ON AN INEQUALITY OF THE LUPAŞ TYPE

N.S. BARNETT AND S.S. DRAGOMIR

Abstract. An inequality of the Lupaş type for Čebyšev functionals is established.

1. Introduction

Let L′_{2} [a, b] be the space of all real functions which are absolutely continuous on [a, b], whose derivatives are square integrable in the Lebesgue sense on [a, b] . Consider the Čebyšev functional defined by

(1.1) T (f, g) := \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.

In 1970, A.M. Ostrowski [3] proved that if f, g ∈ L′_{2} [a, b] , then there exists a constant C with 0 < C < \frac{1}{8} (b − a) such that

(1.2)  |T (f, g)| ≤ C ∥f′∥_{2} ∥g′∥_{2} ,

where

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

is the usual Hilbertian norm in L′_{2} [a, b] .

The best possible constant in (1.2) was obtained by A. Lupaş in 1973 [2] in which he proved that,

(1.3)  |T (f, g)| ≤ \frac{b − a}{π^{2}} ∥f′∥_{2} ∥g′∥_{2} .

There is at least a pair of functions (f_{0}, g_{0}) from L′_{2} [a, b] for which the equality is realised in (1.3). For instance,

f_{0} (x) = A + B sin (\frac{π}{2} · \frac{a + b − 2x}{b − a}) ,
g_{0} (x) = C + D sin (\frac{π}{2} · \frac{a + b − 2x}{b − a})

with A, B, C, D ∈ ℝ and α ∈ [a, b].

Now, assume that w : [a, b] → [0, ∞) is a Lebesgue measurable function and consider the space L′_{2,w} [a, b] of all absolutely continuous functions f : [a, b] → ℝ with the property that,

Date: 10th January, 2007.

2000 Mathematics Subject Classification. 26D15, 26D10.

Key words and phrases. Lupaş type inequality, Čebyšev functionals, Integral inequalities.

(1.4)  ∥f′∥_{2,w} := (∫_{a}^{b} w (x) |f′ (x)|^{2} dx)^{\frac{1}{2}} < ∞.

One can then consider the problem of finding the best possible constant K > 0 such that the following inequality holds:

(1.5)  |T (f, g)| ≤ K ∥f′∥_{2,w} ∥g′∥_{2,w} ,

for any f′, g′ ∈ L′_{w} [a, b] with an appropriate weight w.

In this paper we show that for the weight w : [a, b] → [0, ∞)

(1.6)  w (x) = (b − x) [(b − a)^{2} + (b − a) (x − a) + (x − a)^{2}] , x ∈ [a, b]

we can establish an inequality of the (1.5)-type. We also show that for some examples of elementary functions f and g, the bound in (1.5) for |T (f, g)| is better than the one in (1.3). The problem of the sharpness for the obtained constant is still open.

2. The Results

Lemma 1. Let f : [a, b] → ℝ be an absolutely continuous function on [a, b] , w : [a, b] → [0, ∞) defined by,

(2.1)  w (x) = (b − x) [(b − a)^{2} + (b − a) (x − a) + (x − a)^{2}] , x ∈ [a, b]

and let L′_{2,w} [a, b] be the space defined in the introduction. It follows that,

(2.2) (0 ≤) (b − a) ∫_{a}^{b} f^{2} (x) dx − (∫_{a}^{b} f (x) dx)^{2} ≤ \frac{4}{3π^{2}} · ∫_{a}^{b} w (x) [f′ (x)]^{2} dx

for all f ∈ L′_{2,w} [a, b] .

Proof. We use the following inequality obtained by Diaz and Metcalf in [1]:

(2.3)  ∫_{a}^{b} [v (x) − v (b)]^{2} dx ≤ 4 · \begin{matrix}(b − a) & 2 \\ π & 2 \end{matrix} ∫_{a}^{b} [v′ (x)]^{2} dx,

provided v is absolutely continuous on [a, b] . If

v (x) = v (b) + B sin (\frac{π}{2} · \frac{b − x}{b − a}) , B ∈ ℝ,

then the equality case is realised in (2.3).

Utilising (2.3) we can write,

(2.4)  ∫_{a}^{y} [f (x) − f (y)]^{2} dx ≤ \frac{4}{π^{2}} · (b − a)^{2} ∫_{a}^{y} [f′ (x)]^{2} dx

for any y ∈ [a, b] .

Integrating over y in [a, b] , we deduce,

(2.5) ∫_{a}^{b} (∫_{a}^{y} [f (x) − f (y)]^{2} dx) dy ≤ \frac{4}{π^{2}} ∫_{a}^{b} (y − a)^{2} (∫_{a}^{y} [f′ (x)]^{2} dx) dy.

However,

I := ∫_{a}^{b} (∫_{a}^{y} [f (x) − f (y)]^{2} dx) dy
= ∫_{a}^{b} [∫_{a}^{y} f^{2} (x) dx − 2f (y) ∫_{a}^{y} f (x) dx + f^{2} (y) (y − a)] dy

= [y ∫_{a}^{y} f^{2} (x) dx]|_{a}^{b} − ∫_{a}^{b} yf^{2} (y) dy

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

= b ∫_{a}^{b} f^{2} (x) dx − ∫_{a}^{b} yf^{2} (y) dy − (∫_{a}^{b} f (x) dx)^{2} + ∫_{a}^{b} f^{2} (y) (y − a) dy

= (b − a) ∫_{a}^{b} f^{2} (x) dx − (∫_{a}^{b} f (x) dx)^{2}

and

J := ∫_{a}^{b} (y − a)^{2} (∫_{a}^{y} [f′ (x)]^{2} dx) dy

= \frac{(y − a)^{3}}{3} ∫_{a}^{y} [f′ (x)]^{2} dx|_{a}^{b} − \frac{1}{3} ∫_{a}^{b} (y − a)^{3} [f′ (y)]^{2} dy

= \frac{1}{3} ∫_{a}^{b} [(b − a)^{3} − (y − a)^{3}] [f′ (y)]^{2} dy

= \frac{1}{3} ∫_{a}^{b} w (x) [f′ (x)]^{2} dx.

Replacing I and J in (2.5), we deduce the desired result in (2.2).

Returning to the Čebyšev functional T (f, g) , we are now able to state the following result:

Theorem 1. For any f, g ∈ L′_{2,w} [a, b] with w defined as in (2.1), we have:

(2.6)  |T (f, g)| ≤ \frac{4}{3π^{2}} · \frac{1}{(b − a)^{2}} ∥f′∥_{2,w} ∥g′∥_{2,w} .

Proof. We know that (see for instance [3]),

(2.7)  |T (f, g)|^{2} ≤ T (f, f) T (g, g)

for any f, g ∈ L_{2} [a, b] .

On utilising Lemma 1,

(2.8)  T (f, f) ≤ \frac{4}{3π^{2}} · \frac{1}{(b − a)^{2}} ∥f′∥_{2,w}

and

(2.9)  T (g, g) ≤ \frac{4}{3π^{2}} · \frac{1}{(b − a)^{2}} ∥g′∥_{2,w} ,

for any f, g ∈ L′_{2,w} [a, b] .

Making use of (2.7) – (2.9) we deduce the desired result (2.6).

Open Problem. It is an open problem whether or not the constant \frac{4}{3π^{2}} is the best possible in (2.2) and (2.6) respectively.

3. Numerical Experiments

If we consider a = 0, b = 1 in the above, then,

(3.1)  (0 ≤) ∫_{0}^{1} f^{2} (x) dx − (∫_{0}^{1} f (x) dx)^{2} ≤ \frac{1}{π^{2}} · ∫_{0}^{1} [f′ (x)]^{2} dx

and

(3.2) (0 ≤) ∫_{0}^{1} f^{2} (x) dx − (∫_{0}^{1} f (x) dx)^{2} ≤ \frac{4}{3π^{2}} · ∫_{0}^{1} (1 − x^{3}) [f′ (x)]^{2} dx.

Consider now, f_{r} : [0, 1] → [0, ∞), f_{r} (x) = x^{r}, r ∈ (1/2, ∞) and define

I_{1} (r) := ∫_{0}^{1} [f′_{r} (x)]^{2} dx = \frac{r^{2}}{2r − 1}

and

I_{2} (r) := \frac{4}{3} · ∫_{0}^{1} (1 − x^{3}) [f′_{r} (x)]^{2} dx = \frac{4r^{2}}{(2r − 1) (2r + 2)}.

The plot of the difference Δ (r) := I_{2} (r) − I_{1} (r) on (1/2, ∞) shows that on the interval (1/2, 1] the bound provided by (3.1) is better than the bound provided by (3.2) for the quantity

∫_{0}^{1} f^{2} (x) dx − (∫_{0}^{1} f (x) dx)^{2} ,

while for r ∈ (1, ∞) the conclusion is reversed.

References

[1] J.B. DIAZ and F.T. METCALF, Variations of Wirtinger inequality, in “Inequalities”, Ed. Oved Shisha, Avademic Press, 1967, pp. 79-103.

[2] A. LUPAŞ, The best constant in an integral inequality, Mathematica (Cluj), 15 (38) (1973), No. 2, 219-222.

[3] A.M. OSTROWSKI, On an integral inequality, Aequationes Math., 4 (1970), 358-373.

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

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

LUPAŞ TYPE INEQUALITY 5

Figure 1. Plot of the difference Δ (r)