° °

° °

2

(2.2)

≤ 1

4 kX − xk ² . The constant ¹ ₄ is sharp.

Proof. Define I ₁ :=

¿ X −

Z

Ω

ρ (t) f (t) dµ (t) , Z

Ω

ρ (t) f (t) dµ (t) − x À

and

I ₂ :=

Z

Ω

ρ (t) hX − f (t) , f (t) − xi dµ (t) . Then obviously

I ₁ = Z

Ω

ρ (t) hX, f (t)i dµ (t) − hX, xi

−

° °

° ° Z

Ω

ρ (t) f (t) dµ (t)

° °

° °

2

+ Z

Ω

ρ (t) hf (t) , xi dµ (t) and

I ₂ = Z

Ω

ρ (t) hX, f (t)i dµ (t) − hX, xi

− Z

Ω

ρ (t) kf (t)k ² dµ (t) + Z

Ω

ρ (t) hf (t) , xi dµ (t) . Consequently

(2.3) I ₁ − I ₂ = Z

Ω

ρ (t) kf (t)k ² dµ (t) −

° °

° ° Z

Ω

ρ (t) f (t) dµ (t)

° °

° °

2

.

Taking the real value in (2.3), we can state Z

Ω

ρ (t) kf (t)k ² dµ (t) −

° °

° ° Z

Ω

ρ (t) f (t) dµ (t)

° °

° °

2

(2.4)

= Re

¿ X −

Z

Ω

ρ (t) f (t) dµ (t) , Z

Ω

ρ (t) f (t) dµ (t) − x À

− Z

Ω

ρ (t) Re hX − f (t) , f (t) − xi dµ (t) which is an identity of interest in itself.

Using the assumption (2.1), we may conclude, by (2.4), that Z

Ω

ρ (t) kf (t)k ² dµ (t) −

° °

° ° Z

Ω

ρ (t) f (t) dµ (t)

° °

° °

2

(2.5)

≤ Re

¿ X −

Z

Ω

ρ (t) f (t) dµ (t) , Z

Ω

ρ (t) f (t) dµ (t) − x À

. It is known that if y, z ∈ H, then

(2.6) 4Re hz, yi ≤ kz + yk ² , with equality iff z = y.

Now, by (2.6), we can state that Re

¿ X −

Z

Ω

ρ (t) f (t) dµ (t) , Z

Ω

ρ (t) f (t) dµ (t) − x À

≤ 1 4

° °

° °X − Z

Ω

ρ (t) f (t) dµ (t) + Z

Ω

ρ (t) f (t) dµ (t) − x

° °

° °

2

= 1

4 kX − xk ² . Using (2.5), we deduce (2.2).

To prove the sharpness of the constant ¹ ₄ , let us assume that the inequality (2.2) holds with a constant c > 0. That is,

Z

Ω

ρ (t) kf (t)k ² dµ (t) −

° °

° ° Z

Ω

ρ (t) f (t) dµ (t)

° °

° °

2

≤ c kX − xk ² ,

for any Ω, µ, ρ, f and x, X as above.

Choose Ω = {1, 2}, ρ (1) = ρ (2) = ¹ ₂ , µ is the discrete measure on Ω and f (1) = x, f (2) = X (x 6= X). Then obviously

Z

Ω

ρ (t) kf (t)k ² dµ (t) = kxk ² + kXk ²

2 ,

° °

° ° Z

Ω

ρ (t) f (t) dµ (t)

° °

° °

2

=

° °

° ° x + X 2

° °

° °

2

and by (2.1), we deduce kxk ² + kXk ²

2 −

° °

° ° x + X 2

° °

° °

2

≤ c kX − xk ² which is clearly equivalent to

1

4 kX − xk ² ≤ c kX − xk ² implying that c ≥ ¹ ₄ .

Remark 1. The assumption (2.1) can be replaced by the more gen- eral condition

(2.7)

Z

Ω

ρ (t) Re hX − f (t) , f (t) − xi dµ (t) ≥ 0 and the conclusion (2.2) will still be valid.

The following corollary is natural.

Corollary 1. Let g : Ω → K be Lebesgue integrable on Ω and ρ be as above. If a, A ∈ K are such that

(2.8) Re

h

(A − g (t))

³

g (t) − ¯a

´i

≥ 0 a.e on Ω, then we have the inequality

0 ≤ Z

Ω

ρ (t) |g (t)| ² dµ (t) −

¯ ¯

¯ ¯ Z

Ω

ρ (t) g (t) dµ (t)

¯ ¯

¯ ¯

2

(2.9)

≤ 1

4 |A − a| ² . The constant ¹ ₄ is sharp.

Remark 2. The condition (2.8) can be replaced by the more general assumption

(2.10)

Z

Ω

ρ (t) Re h

(A − g (t))

³

g (t) − ¯a

´i

dµ (t) ≥ 0.

Remark 3. If we assume that K = R, then (2.8) is equivalent with (2.11) a ≤ g (t) ≤ A a.e. on Ω

and then, with the assumption (2.11), we get the Gr¨ uss type inequality 0 ≤

Z

Ω

ρ (t) g ² (t) dµ (t) − µZ

Ω

ρ (t) g (t) dµ (t)

¶ ₂ (2.12)

≤ 1

4 (A − a) ² .

3. An inequality of Gr¨ uss type The following Gr¨ uss type inequality holds.

Theorem 2. Let (H; h·, ·i) be a Hilbert space over K, K = C, R, f, g : Ω → K be Bochner measurable and f, g ∈ L ₂ (Ω, H) and ρ : Ω → [0, ∞) Lebesgue integrable so that R

Ω ρ (t) dµ (t) = 1. If there exists the vectors x, X, y, Y ∈ H such that

Re hX − f (t) , f (t) − xi ≥ 0 and Re hY − g (t) , g (t) − yi ≥ 0 (3.1)

for a.e. t on Ω, then we have the inequality

¯ ¯

¯ ¯

¯ Z

Ω

ρ (t) hf (t) , g (t)i dµ (t)

−

¿Z

Ω

ρ (t) f (t) dµ (t) , Z

Ω

ρ (t) g (t) dµ (t) À ¯¯ ¯

¯ ¯

≤ 1

4 kX − xk kY − yk . (3.2)

The constant ¹ ₄ is sharp.

Proof. A simple calculation shows that Z

Ω

ρ (t) hf (t) , g (t)i dµ (t)

−

¿Z

Ω

ρ (t) f (t) dµ (t) , Z

Ω

ρ (t) g (t) dµ (t) À

= 1 2

Z

Ω

Z

Ω

ρ (t) ρ (s) hf (t) − f (s) , g (t) − g (s)i dµ (t) dµ (s) .

(3.3)

Taking the modulus in both parts of (3.3), and using the generalised triangle inequality, we obtain

¯ ¯

¯ ¯

¯ Z

Ω

ρ (t) hf (t) , g (t)i dµ (t)

−

¿Z

Ω

ρ (t) f (t) dµ (t) , Z

Ω

ρ (t) g (t) dµ (t) À ¯¯ ¯

¯ ¯

≤ 1 2

Z

Ω

Z

Ω

ρ (t) ρ (s) |hf (t) − f (s) , g (t) − g (s)i| dµ (t) dµ (s) . (3.4)

By Schwartz’s inequality in inner product spaces, we have

(3.5) |hf (t) − f (s) , g (t) − g (s)i| ≤ kf (t) − f (s)k kg (t) − g (s)k for all t, s ∈ Ω, and therefore

¯ ¯

¯ ¯

¯ Z

Ω

ρ (t) hf (t) , g (t)i dµ (t)

−

¿Z

Ω

ρ (t) f (t) dµ (t) , Z

Ω

ρ (t) g (t) dµ (t) À ¯¯ ¯ ¯

¯

≤ 1 2

Z

Ω

Z

Ω

ρ (t) ρ (s) kf (t) − f (s)k kg (t) − g (s)k dµ (t) dµ (s) . (3.6)

Using the Cauchy-Buniakowski-Schwartz inequality for double integrals, we can state that

1 2

Z

Ω

Z

Ω

ρ (t) ρ (s) kf (t) − f (s)k kg (t) − g (s)k dµ (t) dµ (s) (3.7)

≤ µ 1

2 Z

Ω

Z

Ω

ρ (t) ρ (s) kf (t) − f (s)k ² dµ (t) dµ (s)

¶

2

× µ 1

2 Z

Ω

Z

Ω

ρ (t) ρ (s) kg (t) − g (s)k ² dµ (t) dµ (s)

¶

2

.

As a simple calculation shows that 1

2 Z

Ω

Z

Ω

ρ (t) ρ (s) kf (t) − f (s)k ² dµ (t) dµ (s)

= Z

Ω

ρ (t) kf (t)k ² dµ (t) −

° °

° ° Z

Ω

ρ (t) f (t) dµ (t)

° °

° °

2

and

1 2

Z

Ω

Z

Ω

ρ (t) ρ (s) kg (t) − g (s)k ² dµ (t) dµ (s)

= Z

Ω

ρ (t) kg (t)k ² dµ (t) −

° °

° ° Z

Ω

ρ (t) g (t) dµ (t)

° °

° °

2

, then we obtain ¯

¯ ¯

¯ ¯ Z

Ω

ρ (t) hf (t) , g (t)i dµ (t)

−

¿Z

Ω

ρ (t) f (t) dµ (t) , Z

Ω

ρ (t) g (t) dµ (t) À ¯¯ ¯

¯ ¯

≤ ÃZ

Ω

ρ (t) kf (t)k ² dµ (t) −

° °

° ° Z

Ω

ρ (t) f (t) dµ (t)

° °

° °

2 !

2

× ÃZ

Ω

ρ (t) kg (t)k ² dµ (t) −

° °

° ° Z

Ω

ρ (t) g (t) dµ (t)

° °

° °

2 !

2

. (3.8)

Using Lemma 1, we know that ÃZ

Ω

ρ (t) kf (t)k ² dµ (t) −

° °

° ° Z

Ω

ρ (t) f (t) dµ (t)

° °

° °

2 !

2

≤ 1

2 kX − xk and ÃZ

Ω

ρ (t) kg (t)k ² dµ (t) −

° °

° ° Z

Ω

ρ (t) g (t) dµ (t)

° °

° °

2 !

2

≤ 1

2 kY − yk and then, by (3.8), we deduce the desired inequality (3.2).

The sharpness of the constant is obvious by Lemma 1 and we omit the details.

Remark 4. The condition (3.1) can be replaced by the more general assumption:

Z

Ω

ρ (t) Re hX − f (t) , f (t) − xi dµ (t) ≥ 0, (3.9)

Z

Ω

ρ (t) Re hY − g (t) , g (t) − yi dµ (t) ≥ 0 and the conclusion (3.2) still remains valid.

The following corollary for real or complex functions holds.

Corollary 2. Let f, g : Ω → K be in L ₂ (Ω, K) and ρ : Ω → [0, ∞) be as above. If a, A, b, B ∈ K are such that

Re h

(A − f (t))

³

f (t) − ¯a

´i

≥ 0, (3.10)

Re h

(B − g (t))

³

g (t) − ¯b

´i

≥ 0 for a.e. t ∈ Ω, then we have the inequality

¯ ¯

¯ ¯ Z

Ω

ρ (t) f (t) ¯ g (t) dµ (t) − Z

Ω

ρ (t) f (t) dµ (t) · Z

Ω

ρ (t) ¯ g (t) dµ (t)

¯ ¯

¯ ¯ (3.11)

≤ 1

4 |A − a| |B − b|

and the constant ¹ ₄ is sharp.

The proof is obvious by Theorem 2 applied for the Hilbert space (K, h·, ·i), hx, yi = x · ¯ y. We omit the details.

Remark 5. The condition (3.10) can be replaced by the more general condition

Z

Ω

ρ (t) Re h

(A − f (t))

³

f (t) − ¯a

´i

dµ (t) ≥ 0, (3.12)

Z

Ω

ρ (t) Re h

(B − g (t))

³

g (t) − ¯b

´i

dµ (t) ≥ 0 and the conclusion of the above corollary will still remain valid.

Remark 6. If we assume that f, g, a, b, A, B are real, then (3.10) is equivalent to

(3.13) a ≤ f (t) ≤ A, b ≤ g (t) ≤ B for a.e. t on Ω, and (3.11) becomes

¯ ¯

¯ ¯ Z

Ω

ρ (t) f (t) g (t) dµ (t) − Z

Ω

ρ (t) f (t) dµ (t) · Z

Ω

ρ (t) g (t) dµ (t)

¯ ¯

¯ ¯ (3.14)

≤ 1

4 (A − a) (B − b)

which is the classical Gr¨ uss inequality for real valued functions.

4. Applications for convex functions

Let (H; h·, ·i) be a real Hilbert space and F : H → R a Fr´echet differ- entiable convex mapping on H. Then we have the “gradient inequality”, that is,

(4.1) F (x) − F (y) ≥ hOF (y) , x − yi

for all x, y ∈ H, where ∇F : H → H is the gradient operator associated to the convex function F .

The following theorem containing a reverse inequality for Jensen’s inequality for the measurable space Ω with µ (Ω) < ∞, holds.

Theorem 3. Let F : H → R be as above, f : Ω → H a Bochner measurable function such that there exists the vectors n, N ∈ H with the property

(4.2) hf (t) − n, N − f (t)i ≥ 0 a.e. on Ω, and the vectors m, M ∈ H such that

(4.3) hOF (f (t)) − m, M − OF (f (t))i ≥ 0 a.e. on Ω.

If ρ : Ω → [0, ∞) is Lebesgue measurable on Ω such that R

Ω ρ (t) dµ (t) >

0 and ρF (f (·)) ∈ L (Ω), ρ · f ∈ L (Ω, H), then we have the inequality

0 ≤ R 1

Ω ρ (t) dµ (t) Z

Ω

ρ (t) F (f (t)) dµ (t) (4.4)

−F

µ 1

R

Ω ρ (t) dµ (t) Z

Ω

ρ (t) f (t) dµ (t)

¶

≤ 1

4 kN − nk kM − mk . Proof. Choose in (4.1)

x = 1

R

Ω ρ (t) dµ (t) Z

Ω

ρ (t) f (t) dµ (t) and y = f (s) to obtain

F

µ R 1

Ω ρ (t) dµ (t) Z

Ω

ρ (t) f (t) dµ (t)

¶

− F (f (s)) (4.5)

≥

¿

∇F (f (s)) , R 1

Ω ρ (t) dµ (t) Z

Ω

ρ (t) f (t) dµ (t) − f (s) À

for a.e. s ∈ Ω.

If we multiply (4.5) by ρ (s) ≥ 0 and integrate on Ω, we have Z

Ω

ρ (s) dµ (s) F

µ R 1

Ω ρ (t) dµ (t) Z

Ω

ρ (t) f (t) dµ (t)

¶

− Z

Ω

ρ (s) (F ◦ f ) (s) dµ (s)

≥ R 1

Ω ρ (t) dµ (t)

¿Z

Ω

ρ (t)

³

(OF ) ◦ (f )

´

(s) dµ (t) , Z

Ω

ρ (t) f (t) dµ (t) À

− Z

Ω

ρ (s) hOF (f (s)) , f (s)i dµ (s) . Dividing by R

Ω ρ (t) dµ (t) > 0, we obtain the inequality

0 ≤ 1

R

Ω ρ (t) dµ (t) Z

Ω

ρ (s) (F ◦ f ) (s) dµ (s) (4.6)

−F

µ R 1

Ω ρ (t) dµ (t) Z

Ω

ρ (t) f (t) dµ (t)

¶

≤ 1

R

Ω ρ (t) dµ (t) Z

Ω

ρ (t) h((OF ) ◦ f ) (t) , f (t)dµ (t)i

−

¿ 1

R

Ω ρ (t) dµ (t) Z

Ω

ρ (t) (OF ◦ f ) (t) dµ (t) , R 1

Ω ρ (t) dµ (t) Z

Ω

ρ (t) f (t) dµ (t) À

which is a generalisation for the case of inner product spaces of the result by Dragomir-Goh established in 1996 for the case of differentiable mappings defined on R ⁿ [10].

Applying Theorem 1 for the case of real inner product spaces, X = N , x = n, g (t) = (∇F ◦ ρ) (t), y = m, Y = M , we easily deduce

R 1

Ω ρ (t) dµ (t) Z

Ω

ρ (t) h(OF ◦ f ) (t) , f (t)i dµ (t) (4.7)

−

¿ 1

R

Ω ρ (t) dµ (t) Z

Ω

ρ (t) (OF ◦ f ) (t) dµ (t) , R 1

Ω ρ (t) dµ (t) Z

Ω

ρ (t) f (t) dµ (t) À

≤ 1

4 kN − nk kM − mk

and then, by (4.6) and (4.7), we can conclude that the desired inequality (4.4) holds.

Remark 7. The conditions (4.2) and (4.3) can be replaced by the more general assumptions

(4.8)

Z

Ω

ρ (t) hf (t) − n, N − f (t)i dµ (t) ≥ 0 and

(4.9)

Z

Ω

ρ (t) h(OF ◦ f ) (t) − m, M − (OF ◦ f ) (t)i dµ (t) ≥ 0 and the conclusion (4.4) will still be valid.

Remark 8. Even if the inequality (4.4) is not as sharp as (4.6), it can be more useful in practice when only some bounds of the gradient operator ∇F and the mapping f are known. On other words, it provides the opportunity to estimate the difference

∆ (F, f, ρ) := R 1

Ω ρ (t) dµ (t) Z

Ω

ρ (t) (F ◦ f ) (t) dµ (t)

−F

µ 1

R

Ω ρ (t) dµ (t) Z

Ω

ρ (t) f (t) dµ (t)

¶ , when the differences kN − nk and kM − mk are known.

For example, if we know that

h(OF ◦ f ) (t) − m, M − (OF ◦ f ) (t)i ≥ 0 for a.e. t ∈ Ω, hf (t) − n, N − f (t)i ≥ 0 for a.e. t ∈ Ω, and

kN − nk ≤ 4ε

kM − mk (ε > 0) , then by (4.4) we can conclude that

0 ≤ ∆ (F, f, ρ) ≤ ε.

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