SOME NEW RESULTS RELATED TO BESSEL AND GRÜSS

INEQUALITIES IN 2-INNER PRODUCT SPACES AND APPLICATIONS

S.S. DRAGOMIR, Y.J. CHO^{⋆}, AND S.S. KIM^{♦}

Abstract. Some new reverses of Bessel’s inequality for orthonormal families in real or complex inner product spaces are pointed out. Applications for some Grüss type inequalities and for determinantal integral inequalities are given as well.

1. Introduction

The concepts of 2-inner products and 2-inner product spaces have been intensively studied by many authors in the last three decades. A systematic presentation of the recent results related to the theory of 2-inner product spaces as well as an extensive list of the related references can be found in [1]. Here we give the basic definitions and the elementary properties of 2-inner product spaces.

Let X be a linear space of dimension greater than 1 over the field 𝕂 = ℝ of real numbers or the field 𝕂 = ℂ of complex numbers. Suppose that (·, ·|·) is a 𝕂-valued function defined on X × X × X satisfying the following conditions:

(2I_{1}) (x, x|z) ≥ 0 and (x, x|z) = 0 if and only if x and z are linearly dependent, (2I_{2}) (x, x|z) = (z, z|x), (2I_{3}) (y, x|z) = \overline{(x, y|z)}, (2I_{4}) (αx, y|z) = α(x, y|z) for any scalar α ∈ 𝕂, (2I_{5}) (x + x′, y|z) = (x, y|z) + (x′, y|z). (·, ·|·) is called a 2-inner product on X and (X, (·, ·|·)) is called a 2-inner product

space (or 2-pre-Hilbert space). Some basic properties of 2-inner product spaces can be immediately obtained as follows [2]:

(1) If 𝕂 = ℝ, then (2I_{3}) reduces to

(y, x|z) = (x, y|z).

(2) From (2I_{3}) and (2I_{4}), we have

(0, y|z) = 0, (x, 0|z) = 0

and also

(1.1)  (x, αy|z) = ᾱ(x, y|z).

(3) Using (2I_{2})–(2I_{5}), we have

Date: September 29, 2003.

2000 Mathematics Subject Classification. 46C05, 46C99, 26D15, 26D10.

Key words and phrases. Bessel’s inequality, Grüss’ inequality, 2-Inner product, Lebesgue integral, Synchronous functions.

2 S.S. DRAGOMIR, Y.J. CHO^{⋆}, AND S.S. KIM^{♦}

(z, z|x ± y) = (x ± y, x ± y|z) = (x, x|z) + (y, y|z) ± 2Re(x, y|z)

and

(1.2)  Re(x, y|z) = \frac{1}{4}[(z, z|x + y) − (z, z|x − y)].

In the real case 𝕂 = ℝ, (1.2) reduces to

(1.3)  (x, y|z) = \frac{1}{4}[(z, z|x + y) − (z, z|x − y)]

and, using this formula, it is easy to see that, for any α ∈ ℝ,

(1.4)  (x, y|αz) = α^{2}(x, y|z).

In the complex case, using (1.1) and (1.2), we have

Im(x, y|z) = Re[−i(x, y|z)] = \frac{1}{4}[(z, z|x + iy) − (z, z|x − iy)],

which, in combination with (1.2), yields

(1.5) (x, y|z) = \frac{1}{4}[(z, z|x + y) − (z, z|x − y)] + \frac{i}{4}[(z, z|x + iy) − (z, z|x − iy)].

Using the above formula and (1.1), we have, for any α ∈ ℂ,

(1.6)  (x, y|αz) = |α|^{2}(x, y|z).

However, for α ∈ ℝ, (1.6) reduces to (1.4).

Also, from (1.6) it follows that

(x, y|0) = 0.

(4) For any three given vectors x, y, z ∈ X, consider the vector u = (y, y|z)x − (x, y|z)y. By (2I_{1}), we know that (u, u|z) ≥ 0 with the equality if and only if u and z are linearly dependent. The inequality (u, u|z) ≥ 0 can be rewritten as,

(1.7)  (y, y|z)[(x, x|z)(y, y|z) − |(x, y|z)|^{2}] ≥ 0.

For x = z, (1.7) becomes

−(y, y|z)|(z, y|z)|^{2} ≥ 0,

which implies that

(1.8)  (z, y|z) = (y, z|z) = 0

provided y and z are linearly independent. Obviously, when y and z are linearly dependent, (1.8) holds too. Thus (1.8) is true for any two vectors y, z ∈ X. Now, if y and z are linearly independent, then (y, y|z) > 0 and, from (1.7), it follows that

(1.9)  |(x, y|z)|^{2} ≤ (x, x|z)(y, y|z).

Using (1.8), it is easy to check that (1.9) is trivially fulfilled when y and z are linearly dependent. Therefore, the inequality (1.9) holds for any three vectors x, y, z ∈ X and is strict unless the vectors u = (y, y|z)x−(x, y|z)y and z are linearly dependent. In fact, we have the equality in (1.9) if and only if the three vectors x, y and z are linearly dependent.

In any given 2-inner product space (X, (·, · | ·)), we can define a function ∥ · | · ∥ on X × X by

(1.10)  ∥x|z∥ = \sqrt{(x, x|z)}

for all x, z ∈ X.

It is easy to see that this function satisfies the following conditions:

(2N_{1}) ∥x|z∥ ≥ 0 and ∥x|z∥ = 0 if and only if x and z are linearly dependent, (2N_{2}) ∥z|x∥ = ∥x|z∥, (2N_{3}) ∥αx|z∥ = |α|∥x|z∥ for any scalar α ∈ 𝕂, (2N_{4}) ∥x + x′|z∥ ≤ ∥x|z∥ + ∥x′|z∥. Any function ∥ · | · ∥ defined on X × X and satisfying the conditions (2N_{1})–(2N_{4})

is called a 2-norm on X and (X, ∥ · | · ∥) is called a linear 2-normed space [6]. Whenever a 2-inner product space (X, (·, ·|·)) is given, we consider it as a linear 2-normed space (X, ∥ · | · ∥) with the 2-norm defined by (1.10).

Let (X; (·, ·|·)) be a 2-inner product space over the real or complex number field 𝕂. If (f_{i})_{1≤i≤n} are linearly independent vectors in the 2-inner product space X, and, for a given z ∈ X, (f_{i}, f_{j}|z) = δ_{ij} for all i, j ∈ {1, …, n} where δ_{ij} is the Kronecker delta (we say that the family (f_{i})_{1≤i≤n} is z−orthonormal), then the following inequality is the corresponding Bessel’s inequality (see for example [2]) for the z−orthonormal family (f_{i})_{1≤i≤n} in the 2-inner product space (X; (·, ·|·)):

(1.11)  ∑_{i=1}^{n} |(x, f_{i}|z)|^{2} ≤ ∥x|z∥^{2}

for any x ∈ X. For more details on this inequality, see the recent paper [2] and the references therein.

The following reverse of Bessel’s inequality in 2-inner product spaces has been obtained in [3]:

Theorem 1. Let {e_{i}}_{i∈I} be a family of z-orthonormal vectors in X and F, ϕ_{i}, Φ_{i} (i ∈ F) and x, z ∈ X be so that either

(i) Re (∑_{i∈F} Φ_{i}e_{i} − x, x − ∑_{i∈F} ϕ_{i}e_{i}|z) ≥ 0

or, equivalently,

(ii) ∥x − ∑_{i∈F} \frac{ϕ_{i}+Φ_{i}}{2} e_{i}|z∥ ≤ \frac{1}{2} (∑_{i∈F} |Φ_{i} − ϕ_{i}|^{2})^{\frac{1}{2}} holds. Then we have the inequality:

0 ≤ ∥x|z∥^{2} − ∑_{i∈F} |(x, e_{i}|z)|^{2}

≤ \frac{1}{4} ∑_{i∈F} |Φ_{i} − ϕ_{i}|^{2} − Re (∑_{i∈F} Φ_{i}e_{i} − x, x − ∑_{i∈F} ϕ_{i}e_{i}|z)

(≤ \frac{1}{4} ∑_{i∈F} |Φ_{i} − ϕ_{i}|^{2}) .

The constant \frac{1}{4} is best possible.

The following different reverse of Bessel’s inequality has been obtained in [4].

Theorem 2. Let {e_{i}}_{i∈I} be a family of z-orthonormal vectors in X, F a finite part of I, ϕ_{i}, Φ_{i}, i ∈ I real or complex numbers. For x ∈ X, if either (i) or (ii) from Theorem 1 holds, then the following reverse of Bessel’s inequality

0 ≤ ∥x|z∥^{2} − ∑_{i∈F} |(x, e_{i}|z)|^{2}

≤ \frac{1}{4} ∑_{i∈F} |Φ_{i} − ϕ_{i}|^{2} − ∑_{i∈F} |\frac{ϕ_{i} + Φ_{i}}{2} − (x, e_{i}|z)|^{2}

(≤ \frac{1}{4} ∑_{i∈F} |Φ_{i} − ϕ_{i}|^{2}) ,

is valid. The constant \frac{1}{4} is best possible.

The main aim of the present paper is to establish a different reverse inequality for (1.11) than those incorporated in the above two theorems. Some companion results and applications for determinantal integral inequalities are also given.

2. A New Reverse of Bessel’s Inequality

The following reverse of Bessel’s inequality holds.

Theorem 3. Let {e_{i}}_{i∈I} be a family of z−orthonormal vectors in X, F a finite part of I, and ϕ_{i}, Φ_{i} (i ∈ F) , real or complex numbers such that ∑_{i∈F} Re (Φ_{i}\overline{ϕ_{i}}) > 0. If x ∈ X is such that either

(i) Re (∑_{i∈F} Φ_{i}e_{i} − x, x − ∑_{i∈F} ϕ_{i}e_{i}|z) ≥ 0 or, equivalently,

(ii) ∥x − ∑_{i∈F} \frac{ϕ_{i}+Φ_{i}}{2} e_{i}|z∥ ≤ \frac{1}{2} (∑_{i∈F} |Φ_{i} − ϕ_{i}|^{2})^{\frac{1}{2}}

holds, then one has the inequality

(2.1)  ∥x|z∥^{2} ≤ \frac{1}{4} · \begin{matrix}∑ & i∈F & |Φ_{i} + ϕ_{i}| & 2 \\ ∑ & i∈F & Re & (Φ_{i}\overline{ϕ_{i}}) \end{matrix} ∑_{i∈F} |(x, e_{i}|z)|^{2} .

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

BESSEL AND GRÜSS INEQUALITIES 5

Proof. Firstly, we observe that for y, a, A ∈ X, the following are equivalent

(2.2)  Re (A − y, y − a|z) ≥ 0

and

(2.3) ∥y − \frac{a + A}{2}|z∥ ≤ \frac{1}{2} ∥A − a|z∥ .

Now, for a = ∑_{i∈F} ϕ_{i}e_{i}, A = ∑_{i∈F} Φ_{i}e_{i}, we have

∥A − a|z∥ = ∥∑_{i∈F} (Φ_{i} − ϕ_{i}) e_{i}|z∥ = [∥∑_{i∈F} (Φ_{i} − ϕ_{i}) e_{i}|z∥^{2}]^{\frac{1}{2}}

= (∑_{i∈F} |Φ_{i} − ϕ_{i}|^{2} ∥e_{i}|z∥^{2})^{\frac{1}{2}} = (∑_{i∈F} |Φ_{i} − ϕ_{i}|^{2})^{\frac{1}{2}} ,

which gives, for y = x, the desired equivalence.

On the other hand, we have the identity

Re (∑_{i∈F} Φ_{i}e_{i} − x, x − ∑_{i∈F} ϕ_{i}e_{i}|z)

= ∑_{i∈F} Re [Φ_{i}\overline{(x, e_{i}|z)} + \overline{ϕ_{i}} (x, e_{i}|z)] − ∥x|z∥^{2} − ∑_{i∈F} Re (Φ_{i}\overline{ϕ_{i}}) ,

which gives, from (i), that

(2.4)  ∥x|z∥^{2} + ∑_{i∈F} Re (Φ_{i}\overline{ϕ_{i}}) ≤ ∑_{i∈F} Re [Φ_{i}\overline{(x, e_{i}|z)} + \overline{ϕ_{i}} (x, e_{i}|z)] .

Utilising the elementary inequality

αp^{2} + \frac{1}{α}q^{2} ≥ 2pq, α > 0, p, q ≥ 0;

we deduce

(2.5)  2 ∥x|z∥ ≤ \begin{matrix}∥x|z∥ & 2 \\ [∑_{i∈F} Re (Φ_{i}\overline{ϕ_{i}})] & \frac{1}{2} \end{matrix} + [∑_{i∈F} Re (Φ_{i}\overline{ϕ_{i}})]^{\frac{1}{2}} .

Dividing (2.4) by [∑_{i∈F} Re (Φ_{i}\overline{ϕ_{i}})]^{\frac{1}{2}} > 0 and using (2.5), we obtain

(2.6)  ∥x|z∥ ≤ \frac{1}{2} \begin{matrix}∑ & i∈F & Re & [Φ_{i}\overline{(x, e_{i}|z)} + \overline{ϕ_{i}} (x, e_{i}|z)] \\ [∑_{i∈F} Re (Φ_{i}\overline{ϕ_{i}})] & \frac{1}{2} \end{matrix}
= \frac{1}{2} \begin{matrix}∑ & i∈F & Re & [(\overline{Φ_{i}} + \overline{ϕ_{i}}) (x, e_{i}|z)] \\ [∑_{i∈F} Re (Φ_{i}\overline{ϕ_{i}})] & \frac{1}{2} \end{matrix},

since it is obvious that

Re [Φ_{i}\overline{(x, e_{i}|z)}] = Re [\overline{Φ_{i}} (x, e_{i}|z)] .

Note that (2.6) is also an interesting inequality in itself.

Utilising the Cauchy-Bunyakovsky-Schwarz inequality for real numbers, we get

(2.7)  ∑_{i∈F} Re [(\overline{Φ_{i}} + \overline{ϕ_{i}}) (x, e_{i}|z)] ≤ ∑_{i∈F} |(\overline{Φ_{i}} + \overline{ϕ_{i}}) (x, e_{i}|z)|

≤ ∑_{i∈F} (|Φ_{i} + ϕ_{i}|) |(x, e_{i}|z)|

≤ [∑_{i∈F} |Φ_{i} + ϕ_{i}|^{2}]^{\frac{1}{2}} [∑_{i∈F} |(x, e_{i}|z)|^{2}]^{\frac{1}{2}} .

Making use of (2.6) and (2.7), we deduce the desired result (2.1).

To prove the sharpness of the constant \frac{1}{4}, let us assume that (2.1) holds with a constant c > 0, i.e.,

(2.8)  ∥x|z∥^{2} ≤ c · \begin{matrix}∑ & i∈F & |Φ_{i} + ϕ_{i}| & 2 \\ ∑ & i∈F & Re & (Φ_{i}\overline{ϕ_{i}}) \end{matrix} ∑_{i∈F} |(x, e_{i}|z)|^{2} ,

provided x, ϕ_{i}, Φ_{i}, i ∈ F satisfy (i).

Suppose that F = {1} , e_{1} = e, ∥e|z∥ = 1, Φ_{1} = Φ > 0, ϕ_{1} = ϕ > 0. If we choose x = Φe, then the condition (i) holds true, and by (2.8) for F = {1} we get

Φ^{2} ≤ c · \frac{(Φ + ϕ)^{2}}{Φϕ} Φ^{2}

i.e., Φϕ ≤ c (Φ + ϕ)^{2} for any Φ, ϕ > 0. Now, if we choose Φ = 1 + ε, ϕ = 1 − ε with ε ∈ (0, 1) in the last inequality and make ε → 0+, then we get c ≥ \frac{1}{4}, and the proof is completed.

Remark 1. By the use of (2.6), the second inequality in (2.7) and the Hölder inequality, we may state the following reverses of Bessel’s inequality as well:

(2.9)  ∥x|z∥ ≤ \frac{1}{2} · \frac{1}{[∑_{i∈F} Re (Φ_{i}\overline{ϕ_{i}})]^{\frac{1}{2}}}

× \{ \begin{matrix}max_{i∈F} & \{|Φ_{i} + ϕ_{i}|\} & ∑_{i∈F} & |(x, e_{i}|z)| \\ [ ∑_{i∈F} |Φ_{i} + ϕ_{i}|^{p}]^{\frac{1}{p}} ( ∑_{i∈F} |(x, e_{i}|z)|^{q})^{\frac{1}{q}} , \\ for p > 1, \frac{1}{p} + \frac{1}{q} = 1; \\ max_{i∈F} |(x, e_{i}|z)| ∑_{i∈F} |Φ_{i} + ϕ_{i}| . \end{matrix}

The following corollary holds.

Corollary 1. With the assumption of Theorem 3 and if either (i) or (ii) holds, then

(2.10)  0 ≤ ∥x|z∥^{2} − ∑_{i∈F} |(x, e_{i}|z)|^{2} ≤ \frac{1}{4} · \begin{matrix}∑ & i∈F & |Φ_{i} − ϕ_{i}| & 2 \\ ∑ & i∈F & Re & (Φ_{i}\overline{ϕ_{i}}) \end{matrix} ∑_{i∈F} |(x, e_{i}|z)|^{2} .

The constant \frac{1}{4} is best possible.

Proof. The inequality (2.10) follows by (2.1) on subtracting the same quantity ∑_{i∈F} |(x, e_{i}|z)|^{2} from both sides.

The best constant may be shown in a similar way to the one in the above Theorem 3 and we omit the details.

Remark 2. If {e_{i}}_{i∈I} is an z−orthonormal family in the real 2-inner product space (X; (·, ·|·)) and M_{i}, m_{i} ∈ ℝ, i ∈ F (F is a finite part of I) and x ∈ X are such that M_{i}, m_{i} ≥ 0 for i ∈ F with ∑_{i∈F} M_{i}m_{i} > 0 and

(∑_{i∈F} M_{i}e_{i} − x, x − ∑_{i∈F} m_{i}e_{i}|z) ≥ 0,

then we have the inequality

(2.11) 0 ≤ ∥x|z∥^{2} − ∑_{i∈F} [(x, e_{i}|z)]^{2} ≤ \frac{1}{4} · \frac{∑_{i∈F} (M_{i} − m_{i})^{2}}{∑_{i∈F} M_{i}m_{i}} · ∑_{i∈F} [(x, e_{i}|z)]^{2} .

The constant \frac{1}{4} is best possible.

The following reverse of the Schwarz’s inequality in 2-inner product spaces holds.

Corollary 2. Let x, y ∈ X and δ, Δ ∈ 𝕂 (𝕂 = ℂ, ℝ) with the property that Re (Δ\overline{δ}) > 0. If either

(2.12)  Re (Δy − x, x − δy|z) ≥ 0

or, equivalently,

(2.13)  ∥x − \frac{δ + Δ}{2} · y|z∥ ≤ \frac{1}{2} |Δ − δ| ∥y|z∥

holds, then we have the inequalities

(2.14)  ∥x|z∥ ∥y|z∥ ≤ \frac{1}{2} · \begin{matrix}Re & [(\overline{Δ} + \overline{δ}) (x, y|z)] \\ √ & \overline{Re (Δ\overline{δ})} \end{matrix}

≤ \frac{1}{2} · \frac{|Δ + δ|}{\sqrt{Re (Δ\overline{δ})}} |(x, y|z)| ,

(2.15)  0 ≤ ∥x|z∥ ∥y|z∥ − |(x, y|z)|

≤ \frac{1}{2} · \frac{|Δ + δ| − 2\sqrt{Re (Δ\overline{δ})}}{\sqrt{Re (Δ\overline{δ})}} |(x, y|z)| ,

(2.16)  ∥x|z∥^{2} ∥y|z∥^{2} ≤ \frac{1}{4} · \begin{matrix}|Δ + δ| & 2 \\ Re & (Δ\overline{δ}) \end{matrix} |(x, y|z)|^{2} ,

and

(2.17)  0 ≤ ∥x|z∥^{2} ∥y|z∥^{2} − |(x, y|z)|^{2} ≤ \frac{1}{4} · \begin{matrix}|Δ − δ| & 2 \\ Re & (Δ\overline{δ}) \end{matrix} |(x, y|z)|^{2} .

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

Proof. The inequality (2.14) follows from (2.6) on choosing F = {1} , e_{1} = e = \frac{y}{∥y|z∥}, Φ_{1} = Φ = Δ ∥y|z∥ , ϕ_{1} = ϕ = δ ∥y|z∥ (y, z linearly independent) . The inequality (2.15) is equivalent with (2.14). The inequality (2.16) follows from (2.1) for F = {1} and the same choices as above. Finally, (2.17) is obviously equivalent with (2.16).

3. Some Grüss Type Inequalities

The following result holds.

Theorem 4. Let {e_{i}}_{i∈I} be a family of z−orthonormal vectors in X, F a finite part of I, ϕ_{i}, Φ_{i}, γ_{i}, Γ_{i} ∈ 𝕂, i ∈ F and x, y ∈ X. If either

(3.1)  Re (∑_{i∈F} Φ_{i}e_{i} − x, x − ∑_{i∈F} ϕ_{i}e_{i}|z) ≥ 0,

Re (∑_{i∈F} Γ_{i}e_{i} − y, y − ∑_{i∈F} γ_{i}e_{i}|z) ≥ 0

or, equivalently,

(3.2)  ∥x − ∑_{i∈F} \frac{Φ_{i} + ϕ_{i}}{2} e_{i}|z∥ ≤ \frac{1}{2} (∑_{i∈F} |Φ_{i} − ϕ_{i}|^{2})^{\frac{1}{2}} ,

∥y − ∑_{i∈F} \frac{Γ_{i} + γ_{i}}{2} e_{i}|z∥ ≤ \frac{1}{2} (∑_{i∈F} |Γ_{i} − γ_{i}|^{2})^{\frac{1}{2}}

hold, then we have the inequality

(3.3)  0 ≤ |(x, y|z) − ∑_{i∈F} (x, e_{i}|z) (e_{i}, y|z)|

≤ \frac{1}{4} (\begin{matrix}∑ & i∈F & |Φ_{i} − ϕ_{i}| & 2 \\ ∑ & i∈F & Re & (Φ_{i}\overline{ϕ_{i}}) \end{matrix} \begin{matrix}∑ & i∈F & |Γ_{i} − γ_{i}| & 2 \\ ∑ & i∈F & Re & (Γ_{i}\overline{γ_{i}}) \end{matrix} )^{1/2}

× (∑_{i∈F} |(x, e_{i}|z)|^{2})^{\frac{1}{2}} (∑_{i∈F} |(y, e_{i}|z)|^{2})^{\frac{1}{2}} .

The constant \frac{1}{4} is best possible.

Proof. Using Schwarz’s inequality in the 2-inner product space (X, (·, ·|·)) one has

|(x − ∑_{i∈F} (x, e_{i}|z) e_{i}, y − ∑_{i∈F} (y, e_{i}|z) e_{i}|z)|^{2} (3.4)

≤ ∥x − ∑_{i∈F} (x, e_{i}|z) e_{i}|z∥^{2} ∥y − ∑_{i∈F} (y, e_{i}|z) e_{i}|z∥^{2}

and, since a simple calculation shows that

(x − ∑_{i∈F} (x, e_{i}|z) e_{i}, y − ∑_{i∈F} (y, e_{i}|z) e_{i}|z) = (x, y|z) − ∑_{i∈F} (x, e_{i}|z) (e_{i}, y|z)

and

∥x − ∑_{i∈F} (x, e_{i}|z) e_{i}|z∥^{2} ≤ ∥x|z∥^{2} − ∑_{i∈F} |(x, e_{i}|z)|^{2}

for any x, y ∈ X, then by (3.4) and by the reverse of Bessel’s inequality in Corollary 1, we have

|(x, y|z) − ∑_{i∈F} (x, e_{i}|z) (e_{i}, y|z)|^{2} (3.5)

≤ (∥x|z∥^{2} − ∑_{i∈F} |(x, e_{i}|z)|^{2}) (∥y|z∥^{2} − ∑_{i∈F} |(y, e_{i}|z)|^{2})

≤ \frac{1}{4} \begin{matrix}∑ & i∈F & |Φ_{i} − ϕ_{i}| & 2 \\ ∑ & i∈F & Re & (Φ_{i}\overline{ϕ_{i}}) \end{matrix} ∑_{i∈F} |(x, e_{i}|z)|^{2} · \frac{1}{4} \begin{matrix}∑ & i∈F & |Γ_{i} − γ_{i}| & 2 \\ ∑ & i∈F & Re & (Γ_{i}\overline{γ_{i}}) \end{matrix} ∑_{i∈F} |⟨y, e_{i}|z⟩|^{2} .

Taking the square root in (3.5), we deduce (3.3).

The fact that \frac{1}{4} is the best possible constant follows by Corollary 1 and we omit the details.

The following corollary for real 2-inner product spaces holds.

Corollary 3. Let {e_{i}}_{i∈I} be a family of z−orthonormal vectors in X, F a finite part of I, M_{i}, m_{i}, N_{i}, n_{i} ≥ 0, i ∈ F and x, y ∈ X such that ∑_{i∈F} M_{i}m_{i} > 0, ∑_{i∈F} N_{i}n_{i} > 0 and

(3.6) (∑_{i∈F} M_{i}e_{i} − x, x − ∑_{i∈F} m_{i}e_{i}|z) ≥ 0, (∑_{i∈F} N_{i}e_{i} − y, y − ∑_{i∈F} n_{i}e_{i}|z) ≥ 0.

Then we have the inequality

0 ≤ |(x, y|z) − ∑_{i∈F} (x, e_{i}|z) (y, e_{i}|z)|^{2} (3.7)

≤ \frac{1}{16} · \frac{∑_{i∈F} (M_{i} − m_{i})^{2} ∑_{i∈F} (N_{i} − n_{i})^{2} ∑_{i∈F} |(x, e_{i}|z)|^{2} ∑_{i∈F} |(y, e_{i}|z)|^{2}}{∑_{i∈F} M_{i}m_{i} ∑_{i∈F} N_{i}n_{i}} .

The constant \frac{1}{16} is best possible.

In the case where the family {e_{i}}_{i∈I} reduces to a single vector, we may deduce from Theorem 4 the following particular case:

Corollary 4. Let e ∈ X, ∥e∥ = 1, ϕ, Φ, γ, Γ ∈ 𝕂 with Re (Φ\overline{ϕ}) , Re (Γ\overline{γ}) > 0 and x, y ∈ X such that either

(3.8)  Re (Φe − x, x − ϕe|z) ≥ 0, Re (Γe − y, y − γe|z) ≥ 0,

or, equivalently,

(3.9)  ∥x − \frac{ϕ + Φ}{2}e|z∥ ≤ \frac{1}{2} |Φ − ϕ| , ∥y − \frac{γ + Γ}{2}e|z∥ ≤ \frac{1}{2} |Γ − γ|

10 S.S. DRAGOMIR, Y.J. CHO^{⋆}, AND S.S. KIM^{♦}

holds, then

(3.10) 0 ≤ |(x, y|z) − (x, e|z) (e, y|z)| ≤ \frac{1}{4} \frac{|Φ − ϕ|}{\sqrt{Re (Φ\overline{ϕ})}} \frac{|Γ − γ|}{\sqrt{Re (Γ\overline{γ})}} |(x, e|z) (e, y|z)| .

The constant \frac{1}{4} is best possible. Remark 3. If X is real, e ∈ X, ∥e|z∥ = 1 and a, b, A, B ∈ ℝ are such that A > a > 0, B > b > 0 and

(3.11) ∥x − \frac{a + A}{2}e|z∥ ≤ \frac{1}{2} (A − a) , ∥y − \frac{b + B}{2} e|z∥ ≤ \frac{1}{2} (B − b) ,

then

(3.12)  |(x, y|z) − (x, e|z) (e, y|z)| ≤ \frac{1}{4} · \begin{matrix}(A − a) & (B − b) \\ √ & \overline{abAB} \end{matrix} |(x, e|z) (e, y|z)| .

The constant \frac{1}{4} is best possible.

If (x, e|z) , (y, e|z) ≠ 0, then the following equivalent form of (3.12) also holds

(3.13)  |\frac{(x, y|z)}{(x, e|z) (e, y|z)} − 1| ≤ \frac{1}{4} · \begin{matrix}(A − a) & (B − b) \\ √ & \overline{abAB} \end{matrix}.

4. Some Companion Inequalities

The following companion of the Grüss inequality also holds. Theorem 5. Let {e_{i}}_{i∈I} be a family of z−orthonormal vectors in X, F a finite part of I, ϕ_{i}, Φ_{i} ∈ 𝕂, (i ∈ F), x, y ∈ X and λ ∈ (0, 1) , such that either

(4.1)  Re (∑_{i∈F} Φ_{i}e_{i} − (λx + (1 − λ) y) , λx + (1 − λ) y − ∑_{i∈F} ϕ_{i}e_{i}|z) ≥ 0

or, equivalently,

(4.2)  ∥λx + (1 − λ) y − ∑_{i∈F} \frac{Φ_{i} + ϕ_{i}}{2} · e_{i}|z∥ ≤ \frac{1}{2} (∑_{i∈F} |Φ_{i} − ϕ_{i}|^{2})^{\frac{1}{2}}

holds. Then we have the inequality

Re [(x, y|z) − ∑_{i∈F} (x, e_{i}|z) (e_{i}, y|z)] (4.3)

≤ \frac{1}{16} · \frac{1}{λ (1 − λ)} \begin{matrix}∑ & i∈F & |Φ_{i} − ϕ_{i}| & 2 \\ ∑ & i∈F & Re & (Φ_{i}\overline{ϕ_{i}}) \end{matrix} ∑_{i∈F} |(λx + (1 − λ) y, e_{i}|z)|^{2} .

The constant \frac{1}{16} is the best possible constant in (4.3) in the sense that it cannot be replaced by a smaller constant. Proof. Using the known inequality

Re (z, u|v) ≤ \frac{1}{4} ∥z + u|v∥^{2}

we may state, for any a, b ∈ X and λ ∈ (0, 1) , that

(4.4)  Re (a, b|z) ≤ \frac{1}{4λ (1 − λ)} ∥λa + (1 − λ) b|z∥^{2} .

BESSEL AND GRÜSS INEQUALITIES 11

Since

(x, y|z) − ∑_{i∈F} (x, e_{i}|z) (e_{i}, y|z) = (x − ∑_{i∈F} (x, e_{i}|z) e_{i}, y − ∑_{i∈F} (y, e_{i}|z) e_{i}|z) ,

for any x, y ∈ X, then, by (4.4), we get

Re [(x, y|z) − ∑_{i∈F} (x, e_{i}|z) (e_{i}, y|z)] (4.5)

= Re [(x − ∑_{i∈F} (x, e_{i}|z) e_{i}, y − ∑_{i∈F} (y, e_{i}|z) e_{i}|z)]

≤ \frac{1}{4λ (1 − λ)} ∥λ (x − ∑_{i∈F} (x, e_{i}|z) e_{i}) + (1 − λ) (y − ∑_{i∈F} (y, e_{i}|z) e_{i}) |z∥^{2}

= \frac{1}{4λ (1 − λ)} ∥λx + (1 − λ) y − ∑_{i∈F} (λx + (1 − λ) y, e_{i}|z) e_{i}|z∥^{2}

= \frac{1}{4λ (1 − λ)} [∥λx + (1 − λ) y|z∥^{2} − ∑_{i∈F} |(λx + (1 − λ) y, e_{i}|z)|^{2}] .

If we apply the reverse of Bessel’s inequality from Corollary 1 for λx + (1 − λ) y, we may state that

∥λx + (1 − λ) y|z∥^{2} − ∑_{i∈F} |(λx + (1 − λ) y, e_{i}|z)|^{2} (4.6)

≤ \frac{1}{4} \begin{matrix}∑ & i∈F & |Φ_{i} − ϕ_{i}| & 2 \\ ∑ & i∈F & Re & (Φ_{i}\overline{ϕ_{i}}) \end{matrix} ∑_{i∈F} |(λx + (1 − λ) y, e_{i}|z)|^{2} .

Now, by making use of (4.5) and (4.6), we deduce (4.3).

The fact that \frac{1}{16} is the best possible constant in (4.3) follows by the fact that if in (4.1) we choose x = y, then it becomes (i) of Theorem 3, implying for λ = \frac{1}{2} the inequality (2.10), for which, we have shown that \frac{1}{4} is the best constant. Remark 4. If in Theorem 5, we choose λ = \frac{1}{2}, then we get

Re [(x, y|z) − ∑_{i∈F} (x, e_{i}|z) (e_{i}, y|z)] (4.7)

≤ \frac{1}{4} \begin{matrix}∑ & i∈F & |Φ_{i} − ϕ_{i}| & 2 \\ ∑ & i∈F & Re & (Φ_{i}\overline{ϕ_{i}}) \end{matrix} ∑_{i∈F} |(\frac{x + y}{2} , e_{i}|z)|^{2} ,

provided

Re (∑_{i∈F} Φ_{i}e_{i} − \frac{x + y}{2} , \frac{x + y}{2} − ∑_{i∈F} ϕ_{i}e_{i}|z) ≥ 0

or, equivalently,

(4.8)  ∥\frac{x + y}{2} − ∑_{i∈F} \frac{Φ_{i} + ϕ_{i}}{2} · e_{i}|z∥ ≤ \frac{1}{2} (∑_{i∈F} |Φ_{i} − ϕ_{i}|^{2})^{\frac{1}{2}} .

5. Applications for Determinantal Integral Inequalities

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

Denote by L_{ρ}^{2} (Ω) the Hilbert space of all real-valued functions f defined on Ω that are 2−ρ−integrable on Ω, i.e., ∫_{Ω} ρ (s) |f (s)|^{2} dμ (s) < ∞, where ρ : Ω → [0, ∞) is a measurable function on Ω.

We can introduce the following 2-inner product on L_{ρ}^{2} (Ω) by formula (5.1)

(f, g|h)_{ρ} := \frac{1}{2} ∫_{Ω} ∫_{Ω} ρ (s) ρ (t) | \begin{matrix}f (s) & f (t) \\ h (s) & h (t) \end{matrix} | | \begin{matrix}g (s) & g (t) \\ h (s) & h (t) \end{matrix} | dμ (s) dμ (t) ,

where

| \begin{matrix}f (s) & f (t) \\ h (s) & h (t) \end{matrix} |

denotes the determinant of the matrix

[ \begin{matrix}f (s) & f (t) \\ h (s) & h (t) \end{matrix} ] ,

which generates the 2-norm on L_{ρ}^{2} (Ω) expressed by

(5.2) ∥f|h∥_{ρ} := (\frac{1}{2} ∫_{Ω} ∫_{Ω} ρ (s) ρ (t) | \begin{matrix}f (s) & f (t) \\ h (s) & h (t) \end{matrix} |^{2} dμ (s) dμ (t))^{1/2} .

A simple calculation with integrals reveals that

(5.3)  (f, g|h)_{ρ} = | \begin{matrix}∫_{Ω} ρfgdμ & ∫_{Ω} ρfhdμ \\ ∫_{Ω} ρghdμ & ∫_{Ω} ρh^{2}dμ \end{matrix} |

and

(5.4)  ∥f|h∥_{ρ} = | \begin{matrix}∫_{Ω} ρf^{2}dμ & ∫_{Ω} ρfhdμ \\ ∫_{Ω} ρfhdμ & ∫_{Ω} ρh^{2}dμ \end{matrix} |^{1/2} ,

where, for simplicity, instead of ∫_{Ω} ρ (s) f (s) g (s) dμ (s) , we have written ∫_{Ω} ρfgdμ.

We recall that the pair of functions (q, p) ∈ L_{ρ}^{2} (Ω)×L_{ρ}^{2} (Ω) is called synchronous if

(q (x) − q (y)) (p (x) − p (y)) ≥ 0

for a.e. x, y ∈ Ω.

We note that, if Ω = [a, b] , then a sufficient condition for synchronicity is that the functions are both monotonic increasing or decreasing. This condition is not necessary.

Now, suppose that h ∈ L_{ρ}^{2} (Ω) is such that h (x) ≠ 0 for μ − a.e. x ∈ Ω. Then, by the definition of 2-inner product (f, g|h)_{ρ} , we have

(f, g|h)_{ρ} (5.5)

= \frac{1}{2} ∫_{Ω} ∫_{Ω} ρ (s) ρ (t) h^{2} (s) h^{2} (t) (\frac{f (s)}{h (s)} − \frac{f (t)}{h (t)}) (\frac{g (s)}{h (s)} − \frac{g (t)}{h (t)}) dμ (s) dμ (t)

and thus a sufficient condition for the inequality

(5.6)  (f, g|h)_{ρ} ≥ 0

to hold, is that, the functions (\frac{f}{h}, \frac{g}{h}) are synchronous. It is obvious that, this condition is not necessary.

Using the representations (5.3), (5.4) and the inequalities for 2-inner products and 2-norms established in the previous sections, we have some interesting determinantal integral inequalities.

Proposition 1. Let h ∈ L_{ρ}^{2} (Ω) be such that h (x) ≠ 0 for μ − a.e. x ∈ Ω, and (f_{i})_{i∈I} a family of functions in L_{ρ}^{2} (Ω) with the property that

| \begin{matrix}∫_{Ω} ρf_{i}f_{j}dμ & ∫_{Ω} ρf_{i}hdμ \\ ∫_{Ω} ρf_{j}hdμ & ∫_{Ω} ρh^{2}dμ \end{matrix} | = δ_{i,j}

for any i, j ∈ I, where δ_{i,j} is the Kronecker delta.

If we assume that there exists the real numbers M_{i}, m_{i}, i ∈ F, with ∑_{i∈F} M_{i}m_{i} > 0, where F is a given finite part of I, such that the functions

∑_{i∈F} M_{i} · \frac{f_{i}}{h} − \frac{f}{h} , \frac{f}{h} − ∑_{i∈F} m_{i} · \frac{f_{i}}{h}

are synchronous on Ω, then we have the inequalities

| \begin{matrix}∫_{Ω} ρf^{2}dμ & ∫_{Ω} ρfhdμ \\ ∫_{Ω} ρfhdμ & ∫_{Ω} ρh^{2}dμ \end{matrix} |

≤ \frac{1}{4} \frac{∑_{i∈F} (M_{i} + m_{i})^{2}}{∑_{i∈F} M_{i}m_{i}} ∑_{i∈F} | \begin{matrix}∫_{Ω} ρf_{i}fdμ & ∫_{Ω} ρf_{i}hdμ \\ ∫_{Ω} ρfhdμ & ∫_{Ω} ρh^{2}dμ \end{matrix} |^{2}

and

0 ≤ | \begin{matrix}∫_{Ω} ρf^{2}dμ & ∫_{Ω} ρfhdμ \\ ∫_{Ω} ρfhdμ & ∫_{Ω} ρh^{2}dμ \end{matrix} | − ∑_{i∈F} | \begin{matrix}∫_{Ω} ρf_{i}fdμ & ∫_{Ω} ρf_{i}hdμ \\ ∫_{Ω} ρfhdμ & ∫_{Ω} ρh^{2}dμ \end{matrix} |^{2}

≤ \frac{1}{4} \frac{∑_{i∈F} (M_{i} − m_{i})^{2}}{∑_{i∈F} M_{i}m_{i}} ∑_{i∈F} | \begin{matrix}∫_{Ω} ρf_{i}fdμ & ∫_{Ω} ρf_{i}hdμ \\ ∫_{Ω} ρfhdμ & ∫_{Ω} ρh^{2}dμ \end{matrix} |^{2} .

The constant \frac{1}{4} is best possible in both inequalities.

The proof follows by Theorem 3 and Corollary 1 applied for the 2-inner product (·, ·|·)_{ρ} and we omit the details.

Similar determinantal integral inequalities may be stated if one uses the other results for 2-inner products obtained above, but we do not present them here.

Acknowledgement: S. S. Dragomir and Y. J. Cho greatly acknowledge the financial support from the Brain Pool Program (2002) of the Korean Federation of Science and Technology Societies. The research was performed under the ”Memorandum of Understanding” between Victoria University and Gyeongsang National University.

References

[1] Y.J. CHO, C.S. LIN, S.S. KIM and A. MISIAK, Theory of 2-Inner Product Spaces, Nova Science Publishers, Inc., New York, 2001.

[2] Y.J. CHO, M. MATIĆ and J.E. PEČARIĆ, On Gram’s determinant in 2-inner product spaces, J. Korean Math. Soc., 38(2001), No. 6, pp. 1125-1156.

[3] S.S. DRAGOMIR, Y.J. CHO, S.S. KIM and J. ROUMELIOTIS, A reverse of Bessel’s inequality in 2-inner product spaces and some Grüss type related results with applications, Preprint on line: http://www.mathpreprints.com/math/Preprint/Sever/20030923.1/2

[4] S.S. DRAGOMIR, Y.J. CHO, S.S. KIM and Y.-H. KIM, On Bessel’s and Gruss’ inequalities for orthonormal families in 2-inner product spaces and applications, Preprint on line: http://www.mathpreprints.com/math/Preprint/Sever/20030929/1/

[5] D.S. MITRINOVIĆ, J.E. PEČARIĆ and A.M. FINK, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, 1993.

[6] R.W. FREESE and Y.J. CHO, Geometry of Linear 2-Normed Spaces, Nova Science Publishers, Inc., New York, 2001.

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

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

Department of Mathematics Education, The Research Institute of Natural Sciences, College of Education, Gyeongsang National University, Chinju 660-701, Korea

   E-mail address: yjcho@nongae.gsnu.ac.kr

Department of Mathematics, Dongeui University, Pusan, 614-714, Korea

   E-mail address: sskim@dongeui.ac.kr