A GRÜSS TYPE INEQUALITY FOR ISOTONIC LINEAR FUNCTIONALS AND APPLICATIONS

S.S. DRAGOMIR

Abstract. An inequality for a normalised isotonic linear functional of Grüss type and particular cases for integrals and norms are established. Applications in obtaining a counterpart for the Cauchy-Buniakowski-Schwartz inequality for functionals and Jessen’s inequality for convex functions are also given.

1. Introduction

Let L be a linear class of real-valued functions g : E → ℝ having the properties (L1) f, g ∈ L imply (αf + βg) ∈ L for all α, β ∈ ℝ; (L2) 1 ∈ L, i.e., if f (t) = 1, t ∈ E, then f ∈ L. An isotonic linear functional A : L → ℝ is a functional satisfying (A1) A (αf + βg) = αA (f) + βA (g) for all f, g ∈ L and α, β ∈ ℝ; (A2) If f ∈ L and f ≥ 0, then A (f) ≥ 0. The mapping A is said to be normalised if (A3) A (1) = 1. Usual examples of isotonic linear functionals that are normalised are the following

ones

A (f) = \frac{1}{μ (X)} ∫_{X} f (x) dμ (x) if μ (X) < ∞

or

A_{w} (f) := \frac{1}{∫_{X} w (x) dμ (x)} ∫_{X} w (x) f (x) dμ (x) ,

where w (x) ≥ 0, ∫_{X} w (x) dμ (x) > 0, X is a measurable space and μ a positive measure on X.

In particular, for x̄ = (x_{1}, …, x_{n}) , w̄ := (w_{1}, …, w_{n}) ∈ ℝ^{n} with w_{i} ≥ 0, W_{n} := ∑_{i=1}^{n} w_{i} > 0, we have

A \binom{}{x̄} := \frac{1}{n} ∑_{i=1}^{n} x_{i}

and

Aw̄ \binom{}{x̄} := \frac{1}{W_{n}} ∑_{i=1}^{n} w_{i}x_{i},

are normalised isotonic functionals on ℝ^{n}.

Date: May 07, 2002.

1991 Mathematics Subject Classification. Primary 26D15; Secondary 26D10.

Key words and phrases. Grüss inequality, Isotonic linear functionals, Cauchy-Buniakowski-Schwartz inequality, Jessen’s inequality, Jensen Discrete and Integral Inequalities.

In 1988, D. Andrica and C. Badea [1], proved the following generalisation of the Grüss inequality for isotonic linear functionals.

Theorem 1. If f, g ∈ L so that fg ∈ L and m ≤ f ≤ M, n ≤ g ≤ N where m, M, n, N are given real numbers, then for any normalised isotonic linear functional A : L → ℝ one has the inequality

(1.1)  |A (fg) − A (f) A (g)| ≤ \frac{1}{4} (M − m) (N − n) .

The constant \frac{1}{4} in (1.1) is best possible in the sense that it cannot be replaced by a smaller constant.

In this paper we point out a refinement of the Grüss inequality (1.1) for isotonic linear functionals. Applications for the Cauchy-Buniakowski-Schwartz and Jessen’s inequality are also provided.

2. A Grüss Type Inequality

The following result holds.

Theorem 2. Let f, g ∈ L be such that fg ∈ L and assume that there exists the real numbers n and N so that

(2.1)  n ≤ g ≤ N.

Then for any normalised isotonic linear functional A : L → ℝ for which |f − A (f) · 1| ∈ L one has the inequality

(2.2)  |A (fg) − A (f) A (g)| ≤ \frac{1}{2} (N − n) A (|f − A (f) · 1|) .

The constant \frac{1}{2} in (2.2) is best possible in the sense that it cannot be replaced by a smaller constant.

Proof. Using the linearity property of A, we have

A [(f − A (f) · 1) (g − \frac{n + N}{2} · 1)] (2.3)

= A [(f − A (f) · 1) g] − \frac{n + N}{2} A [f − A (f) · 1]
= A (fg) − A (f) A (g) − \frac{n + N}{2} [A (f) − A (f) · A (1)]
= A (fg) − A (f) A (g)

since, by the normality property of A, A (1) = 1.

From (2.1) we may easily deduce that

(2.4)  |g − \frac{n + N}{2} · 1| ≤ \frac{M − n}{2} · 1.

It is known that if h ∈ L so that |h| ∈ L, then, by the monotonicity and linearity of A, one has

(2.5)  |A (h)| ≤ A (|h|) .

Using this property, the monotonicity property of A and condition (2.4), we deduce

|A [(f − A (f) · 1) (g − \frac{n + N}{2} · 1)]| (2.6)

≤ A (|(f − A (f) · 1) (g − \frac{n + N}{2} · 1)|)
≤ \frac{N − n}{2}A (|f − A (f) · 1|) .

Utilising (2.3) and (2.6) we deduce the desired result (2.2).

To prove the sharpness of the constant \frac{1}{2}, we assume that (2.2) holds with a constant c > 0 for A = \frac{1}{b−a} ∫_{a}^{b}, L = L [a, b] (the Lebesgue space of integrable functions on [a, b]) and g satisfying the condition (2.1) on the interval [a, b] , i.e., one has the inequality

(2.7) |\frac{1}{b − a} ∫_{a}^{b} f (x) g (x) dx − \frac{1}{b − a} ∫_{a}^{b} f (x) dx · \frac{1}{b − a} ∫_{a}^{b} g (x) dx|

≤ c (N − n) · \frac{1}{b − a} ∫_{a}^{b} |f (x) − \frac{1}{b − a} ∫_{a}^{b} f (y) dy| dx.

If we choose g = f and f : [a, b] → ℝ,

f (x) = \{ \begin{matrix}−1 & if x ∈ [a, \frac{a+b}{2}] \\ 1 & if x ∈ (\frac{a+b}{2}, b] \end{matrix}

then

\frac{1}{b − a} ∫_{a}^{b} f^{2} (x) dx − (\frac{1}{b − a} ∫_{a}^{b} f (x) dx)^{2} = 1,

\frac{1}{b − a} ∫_{a}^{b} |f (x) − \frac{1}{b − a} ∫_{a}^{b} f (y) dy| dx = 1,
m = − 1, M = 1.

and by (2.7) we deduce c ≥ \frac{1}{2}.

The following corollaries are natural consequences of the above result.

Corollary 1. Let f ∈ L be such that f^{2} ∈ L and there exists the real numbers m, M so that

(2.8)  m ≤ f ≤ M.

Then for any A : L → ℝ a normalised isotonic linear functional so that |f − A (f) · 1| ∈ L one has the inequality

(2.9)  0 ≤ A (f^{2}) − [A (f)]^{2} ≤ \frac{1}{2} (M − m) A (|f − A (f) · 1|) .

The constant \frac{1}{2} is sharp.

Corollary 2. Let f, g ∈ L so that fg ∈ L and f satisfy (2.8) while g satisfies (2.1). Then for any normalised isotonic linear functional A : L → ℝ so that |f − A (f) · 1| , |g − A (g) · 1| ∈ L one has the inequality:

(2.10) |A (fg) − A (f) A (g)|

≤ \frac{1}{2} [(M − m) (N − n)]^{\frac{1}{2}} [A (|f − A (f) · 1|) A (|g − A (g) · 1|)]^{\frac{1}{2}} .

The constant \frac{1}{2} is sharp. Remark 1. Using Hölder’s inequality for isotonic linear functionals, we may state the following inequalities as well

|A (fg) − A (f) A (g)|
≤ \frac{1}{2} (N − n) A (|f − A (f) · 1|) if |f − A (f) · 1| ∈ L,
≤ \frac{1}{2} (N − n) [A (|f − A (f) · 1|^{p})]^{\frac{1}{p}} if |f − A (f) · 1|^{p} ∈ L, p > 1
≤ sup_{t∈E} |f (t) − A (f)|;

provided f, g ∈ L and fg ∈ L while g satisfies the condition (2.1).

If f and g fulfill the conditions (2.8) and (2.1), then we have the following refinement of the Grüss inequality (1.1)

(2.11)  |A (fg) − A (f) A (g)| ≤ \frac{1}{2} (N − n) A (|f − A (f) · 1|)
≤ \frac{1}{2} (N − n) [A (f^{2}) − [A (f)]^{2}]^{\frac{1}{2}}
≤ \frac{1}{4} (M − m) (N − n) .

The constants \frac{1}{2}, \frac{1}{2} and \frac{1}{4} are sharp in (2.11).

The following weighted version of Theorem 2 also holds. Theorem 3. Let f, g, h ∈ L be such that h ≥ 0, fh, gh, fgh ∈ L and there exists the real constants n, N so that (2.1) holds. Then for any B : L → ℝ an isotonic linear functional so that B (h) > 0, h |f − \frac{1}{B(h)} · 1| ∈ L one has the inequality:

(2.12) |\frac{B (fgh)}{B (h)} − \frac{B (fh)}{B (h)} · \frac{B (gh)}{B (h)}|

≤ \frac{1}{2} (N − n) \frac{1}{B (h)}B [h |f − \frac{1}{B (h)}B (hf) · 1|] .

The constant \frac{1}{2} is best possible.

Proof. Apply Theorem 1 for the functional A_{h} : L → ℝ,

A_{h} (f) := \frac{1}{B (h)}B (hf) ,

that is a normalised isotonic linear functional on L.

Similar corollaries may be stated from the weighted inequality (2.12), but we omit the details.

3. Applications for Integral and Discrete Inequalities

Let (Ω, A, μ) be a measurable space consisting of a set Ω, a σ−algebra of parts of Ω and a countably additive and positive measure μ on A with values in ℝ∪ {∞} .

For a μ−measurable function w : Ω → ℝ with w (x) ≥ 0 for μ−a.e. x ∈ Ω, assume ∫_{Ω} w (x) dμ (x) > 0. Consider the Lebesgue space L_{w} (Ω, μ) := {f : Ω → ℝ, f is measurable on ∫_{Ω} w (x) |f (x)| dμ (x) < ∞}.

If f, g : Ω → ℝ are μ−measurable functions and f, g, fg ∈ L_{w} (Ω, μ) , then we may consider the Čebyšev functional

T_{w} (f, g) := \frac{1}{∫_{Ω} w (x) dμ (x)} ∫_{Ω} w (x) f (x) g (x) dμ (x)

− \frac{1}{∫_{Ω} w (x) dμ (x)} ∫_{Ω} w (x) f (x) dμ (x)

× \frac{1}{∫_{Ω} w (x) dμ (x)} ∫_{Ω} w (x) g (x) dμ (x) .

We may also consider the functional

D_{w} (f) := \frac{1}{∫_{Ω} w (x) dμ (x)}

× ∫_{Ω} w (x) |f (x) − \frac{1}{∫_{Ω} w (y) dμ (y)} ∫_{Ω} w (y) f (y) dμ (y)| dμ (x) .

Applying Theorem 2 for the normalised isotonic linear functional

A (f) := \frac{1}{∫_{Ω} w (x) dμ (x)} ∫_{Ω} w (x) f (x) dμ (x) ,

A : L_{w} (Ω, μ) → ℝ, we may recapture the following result due to Cerone and Dragomir [2]. Note that the proof of this result in [2] is different to the one in Theorem 2. Theorem 4. Let w, f, g : Ω → ℝ be μ−measurable functions with w ≥ 0 μ−a.e. on Ω and ∫_{Ω} w (x) dμ (x) > 0. If f, g, fg ∈ L_{w} (Ω, μ) and there exists the constants n, N so that

(3.1)  −∞ < n ≤ g (x) ≤ N < ∞ for μ-a.e. x ∈ Ω,

then we have the inequality

(3.2)  |T_{w} (f, g)| ≤ \frac{1}{2} (N − n) D_{w} (f) .

The constant \frac{1}{2} is sharp in the sense that it cannot be replaced by a smaller constant. Remark 2. If Ω = [a, b] and w (x) = 1 in Theorem 4, then we recapture the result obtained in [3]

(3.3) |\frac{1}{b − a} ∫_{a}^{b} f (x) g (x) dx − \frac{1}{b − a} ∫_{a}^{b} f (x) dx · \frac{1}{b − a} ∫_{a}^{b} g (x) dx|

≤ \frac{1}{2} (N − n) · \frac{1}{b − a} ∫_{a}^{b} |f (x) − \frac{1}{b − a} ∫_{a}^{b} f (y) dy| dx

provided n ≤ g (x) ≤ N for a.e. x ∈ [a, b] .

Note that the proof in Theorem 2 is different to the one in [3], using only the linearity and monotonicity properties of the functional A. We should also remark that in [3] the authors did not show the sharpness of the constant \frac{1}{2}.

Now, if we consider the normalised isotonic linear functional

(3.4)  Aw̄ \binom{}{x̄} := \frac{1}{W_{n}} ∑_{i=1}^{n} w_{i}x_{i},

Aw̄ : ℝ^{n} → ℝ, where w_{i} ≥ 0 (i = \overline{1, n}) and W_{n} := ∑_{i=1}^{n} w_{i} > 0, the by Theorem 2 we may obtain the following discrete inequality obtained by Cerone and Dragomir in [2]. Theorem 5. Let ā = (a_{1}, …, a_{n}) , b̄ = (b_{1}, …, b_{n}) ∈ ℝ be such that there exists the constants b, B ∈ ℝ so that

(3.5)  b ≤ b_{i} ≤ B for each i ∈ {1, …, n} .

Then one has the inequality

(3.6) |\frac{1}{W_{n}} ∑_{i=1}^{n} w_{i}a_{i}b_{i} − \frac{1}{W_{n}} ∑_{i=1}^{n} w_{i}a_{i} · \frac{1}{W_{n}} ∑_{i=1}^{n} w_{i}b_{i}|

≤ \frac{1}{2} (B − b) \frac{1}{W_{n}} ∑_{i=1}^{n} w_{i} |a_{i} − \frac{1}{W_{n}} ∑_{j=1}^{n} w_{j}a_{j}| .

The constant \frac{1}{2} is sharp in (3.6).

4. A Counterpart of the (CBS)-Inequality

The following inequality is known in the literature as the Cauchy-Buniakowski-Schwartz’s inequality for isotonic linear functionals or the (CBS)-inequality, for short,

(4.1)  [A (fg)]^{2} ≤ A (f^{2}) A (g^{2}) ,

provided f, g : E → ℝ are with the property that fg, f^{2}, g^{2} ∈ L and A : L → ℝ is any isotonic linear functional.

Making use of the Grüss inequality (2.12), we may prove the following counterpart of the (CBS)-inequality for isotonic linear functionals. Theorem 6. Let k, l : E → ℝ be such that k^{2}, l^{2}, kl ∈ L and there exists the real constants γ, Γ ∈ ℝ so that

(4.2)  γ ≤ \frac{k}{l} ≤ Γ.

Then for any isotonic linear functional A : L → ℝ so that |l| |A (l^{2}) k − A (kl) l| ∈ L, one has the inequality:

0 ≤ A (k^{2}) A (l^{2}) − [A (kl)]^{2}
≤ \frac{1}{2} (Γ − γ) A [|l| |A (l^{2}) k − A (kl) l|] .

The constant \frac{1}{2} is sharp.

Proof. We choose in (2.12) f = g = \frac{k}{l} , h = l^{2} and B = A to get

0 ≤ \frac{A (k^{2})}{A (l^{2})} − \begin{matrix}[A (kl)] & 2 \\ [A (l^{2})] & 2 \end{matrix}
≤ \frac{1}{2} (Γ − γ) \frac{1}{A (l^{2})}A [l^{2} |\frac{k}{l} − \frac{1}{A (l^{2})}A (kl)|] ,

provided A (l^{2}) ≠ 0, which is equivalent to

0 ≤ A (k^{2}) A (l^{2}) − [A (kl)]^{2}
≤ \frac{1}{2} (Γ − γ) A (l^{2}) A [|kl − \frac{l^{2}}{A (l^{2})}A (kl)|] ,

which is clearly equivalent to (4.3).

The following integral inequality holds. Corollary 3. Let w, f, g : Ω → ℝ be a μ−measurable function with w ≥ 0 μ-a.e. on Ω. If f, g ∈ L_{w}^{2} (Ω, μ) := {f : Ω → ℝ, ∫_{Ω} w (y) f^{2} (y) dμ (y) < ∞} and there exists γ, Γ so that

(4.3)  −∞ < γ ≤ \frac{f}{g} ≤ Γ < ∞ for μ-a.e. x ∈ Ω,

then one has the inequality:

(4.4) 0 ≤ ∫_{Ω} w (x) f^{2} (x) dμ (x) ∫_{Ω} w (x) g^{2} (x) dμ (x)
− [∫_{Ω} w (x) f (x) g (x) dμ (x)]^{2}

≤ \frac{1}{2} (Γ − γ) ∫_{Ω} w (x) |g (x)| |(∫_{Ω} w (y) g^{2} (y) dμ (y)) f (x)

− g (x) ∫_{Ω} w (y) f (y) g (y) dμ (y)| dμ (x)

= \frac{1}{2} (Γ − γ) ∫_{Ω} w (x) |g (x)| |∫_{Ω} w (y) g (y) | \begin{matrix}f (x) & g (x) \\ f (y) & g (y) \end{matrix} | dμ (y)| dμ (x) .

The constant \frac{1}{2} is sharp. Remark 3. In particular, if f, g ∈ L^{2} (Ω, μ) and the condition (4.3) holds, then

(4.5) 0 ≤ ∫_{Ω} f^{2} (x) dμ (x) ∫_{Ω} g^{2} (x) dμ (x) − [∫_{Ω} f (x) g (x) dμ (x)]^{2}
≤ \frac{1}{2} (Γ − γ) ∫_{Ω} |g (x)| |∫_{Ω} g (y) | \begin{matrix}f (x) & g (x) \\ f (y) & g (y) \end{matrix} | dμ (y)| dμ (x) .

The constant \frac{1}{2} is sharp.

The following discrete inequality also holds. Corollary 4. Let ā = (a_{1}, …, a_{n}) , b̄ = (b_{1}, …, b_{n}) and w̄ = (w_{1}, …, w_{n}) be the sequences of real numbers so that w_{i} ≥ 0 (i = 1, …, n) , W_{n} := ∑_{i=1}^{n} w_{i} > 0 and

(4.6)  γ ≤ \frac{a_{i}}{b_{i}} ≤ Γ for each i ∈ \{1, …, n\} .

Then one has the inequality

(4.7)  0 ≤ ∑_{i=1}^{n} w_{i}a_{i}^{2} ∑_{i=1}^{n} w_{i}b_{i}^{2} − (∑_{i=1}^{n} w_{i}a_{i}b_{i})^{2}

≤ \frac{1}{2} (Γ − γ) ∑_{i=1}^{n} w_{i}b_{i} |∑_{j=1}^{n} w_{j}b_{j} | \begin{matrix}a_{i} & b_{i} \\ a_{j} & b_{j} \end{matrix} || .

The constant \frac{1}{2} is sharp. Remark 4. If ā, b̄ satisfy (4.6), then one has the inequality

(4.8)  0 ≤ ∑_{i=1}^{n} a_{i}^{2} ∑_{i=1}^{n} b_{i}^{2} − (∑_{i=1}^{n} a_{i}b_{i})^{2}

≤ \frac{1}{2} (Γ − γ) ∑_{i=1}^{n} b_{i} |∑_{j=1}^{n} b_{j} | \begin{matrix}a_{i} & b_{i} \\ a_{j} & b_{j} \end{matrix} || .

The constant \frac{1}{2} is sharp.

5. A Converse for Jessen’s Inequality

In [4], the author has proved the following converse of Jessen’s inequality for normalized isotonic linear functionals. Theorem 7. Let Φ : (α, β) ⊆ ℝ → ℝ be a differentiable convex function on (α, β) , f : E → (α, β) so that Φ ∘ f, f, Φ′ ∘ f, (Φ′ ∘ f) · f ∈ L. If A : L → ℝ is an isotonic linear and normalised functional, then

(5.1)  0 ≤ A (Φ ∘ f) − Φ (A (f))
≤ A [(Φ′ ∘ f) · f] − A (f) A (Φ′ ∘ f)
≤ \frac{1}{4} [Φ′ (β) − Φ′ (α)] (β − α) (if α, β are finite).

We can state the following result improving the inequality (5.1). Theorem 8. Let Φ : [α, β] → ℝ with −∞ < α < β < ∞, and f, A are as in Theorem 7, then one has the inequality

(5.2)  0 ≤ A (Φ ∘ f) − Φ (A (f))
≤ A [(Φ′ ∘ f) · f] − A (f) A (Φ′ ∘ f)
≤ \frac{1}{2} [Φ′ (β) − Φ′ (α)] A (|f − A (f) · 1|) ,

provided |f − A (f) · 1| ∈ L.

Proof. Taking into account that α ≤ f ≤ β and Φ′ is monotonic on [α, β] , we have Φ′ (α) ≤ Φ′ ∘ f ≤ Φ′ (β) . Applying Theorem 2, we deduce

A [(Φ′ ∘ f) · f] − A (f) A (Φ′ ∘ f)
≤ \frac{1}{2} [Φ′ (β) − Φ′ (α)] A (|f − A (f) · 1|) ,

and the theorem is proved.

The following corollary addressing the integral case also holds.

Corollary 5. Let Φ : [α, β] ⊂ ℝ → ℝ be a differentiable convex function on (α, β) and f : Ω → [α, β] so that Φ ∘ f, f, Φ′ ∘ f, (Φ′ ∘ f) · f ∈ L_{w} (Ω, μ) , where w ≥ 0 μ-a.e. on Ω with ∫_{Ω} w (x) dμ (x) > 0. Then we have the inequality:

(5.3) 0 ≤ \frac{1}{∫_{Ω} w (x) dμ (x)} ∫_{Ω} w (x) Φ (f (x)) dμ (x)

− Φ (\frac{1}{∫_{Ω} w (x) dμ (x)} ∫_{Ω} w (x) f (x) dμ (x))

≤ \frac{1}{∫_{Ω} w (x) dμ (x)} ∫_{Ω} w (x) Φ′ (f (x)) f (x) dμ (x)

− \frac{1}{∫_{Ω} w (x) dμ (x)} ∫_{Ω} w (x) Φ′ (f (x)) dμ (x)

× \frac{1}{∫_{Ω} w (x) dμ (x)} ∫_{Ω} w (x) f (x) dμ (x)

≤ \frac{1}{2} [Φ′ (β) − Φ′ (α)] \frac{1}{∫_{Ω} w (x) dμ (x)}

× ∫_{Ω} w (x) |f (x) − \frac{1}{∫_{Ω} w (y) dμ (y)} ∫_{Ω} w (y) f (y) dμ (y)| dμ (x) .

Remark 5. If μ (Ω) < ∞ and Φ ∘ f, f, Φ′ ∘ f, (Φ′ ∘ f) · f ∈ L (Ω, μ) , then we have the inequality:

(5.4) 0 ≤ \frac{1}{μ (Ω)} ∫_{Ω} Φ (f (x)) dμ (x) − Φ (\frac{1}{μ (Ω)} ∫_{Ω} f (x) dμ (x))

≤ \frac{1}{μ (Ω)} ∫_{Ω} Φ′ (f (x)) f (x) dμ (x)

− \frac{1}{μ (Ω)} ∫_{Ω} Φ′ (f (x)) dμ (x) · \frac{1}{μ (Ω)} ∫_{Ω} f (x) dμ (x)

≤ \frac{1}{2} [Φ′ (β) − Φ′ (α)] \frac{1}{μ (Ω)} ∫_{Ω} |f (x) − \frac{1}{μ (Ω)} ∫_{Ω} f (y) dμ (y)| dμ (x) .

The case of functions of a real variable is embodied in the following inequality that provides a counterpart for the Jensen’s integral inequality

(5.5)  0 ≤ \frac{1}{b − a} ∫_{a}^{b} Φ (f (x)) dx − Φ (\frac{1}{b − a} ∫_{a}^{b} f (x) dx)

≤ \frac{1}{b − a} ∫_{a}^{b} Φ′ (f (x)) f (x) dx

− \frac{1}{b − a} ∫_{a}^{b} Φ′ (f (x)) dx · \frac{1}{b − a} ∫_{a}^{b} f (x) dx

≤ \frac{1}{2} [Φ′ (β) − Φ′ (α)] \frac{1}{b − a} ∫_{a}^{b} |f (x) − \frac{1}{b − a} ∫_{a}^{b} f (y) dy| dx.

The following discrete inequality is valid as well. Corollary 6. Let Φ : [α, β] → ℝ be a differentiable convex function on (α, β) . If x_{i} ∈ [α, β] and w_{i} ≥ 0 (i = 1, …, n) with W_{n} > 0, then one has the counterpart of Jensen’s discrete inequality:

(5.6)  0 ≤ \frac{1}{W_{n}} ∑_{i=1}^{n} w_{i}Φ (x_{i}) − Φ (\frac{1}{W_{n}} ∑_{i=1}^{n} w_{i}x_{i})

≤ \frac{1}{W_{n}} ∑_{i=1}^{n} w_{i}Φ′ (x_{i}) x_{i} − \frac{1}{W_{n}} ∑_{i=1}^{n} w_{i}Φ′ (x_{i}) \frac{1}{W_{n}} ∑_{i=1}^{n} w_{i}x_{i}

≤ \frac{1}{2} [Φ′ (β) − Φ′ (α)] \frac{1}{W_{n}} ∑_{i=1}^{n} w_{i} |x_{i} − \frac{1}{W_{n}} ∑_{j=1}^{n} w_{j}x_{j}| .

Remark 6. In particular, we get the discrete inequality:

(5.7)  0 ≤ \frac{1}{n} ∑_{i=1}^{n} Φ (x_{i}) − Φ (\frac{1}{n} ∑_{i=1}^{n} x_{i})

≤ \frac{1}{n} ∑_{i=1}^{n} Φ′ (x_{i}) x_{i} − \frac{1}{n} ∑_{i=1}^{n} Φ′ (x_{i}) \frac{1}{n} ∑_{i=1}^{n} x_{i}

≤ \frac{1}{2} [Φ′ (β) − Φ′ (α)] \frac{1}{n} ∑_{i=1}^{n} |x_{i} − \frac{1}{n} ∑_{j=1}^{n} x_{j}| .

References

[1] D. ANDRICA and C. BADEA, Grüss’ inequality for positive linear functionals, Periodica Math. Hungarica, 19(2)(1988), 155-167.

[2] P. CERONE and S.S. DRAGOMIR, A refinement of the Grüss inequality and applications, submitted.

[3] X.-L. CHENG and J. SUN, A note on the perturbed trapezoid inequality, J. Ineq. Pure & Appl. Math., 3(2002), No. 2, Article 29. [ON LINE: http://jipam.vu.edu.au].

[4] S.S. DRAGOMIR, On a reverse of Jessen’s inequality for isotonic linear functionals, J. Ineq. Pure & Appl. Math., 2(2001), No. 3, Article 36. [ON LINE: http://jipam.vu.edu.au].

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