NOTE ON INTEGRAL VERSION OF THE GRÜSS INEQUALITY FOR COMPLEX FUNCTIONS

S.S. DRAGOMIR, J.E. PEČARIĆ, AND B. TEPEŠ

Abstract. The corresponding version for complex-valued functions of a recent refinement of Grüss integral inequality due to Cerone & Dragomir is obtained.

1. Introduction

Let (Ω, Σ, μ) be a measure space consisting of a set Ω, Σ a σ−algebra of subsets of Ω and a countably additive and positive measure μ on Σ with values in ℝ ∪ {∞}. Denote by L_{ρ}^{2} (Ω, 𝕂) the Hilbert space of all measurable functions f : Ω → 𝕂 (𝕂 = ℂ, ℝ) that are 2 –ρ−integrable on Ω, i.e., ∫_{Ω} ρ (x) |f (x)|^{2} dμ (x) < 0, where ρ : Ω → [0, ∞) is a given measurable function on Ω.

Dragomir and Gomm [3] have proved the following Grüss type result for complex valued functions:

Theorem 1. Let φ, γ, Φ, Γ ∈ 𝕂 and h, f, g ∈ L_{ρ}^{2} (Ω,𝕂) be such that

(1.1)  Re [(Φh (x) − f (x)) (f̄(x) − φ̄ h̄ (x))] ≥ 0,
Re [(Γh (x) − g (x)) (ḡ (x) − γ̄ h̄ (x))] ≥ 0,

for μ−a.e. x ∈ Ω and ∫_{Ω} ρ (x) |h (x)|^{2} dμ (x) = 1. Then we have inequality

(1.2) |∫_{Ω} ρ (x) f (x) ḡ (x) dμ (x)
− ∫_{Ω} ρ (x) f (x) h̄ (x) dμ (x) ∫_{Ω} ρ (x) ḡ (x) h (x) dμ (x)|

≤ \frac{1}{4} |Φ − φ| |Γ − γ| .

The constant \frac{1}{4} is best possible in (1.2) in the sense that it cannot be replaced by a smaller quantity.

Denote by T (f, g; ρ) the Čebyšev functional

T (f, g; ρ)

: = \frac{1}{R} ∫_{Ω} ρ (x) f (x)ḡ (x) dμ (x) − \frac{1}{R} ∫_{Ω} ρ (x) f (x)dμ (x) \frac{1}{R} ∫_{Ω} ρ (x)ḡ (x) dμ (x) ,

where, for simplicity, we denote R := ∫_{Ω} ρ (x) dμ (x) > 0.

The following result was obtained in [3] as a simple consequence of Theorem 1.

1991 Mathematics Subject Classification. 26D15, 26D10.

Key words and phrases. Grüss inequality, Complex-Vauled functions.

Corollary 1. Let φ, γ, Φ, Γ ∈ 𝕂 and h, f, g ∈ L_{ρ}^{2} (Ω,𝕂) be such that

(1.3)  Re [(Φ − f (x)) (f̄(x) − φ̄ )] ≥ 0,
Re [(Γ − g (x)) (ḡ (x) − γ̄)] ≥ 0,

for μ−a.e. x ∈ Ω. Then we have the inequality

(1.4)  |T (f, g; ρ)| ≤ \frac{1}{4} |Φ − φ| |Γ − γ| .

Here the constant \frac{1}{4} is also best possible.

Let us note that we can prove the reverse implications as well, that is, Theorem 1 can be obtained as a consequence of Corollary 1.

Indeed, we have from the condition (1.1):

(1.5)  Re [\frac{(Φh (x) − f (x)) (f̄(x) − φ̄ h̄ (x))}{h (x) h̄ (x)}] ≥ 0,

Re [\frac{(Γh (x) − g (x)) (ḡ (x) − γ̄ h̄ (x))}{h (x) h̄ (x)}] ≥ 0,

or

(1.6)  Re [(Φ − \frac{f (x)}{h (x)}) (\begin{matrix}f̄ & (x) \\ h̄ & (x) \end{matrix} − φ̄ )] ≥ 0,
Re [(Γ − \frac{g (x)}{h (x)}) (\begin{matrix}ḡ & (x) \\ h̄ & (x) \end{matrix} − γ̄ )] ≥ 0.

Using conditions (1.6) and inequality (1.4) in Corollary 1 with ρ (x) := ρ (x) |h (x)|^{2} , f (x) := \frac{f(x)}{h(x)}, g (x) := \frac{g(x)}{h(x)} we get

(1.7) |\frac{1}{R} ∫_{Ω} ρ (x) f (x) ḡ (x) dμ (x)
− \frac{1}{R} ∫_{Ω} ρ (x) f (x) h̄ (x) dμ (x)\frac{1}{R} ∫_{Ω} ρ (x) ḡ (x) h (x) dμ (x)|

≤ \frac{1}{4} |Φ − φ| |Γ − γ| ,

i.e. the inequality (1.2) holds true.

For the real case, we also recall the following refinement of the classical Grüss inequality obtained in [1].

With the assumptions presented above and if

f ∈ L_{ρ} (Ω, μ) := {f : Ω → ℝ, ∫_{Ω} ρ (x) |f (x)| dμ (x) < 0}

then we may define the functional

D_{ρ} (f) (1.8)

:= \frac{1}{∫_{Ω} ρ (x) dμ (x)} ∫_{Ω} ρ (x) |f (x) − \frac{1}{∫_{Ω} ρ (y) dμ (y)} ∫_{Ω} ρ (y) f (y) dμ (y)| dμ (x) .

The following result holds [1].

Theorem 2. Let ρ, f, g : Ω → ℝ be μ−measurable functions with ρ ≥ 0 μ−a.e. on Ω and ∫_{Ω} ρ (y) dμ (y) > 0. If f, g, fg ∈ L_{ρ} (Ω, μ) and there exists the constants δ, Δ such that

(1.9)  −∞ < δ ≤ g (x) ≤ Δ < ∞ for μ − a.e. x ∈ Ω,

then we have the inequality

(1.10)  |T (f, g; ρ)| ≤ \frac{1}{2} (Δ − δ) D_{ρ} (f) .

The constant \frac{1}{2} is sharp in the sense that it cannot be replaced by a smaller quantity.

The above result naturally extend the one proved in Cheng and Sun [2] for functions of a real variable. The sharpness of the constant was not considered in [2]. The technique employed in [1] to prove the main results is different from the technique used by Cheng and Sun to obtain the inequality below:

Theorem 3. Let f, g : [a, b] → ℝ be two integrable functions such that δ ≤ g (x) ≤ Δ for some constants δ, Δ and for all x ∈ [a, b] , then

(1.11) |\frac{1}{b − a} ∫_{a}^{b} f (x) g (x) dx − \frac{1}{(b − a)^{2}} ∫_{a}^{b} f (x) dx ∫_{a}^{b} g (x) dx|

≤ \frac{Δ − δ}{2} ∫_{a}^{b} |f (x) − \frac{1}{b − a} ∫_{a}^{b} f (y) dy| dx.

The main aim of this note is to provide the corresponding refinement of Grüss inequality for complex-valued functions. Some related results are also pointed out.

2. The Results

The following preposition holds.

Proposition 1. Let φ, γ, Φ, Γ ∈ 𝕂 and f, g ∈ L_{ρ}^{2} (Ω,𝕂). Then we have inequality

(2.1) |T (f, g; ρ)| ≤ \frac{1}{R} ∫_{Ω} ρ (x) |f (x) − α| |ḡ (x) − \frac{1}{R} ∫_{Ω} ρ (t) ḡ (t) dμ (t)| dμ (x),

and every α ∈ 𝕂.

Proof. The following identity of Sonin type is valid:

(2.2) T (f, g; ρ) = \frac{1}{R} ∫_{Ω} ρ (x) (f (x) − α) (ḡ (x) − \frac{1}{R} ∫_{Ω} ρ (t) ḡ (t) dμ (t)) dμ (x).

Using well-known triangle inequality we get (2.1).

Corollary 2. Let φ, γ, Φ, Γ ∈ 𝕂 and f, g ∈ L_{ρ}^{2} (Ω,𝕂). Then we have inequality

(2.3) |T (f, g; ρ)| ≤ \frac{1}{R} ∫_{Ω} ρ (x) |f (x) − \frac{1}{R} ∫_{Ω} ρ (t) f (t) dμ (t)|

× |ḡ (x) − \frac{1}{R} ∫_{Ω} ρ (t) ḡ (t) dμ (t)| dμ (x).

Proof. Applying inequality (2.2) in Theorem 1 with α = \frac{1}{R} ∫_{Ω} ρ (x) dμ (x), we deduce the desired inequality (2.3).

Remark 1. Inequality (2.3) is a generalization of an inequality of S. S. Dragomir and McAndrew in [4] holding for functions of a real variable.

The following pre-Grüss type inequality for complex-valued functions holds.

Theorem 4. Let φ, Φ ∈ 𝕂 and f, g ∈ L_{ρ}^{2} (Ω,𝕂) be such that

(2.4)  Re [(Φ − f (x)) (f̄(x) − φ̄ )] ≥ 0,

for μ−a.e. x ∈ Ω, or equivalently,

(2.5)  |f (x) − \frac{φ + Φ}{2}| ≤ \frac{1}{2} |Φ − φ| ,

for μ−a.e. x ∈ Ω. Then we have the inequalities

|T (f, g; ρ)| (2.6)

≤ \frac{1}{2} |Φ − φ| \frac{1}{R} ∫_{Ω} ρ (x) |ḡ (x) − \frac{1}{R} ∫_{Ω} ρ (t) ḡ (t) dμ (t)| dμ (x)

≤ \frac{1}{2} |Φ − φ| (\frac{1}{R} ∫_{Ω} ρ (x) |ḡ (x) − \frac{1}{R} ∫_{Ω} ρ (t) ḡ (t) dμ (t)|^{2} dμ (x))^{\frac{1}{2}}

= \frac{1}{2} |Φ − φ| (\frac{1}{R} ∫_{Ω} ρ (x) |ḡ (x)|^{2} dμ (x) − |\frac{1}{R} ∫_{Ω} ρ (t) ḡ (t) dμ (t)|^{2})^{\frac{1}{2}} .

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

Proof. The equivalence between (2.4) and (2.5) can be easily proved (see, for instance [5] where a proof in the general settings of inner product spaces is given). Using inequality (2.1) with α = \frac{φ+Φ}{2} and (2.5), we have:

|T (f, g; ρ)| (2.7)

≤ \frac{1}{R} ∫_{Ω} ρ (x) |f (x) − \frac{φ + Φ}{2}| |ḡ (x) − \frac{1}{R} ∫_{Ω} ρ (t) ḡ (t) dμ (t)| dμ (x)

≤ \frac{1}{2} |Φ − φ| \frac{1}{R} ∫_{Ω} ρ (x) |ḡ (x) − \frac{1}{R} ∫_{Ω} ρ (t) ḡ (t) dμ (t)| dμ (x).

Now using the fundamental inequality for means ([6, p. 15]), we have

\frac{1}{2} |Φ − φ| \frac{1}{R} ∫_{Ω} ρ (x) |ḡ (x) − \frac{1}{R} ∫_{Ω} ρ (t) ḡ (t) dμ (t)| dμ (x) (2.8)

≤ \frac{1}{2} |Φ − φ| (\frac{1}{R} ∫_{Ω} ρ (x) |ḡ (x) − \frac{1}{R} ∫_{Ω} ρ (t) ḡ (t) dμ (t)|^{2} dμ (x))^{\frac{1}{2}}

= \frac{1}{2} |Φ − φ| (\frac{1}{R} ∫_{Ω} ρ (x) |ḡ (x)|^{2} dμ (x) − |\frac{1}{R} ∫_{Ω} ρ (t) ḡ (t) dμ (t)|^{2})^{\frac{1}{2}} .

The last identity in (2.8) is obvious.

The sharpness of the best constant follows from the real case, see for instance, [1], and we omit the details Corollary 3. Let φ, γ, Φ, Γ ∈ 𝕂 and h, f, g ∈ L_{ρ}^{2} (Ω,𝕂) be such that either

Re [(Φ − f (x)) (\overline{f}(x) −\overline{φ})] ≥ 0,
Re [(Γ−g (x)) (\overline{g}(x) −\overline{γ})] ≥ 0,

for μ−a.e. x ∈ Ω, or equivalently

(2.9)  |f (x) − \frac{φ + Φ}{2}| ≤ \frac{1}{2} |Φ − φ| ,

|g (x) − \frac{γ + Γ}{2}| ≤ \frac{1}{2} |Γ − γ| ,

for μ−a.e. x ∈ Ω. Then we have the inequalities

|T (f, g; ρ)| (2.10)

≤ \frac{1}{2} |Φ − φ| \frac{1}{R} ∫_{Ω} ρ (x) |ḡ (x) − \frac{1}{R} ∫_{Ω} ρ (t) ḡ (t) dμ (t)| dμ (x)

≤ \frac{1}{2} |Φ − φ| (\frac{1}{R} ∫_{Ω} ρ (x) |ḡ (x) − \frac{1}{R} ∫_{Ω} ρ (t) ḡ (t) dμ (t)|^{2} dμ (x))^{\frac{1}{2}}

= \frac{1}{2} |Φ − φ| (\frac{1}{R} ∫_{Ω} ρ (x) |g (x)|^{2} dμ (x) − |\frac{1}{R} ∫_{Ω} ρ (t)\overline{g}(t) dμ (t)|^{2})^{\frac{1}{2}}

≤ \frac{1}{4} |Φ − φ| |Γ − γ|

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

Proof. Using (2.6) we have

(2.11)  |T (g, g; ρ)|^{2} ≤ |\frac{1}{R} ∫_{Ω} ρ (x) |g (x)|^{2} dμ (x) − |\frac{1}{R} ∫_{Ω} ρ (t)\overline{g}(t) dμ (t)|^{2}|^{2}

≤ \frac{1}{4} |Γ − γ|^{2} |T (g, g; ρ)| .

Using (2.11) and (2.9), we may state that:

(2.12)  |T (g, g; ρ)| ≤ \frac{1}{R} ∫_{Ω} ρ (x) |g (x)|^{2} dμ (x) − |\frac{1}{R} ∫_{Ω} ρ (t)\overline{g}(t) dμ (t)|^{2}

≤ \frac{1}{4} |Γ − γ|^{2} .

Now, the inequality (2.12) and inequality (2.8) give the desired inequality (2.10).

References

[1] P. Cerone and S.S. Dragomir, A refinement of the Grüss inequality and applications, RGMIA Res. Rep. Coll., 5(2002), No. 2., Article 14 [ONLINE:http://rgmia.vu.edu.au/v5n2.html].

[2] X.L. Cheng and J. Sun, A note on the perturbed trapezoid inequality, J. Ineq. Pure. & Appl.

   Math., 3(2) (2002), Article 29. [ONLINE: http://jipam.vu.edu.au/v3n2/046 01.html]

[3] S.S. Dragomir and I. Gomm, Some integral and discrete versions of the Grüss inequality for real and complex functions and sequences, Tamsui Oxford Journal of Mathematical Sciences, 19(1) (2003), 67-77. Preprint: RGMIA Res. Rep. Coll., 5 (2003), No. 3, Article 9 [ONLINE:http://rgmia.vu.edu.au/v5n3.html].

[4] S.S. Dragomir and A. McAndrew, On trapezoid inequality via a Grüss type result and applications, Tamkang J. Math., 31(3) (2000), 193-201. Preprint: RGMIA Res. Rep. Coll., 2(2), Article 6, 1999 [ONLINE:http://rgmia.vu.edu.au/v2n2.html].

[5] S.S. Dragomir, Some Grüss type inequalities in inner product spaces, J. Inequal. Pure and Appl. Math., 4 (2003), No. 2, Article 42.

[6] D.S. Mitrinović, J.E. Pečarić and A.M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht/Boston/London, 1993.

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

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

Department of Mathematics, Faculty of Textile Technology, University of Zagreb, Croatia.

E-mail address: pecaric@mahazu.hazu.hr
URL: http://mahazu.hazu.hr/DepMPCS/indexJP.html

Department of Mathematics, Faculty of Textile Technology, University of Zagreb, Croatia.