SOME PEČARIĆ’S TYPE INEQUALITIES IN 2-INNER PRODUCT SPACES AND APPLICATIONS

Y.J. CHO^{⋆}, S.S. DRAGOMIR, C.-S. LIN, S.S. KIM^{♦}, AND Y.-H. KIM

Abstract. Some results related to the Pečarić’s type generalisation of Bessel’s inequality in 2-inner product spaces are given. Applications for determinantal integral inequalities are also provided.

1. Introduction

In 1992, J.E. Pečarić [5] proved the following inequality for vectors in complex inner product spaces (H; (·, ·)).

Theorem 1. Suppose that x, y_{1}, …, y_{n} are vectors in H and c_{1}, …, c_{n} are complex numbers. Then the following inequalities

(1.1)  |∑_{i=1}^{n} c_{i} (x, y_{i})|^{2} ≤ ∥x∥^{2} ∑_{i=1}^{n} |c_{i}|^{2} (∑_{j=1}^{n} |(y_{i}, y_{j})|)

≤ ∥x∥^{2} ∑_{i=1}^{n} |c_{i}|^{2} max_{1≤i≤n} (∑_{j=1}^{n} |(y_{i}, y_{j})|)

hold.

He also showed that for c_{i} = \overline{(x, y_{i})}, i ∈ \{1, ..., n\} , one gets

(1.2)  (∑_{i=1}^{n} |(x, y_{i})|^{2})^{2} ≤ ∥x∥^{2} ∑_{i=1}^{n} |(x, y_{i})|^{2} (∑_{j=1}^{n} |(y_{i}, y_{j})|)

≤ ∥x∥^{2} ∑_{i=1}^{n} |(x, y_{i})|^{2} max_{1≤i≤n} (∑_{j=1}^{n} |(y_{i}, y_{j})|) ,

which improves Bombieri’s inequality [1]

(1.3)  ∑_{i=1}^{n} |(x, y_{i})|^{2} ≤ ∥x∥^{2} max_{1≤i≤n} (∑_{j=1}^{n} |(y_{i}, y_{j})|) .

Date: 26 August, 2003.

1991 Mathematics Subject Classification. 26D15, 26D10, 46C05, 46C99.

Key words and phrases. 2-Inner products, 2-Normed spaces, Bessel’s inequality in 2-inner product spaces, Pečarić’s type inequalities in 2-inner product spaces, Determinantal integral inequalities.

⋆, ♦ Corresponding authors.

2 Y.J. CHO^{⋆}, S.S. DRAGOMIR, C.-S. LIN, S.S. KIM^{♦}, AND Y.-H. KIM

Note that (1.3) is in its turn a natural generalization of Bessel’s inequality

(1.4)  ∑_{i=1}^{n} |(x, e_{i})|^{2} ≤ ∥x∥^{2}

for any x ∈ H, which holds for the orthornormal vectors (e_{i})_{1≤i≤n} .

In this paper we point out some results of Pečarić’s type for 2-inner products spaces. Some inequalities of Bombieri type holding in these spaces are also mentioned. Natural applications for determinantal integral inequalities are given as well.

2. Some Preliminary Results in 2-Inner Product Spaces

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 the book [2]. 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 (·, ·|·) can be immediately obtained as follows [3]:

(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

(2.1)  (x, αy|z) = ᾱ(x, y|z).

(3) Using (2I_{2})–(2I_{5}), we have

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

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

In the real case 𝕂 = ℝ, (2.2) reduces to

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

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

(2.4)  (x, y|αz) = α^{2}(x, y|z).

In the complex case, using (2.1) and (2.2), 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 (2.2), yields

(2.5) (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 (2.1), we have, for any α ∈ ℂ, that

(2.6)  (x, y|αz) = |α|^{2}(x, y|z).

However, for α ∈ ℝ, (2.6) reduces to (2.4). Also, from (2.6) 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

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

For x = z, (2.7) becomes

−(y, y|z)|(z, y|z)|^{2} ≥ 0,

which implies that

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

provided y and z are linearly independent. Obviously, when y and z are linearly dependent, (2.8) holds too. Thus (2.8) is true for any two vectors y, z ∈ X. Now, if y and z are linearly independent, then (y, y|z) > 0 and, from (2.7), it follows

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

Using (2.8), it is easy to check that (2.9) is trivially fulfilled when y and z are linearly dependent. Therefore, the inequality (2.9) 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 (2.9) 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

(2.10)  ∥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. For recent result devoted to the geometry of linear 2-normed spaces, see [4].

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 (2.10).

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 [3]) for z−orthonormal family (f_{i})_{1≤i≤n} in the 2-inner product space (X; (·, ·|·)):

(2.11)  ∑_{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 [3] and the references therein.

3. Some Inequalities for 2-Norms

We start with the following lemma that is interesting in its own right.

Lemma 1. Let (X, (·, ·|·)) be a 2-inner product space on 𝕂 and z_{1}, …, z_{n}, z ∈ X, α_{1}, …, α_{n} ∈ 𝕂. Then one has the inequalities:

∥∑_{i=1}^{n} α_{i}z_{i}|z∥^{2} (3.1)

≤ (∑_{i=1}^{n} |α_{i}|^{p} (∑_{j=1}^{n} |(z_{i}, z_{j}|z)|))^{\frac{1}{p}} (∑_{i=1}^{n} |α_{i}|^{q} (∑_{j=1}^{n} |(z_{i}, z_{j}|z)|))^{\frac{1}{q}}

≤ \{ \begin{matrix}A; \\ B; \\ C; \end{matrix}

where

A := \{ \begin{matrix}max_{1≤i≤n} |α_{i}|^{2} ∑_{i,j=1}^{n} |(z_{i}, z_{j}|z)|; \\ max_{1≤i≤n} |α_{i}| (∑_{i=1}^{n} |α_{i}|^{γq})^{\frac{1}{γq}} ( ∑_{i,j=1}^{n} |(z_{i}, z_{j}|z)|)^{\frac{1}{p}} (∑_{i=1}^{n} ( ∑_{j=1}^{n} |(z_{i}, z_{j}|z)|)^{δ})^{\frac{1}{δq}} , \\ if γ > 1, \frac{1}{γ} + \frac{1}{δ} = 1; \\ max_{1≤i≤n} |α_{i}| (∑_{i=1}^{n} |α_{i}|^{q})^{\frac{1}{q}} ( ∑_{i,j=1}^{n} |(z_{i}, z_{j}|z)|)^{\frac{1}{p}} max_{1≤i≤n} ( ∑_{j=1}^{n} |(z_{i}, z_{j}|z)|)^{\frac{1}{q}} ; \end{matrix}

B := \{ \begin{matrix}max_{1≤i≤n} & |α_{i}| & (∑_{i=1}^{n} |α_{i}|^{αp}) & \frac{1}{αp} & ( ∑_{i,j=1}^{n} |(z_{i}, z_{j}|z)|) & \frac{1}{q} \\ × (∑_{i=1}^{n} ( ∑_{j=1}^{n} |(z_{i}, z_{j}|z)|)^{β})^{\frac{1}{βq}} ,  if α > 1, \frac{1}{α} + \frac{1}{β} = 1; \\ (∑_{i=1}^{n} |α_{i}|^{αp}) & \frac{1}{αp} & (∑_{i=1}^{n} |α_{i}|^{γq}) & \frac{1}{γq} & (∑_{i=1}^{n} ( ∑_{j=1}^{n} |(z_{i}, z_{j}|z)|)^{β}) & \frac{1}{pβ} \\ × (∑_{i=1}^{n} ( ∑_{j=1}^{n} |(z_{i}, z_{j}|z)|)^{δ})^{\frac{1}{δq}} ,  if α > 1, \frac{1}{α} + \frac{1}{β} = 1 \\ and γ > 1, \frac{1}{γ} + \frac{1}{δ} = 1; \\ (∑_{i=1}^{n} |α_{i}|^{q}) & \frac{1}{q} & (∑_{i=1}^{n} |α_{i}|^{αp}) & \frac{1}{αp} & max_{1≤i≤n} & ( ∑_{j=1}^{n} |(z_{i}, z_{j}|z)|) & \frac{1}{q} \\ × (∑_{i=1}^{n} ( ∑_{j=1}^{n} |(z_{i}, z_{j}|z)|)^{β})^{\frac{1}{pβ}} , if α > 1, \frac{1}{α} + \frac{1}{β} = 1; \end{matrix}

C := \{ \begin{matrix}max_{1≤i≤n} |α_{i}| (∑_{i=1}^{n} |α_{i}|^{p})^{\frac{1}{p}} max_{1≤i≤n} ( ∑_{j=1}^{n} |(z_{i}, z_{j}|z)|)^{\frac{1}{p}} ( ∑_{i,j=1}^{n} |(z_{i}, z_{j}|z)|)^{\frac{1}{q}} ; \\ (∑_{i=1}^{n} |α_{i}|^{p}) & \frac{1}{p} & (∑_{i=1}^{n} |α_{i}|^{γq}) & \frac{1}{γq} & max_{1≤i≤n} & ( ∑_{j=1}^{n} |(z_{i}, z_{j}|z)|) & \frac{1}{p} \\ × (∑_{i=1}^{n} ( ∑_{j=1}^{n} |(z_{i}, z_{j}|z)|)^{δ})^{\frac{1}{δq}} , if γ > 1, \frac{1}{γ} + \frac{1}{δ} = 1; \\ (∑_{i=1}^{n} |α_{i}|^{p})^{\frac{1}{p}} (∑_{i=1}^{n} |α_{i}|^{q})^{\frac{1}{q}} max_{1≤i≤n} ( ∑_{j=1}^{n} |(z_{i}, z_{j}|z)|) , \end{matrix}

6 Y.J. CHO^{⋆}, S.S. DRAGOMIR, C.-S. LIN, S.S. KIM^{♦}, AND Y.-H. KIM

and p > 1, \frac{1}{p} + \frac{1}{q} = 1. Proof. We observe that

(3.2) ∥∑_{i=1}^{n} α_{i}z_{i}|z∥^{2} = (∑_{i=1}^{n} α_{i}z_{i}, ∑_{j=1}^{n} α_{j}z_{j}|z)
= ∑_{i=1}^{n} ∑_{j=1}^{n} α_{i}\overline{α_{j}} (z_{i}, z_{j}|z) = |∑_{i=1}^{n} ∑_{j=1}^{n} α_{i}\overline{α_{j}} (z_{i}, z_{j}|z)|
≤ ∑_{i=1}^{n} ∑_{j=1}^{n} |α_{i}| |α_{j}| |(z_{i}, z_{j}|z)| =: M.

If one uses the Hölder inequality for double sums, i.e., we recall it

(3.3)  ∑_{i,j=1}^{n} m_{ij}a_{ij}b_{ij} ≤ ( ∑_{i,j=1}^{n} m_{ij}a_{ij}^{p})^{\frac{1}{p}} ( ∑_{i,j=1}^{n} m_{ij}b_{ij}^{q})^{\frac{1}{q}} ,

where m_{ij}, a_{ij}, b_{ij} ≥ 0, \frac{1}{p} + \frac{1}{q} = 1, p > 1; then

(3.4) M ≤ ( ∑_{i,j=1}^{n} |(z_{i}, z_{j}|z)| |α_{i}|^{p})^{\frac{1}{p}} ( ∑_{i,j=1}^{n} |(z_{i}, z_{j}|z)| |α_{i}|^{q})^{\frac{1}{q}}

= (∑_{i=1}^{n} |α_{i}|^{p} (∑_{j=1}^{n} |(z_{i}, z_{j}|z)|))^{\frac{1}{p}} (∑_{i=1}^{n} |α_{i}|^{q} (∑_{j=1}^{n} |(z_{i}, z_{j}|z)|))^{\frac{1}{q}}

and the first inequality in (3.1) is proved.

Observe, by Hölder inequality, that

∑_{i=1}^{n} |α_{i}|^{p} (∑_{j=1}^{n} |(z_{i}, z_{j}|z)|) ≤ \{ \begin{matrix}max_{1≤i≤n} |α_{i}|^{p} ∑_{i,j=1}^{n} |(z_{i}, z_{j}|z)|; \\ (∑_{i=1}^{n} |α_{i}|^{αp}) & \frac{1}{α} & (∑_{i=1}^{n} ( ∑_{j=1}^{n} |(z_{i}, z_{j}|z)|)^{β}) & \frac{1}{β} \\ if α > 1, \frac{1}{α} + \frac{1}{β} = 1; \\ ∑_{i=1}^{n} |α_{i}|^{p} max_{1≤i≤n} ( ∑_{j=1}^{n} |(z_{i}, z_{j}|z)|); \end{matrix}

which gives

(3.5) (∑_{i=1}^{n} |α_{i}|^{p} (∑_{j=1}^{n} |(z_{i}, z_{j}|z)|))^{\frac{1}{p}}

≤ \{ \begin{matrix}max_{1≤i≤n} |α_{i}| ( ∑_{i,j=1}^{n} |(z_{i}, z_{j}|z)|)^{\frac{1}{p}} ; \\ (∑_{i=1}^{n} |α_{i}|^{αp})^{\frac{1}{αp}} (∑_{i=1}^{n} ( ∑_{j=1}^{n} |(z_{i}, z_{j}|z)|)^{β})^{\frac{1}{βp}} if α > 1, \frac{1}{α} + \frac{1}{β} = 1; \\ (∑_{i=1}^{n} |α_{i}|^{p})^{\frac{1}{p}} max_{1≤i≤n} ( ∑_{j=1}^{n} |(z_{i}, z_{j}|z)|)^{\frac{1}{p}} . \end{matrix}

Similarly, we have

(3.6) (∑_{i=1}^{n} |α_{i}|^{q} (∑_{j=1}^{n} |(z_{i}, z_{j}|z)|))^{\frac{1}{q}}

≤ \{ \begin{matrix}max_{1≤i≤n} |α_{i}| ( ∑_{i,j=1}^{n} |(z_{i}, z_{j}|z)|)^{\frac{1}{q}} ; \\ (∑_{i=1}^{n} |α_{i}|^{γq})^{\frac{1}{γq}} (∑_{i=1}^{n} ( ∑_{j=1}^{n} |(z_{i}, z_{j}|z)|)^{δ})^{\frac{1}{δq}} if γ > 1, \frac{1}{γ} + \frac{1}{δ} = 1; \\ (∑_{i=1}^{n} |α_{i}|^{q})^{\frac{1}{q}} max_{1≤i≤n} ( ∑_{j=1}^{n} |(z_{i}, z_{j}|z)|)^{\frac{1}{q}} . \end{matrix}

Using (3.1) and (3.5)–(3.6), we deduce the 9 inequalities in the second part of (3.2).

If we choose p = q = 2, then the following result holds. Corollary 1. If z_{1}, …, z_{n}, z ∈ X and α_{1}, …, α_{n} ∈ 𝕂, then one has

∥∑_{i=1}^{n} α_{i}z_{i}|z∥^{2} ≤ ∑_{i=1}^{n} |α_{i}|^{2} (∑_{j=1}^{n} |(z_{i}, z_{j}|z)|)
≤ \{ \begin{matrix}D; \\ E; \\ F; \end{matrix}

8 Y.J. CHO^{⋆}, S.S. DRAGOMIR, C.-S. LIN, S.S. KIM^{♦}, AND Y.-H. KIM

where

D := \{ \begin{matrix}max_{1≤i≤n} |α_{i}|^{2} ∑_{i,j=1}^{n} |(z_{i}, z_{j}|z)|; \\ max_{1≤i≤n} |α_{i}| (∑_{i=1}^{n} |α_{i}|^{2γ})^{\frac{1}{2γ}} ( ∑_{i,j=1}^{n} |(z_{i}, z_{j}|z)|)^{\frac{1}{2}} (∑_{i=1}^{n} ( ∑_{j=1}^{n} |(z_{i}, z_{j}|z)|)^{δ})^{\frac{1}{2δ}} , \\ if γ > 1, \frac{1}{γ} + \frac{1}{δ} = 1; \\ max_{1≤i≤n} |α_{i}| (∑_{i=1}^{n} |α_{i}|^{2})^{\frac{1}{2}} ( ∑_{i,j=1}^{n} |(z_{i}, z_{j}|z)|)^{\frac{1}{2}} max_{1≤i≤n} ( ∑_{j=1}^{n} |(z_{i}, z_{j}|z)|)^{\frac{1}{2}} ; \end{matrix}

E := \{ \begin{matrix}max_{1≤i≤n} |α_{i}| (∑_{i=1}^{n} |α_{i}|^{2α})^{\frac{1}{2α}} ( ∑_{i,j=1}^{n} |(z_{i}, z_{j}|z)|)^{\frac{1}{2}} (∑_{i=1}^{n} ( ∑_{j=1}^{n} |(z_{i}, z_{j}|z)|)^{β})^{\frac{1}{2β}} , \\ if α > 1, \frac{1}{α} + \frac{1}{β} = 1; \\ (∑_{i=1}^{n} |α_{i}|^{2α}) & \frac{1}{2α} & (∑_{i=1}^{n} |α_{i}|^{2γ}) & \frac{1}{2γ} & (∑_{i=1}^{n} ( ∑_{j=1}^{n} |(z_{i}, z_{j}|z)|)^{β}) & \frac{1}{2β} \\ × (∑_{i=1}^{n} ( ∑_{j=1}^{n} |(z_{i}, z_{j}|z)|)^{δ})^{\frac{1}{2δ}} , if α > 1, \frac{1}{α} + \frac{1}{β} = 1 \\ and γ > 1, \frac{1}{γ} + \frac{1}{δ} = 1; \\ (∑_{i=1}^{n} |α_{i}|^{2})^{\frac{1}{2}} (∑_{i=1}^{n} |α_{i}|^{2α})^{\frac{1}{2α}} max_{1≤i≤n} ( ∑_{j=1}^{n} |(z_{i}, z_{j}|z)|)^{\frac{1}{2}} , \\ × (∑_{i=1}^{n} ( ∑_{j=1}^{n} |(z_{i}, z_{j}|z)|)^{β})^{\frac{1}{2β}} , if α > 1, \frac{1}{α} + \frac{1}{β} = 1; \end{matrix}

and

F := \{ \begin{matrix}max_{1≤i≤n} |α_{i}| (∑_{i=1}^{n} |α_{i}|^{2})^{\frac{1}{2}} max_{1≤i≤n} ( ∑_{j=1}^{n} |(z_{i}, z_{j}|z)|)^{\frac{1}{2}} ( ∑_{i,j=1}^{n} |(z_{i}, z_{j}|z)|)^{\frac{1}{2}} ; \\ (∑_{i=1}^{n} |α_{i}|^{2}) & \frac{1}{2} & (∑_{i=1}^{n} |α_{i}|^{2γ}) & \frac{1}{2γ} & max_{1≤i≤n} & ( ∑_{j=1}^{n} |(z_{i}, z_{j}|z)|) & \frac{1}{2} \\ × (∑_{i=1}^{n} ( ∑_{j=1}^{n} |(z_{i}, z_{j}|z)|)^{δ})^{\frac{1}{2δ}} , if γ > 1, \frac{1}{γ} + \frac{1}{δ} = 1; \\ ∑_{i=1}^{n} |α_{i}|^{2} max_{1≤i≤n} ( ∑_{j=1}^{n} |(z_{i}, z_{j}|z)|) . \end{matrix}

4. Some Pečarić Type Inequalities for 2-Inner Products

We are now able to point out the following result which complements and generalizes the Bessel inequality (2.11) in 2-inner product spaces.

Theorem 2. Let x, y_{1}, …, y_{n}, z be vectors of an inner product space (X; (·, ·)) and c_{1}, …, c_{n} ∈ 𝕂. Then we have

|∑_{i=1}^{n} c_{i} (x, y_{i}|z)|^{2} (4.1)

≤ ∥x|z∥^{2} (∑_{i=1}^{n} |c_{i}|^{p} (∑_{j=1}^{n} |(y_{i}, y_{j}|z)|))^{\frac{1}{p}} (∑_{i=1}^{n} |c_{i}|^{q} (∑_{j=1}^{n} |(y_{i}, y_{j}|z)|))^{\frac{1}{q}}

≤ ∥x|z∥^{2} × \{ \begin{matrix}G; \\ H; \\ L; \end{matrix}

where

G := \{ \begin{matrix}max_{1≤i≤n} |c_{i}|^{2} ∑_{i,j=1}^{n} |(y_{i}, y_{j}|z)|; \\ max_{1≤i≤n} |c_{i}| (∑_{i=1}^{n} |c_{i}|^{γq})^{\frac{1}{γq}} ( ∑_{i,j=1}^{n} |(y_{i}, y_{j}|z)|)^{\frac{1}{p}} (∑_{i=1}^{n} ( ∑_{j=1}^{n} |(y_{i}, y_{j}|z)|)^{δ})^{\frac{1}{δq}} , \\ if γ > 1, \frac{1}{γ} + \frac{1}{δ} = 1; \\ max_{1≤i≤n} |c_{i}| (∑_{i=1}^{n} |c_{i}|^{q})^{\frac{1}{q}} ( ∑_{i,j=1}^{n} |(y_{i}, y_{j}|z)|)^{\frac{1}{p}} max_{1≤i≤n} ( ∑_{j=1}^{n} |(y_{i}, y_{j}|z)|)^{\frac{1}{q}} ; \end{matrix}

10 Y.J. CHO^{⋆}, S.S. DRAGOMIR, C.-S. LIN, S.S. KIM^{♦}, AND Y.-H. KIM

H := \{ \begin{matrix}max_{1≤i≤n} |c_{i}| (∑_{i=1}^{n} |c_{i}|^{αp})^{\frac{1}{αp}} ( ∑_{i,j=1}^{n} |(y_{i}, y_{j}|z)|)^{\frac{1}{q}} (∑_{i=1}^{n} ( ∑_{j=1}^{n} |(y_{i}, y_{j}|z)|)^{β})^{\frac{1}{pβ}} , \\ if α > 1, \frac{1}{α} + \frac{1}{β} = 1; \\ (∑_{i=1}^{n} |c_{i}|^{αp}) & \frac{1}{αp} & (∑_{i=1}^{n} |c_{i}|^{γq}) & \frac{1}{γq} & (∑_{i=1}^{n} ( ∑_{j=1}^{n} |(y_{i}, y_{j}|z)|)^{β}) & \frac{1}{pβ} \\ × (∑_{i=1}^{n} ( ∑_{j=1}^{n} |(y_{i}, y_{j}|z)|)^{δ})^{\frac{1}{δq}} , if α > 1, \frac{1}{α} + \frac{1}{β} = 1 \\ and γ > 1, \frac{1}{γ} + \frac{1}{δ} = 1; \\ (∑_{i=1}^{n} |c_{i}|^{q}) & \frac{1}{q} & (∑_{i=1}^{n} |c_{i}|^{αp}) & \frac{1}{αp} & max_{1≤i≤n} & ( ∑_{j=1}^{n} |(y_{i}, y_{j}|z)|) & \frac{1}{q} \\ × (∑_{i=1}^{n} ( ∑_{j=1}^{n} |(y_{i}, y_{j}|z)|)^{β})^{\frac{1}{pβ}} , if α > 1, \frac{1}{α} + \frac{1}{β} = 1; \end{matrix}

and

L := \{ \begin{matrix}max_{1≤i≤n} |c_{i}| (∑_{i=1}^{n} |c_{i}|^{p})^{\frac{1}{p}} max_{1≤i≤n} ( ∑_{j=1}^{n} |(y_{i}, y_{j}|z)|)^{\frac{1}{p}} ( ∑_{i,j=1}^{n} |(y_{i}, y_{j}|z)|)^{\frac{1}{q}} ; \\ (∑_{i=1}^{n} |c_{i}|^{p}) & \frac{1}{p} & (∑_{i=1}^{n} |c_{i}|^{γq}) & \frac{1}{γq} & max_{1≤i≤n} & ( ∑_{j=1}^{n} |(y_{i}, y_{j}|z)|) & \frac{1}{p} \\ × (∑_{i=1}^{n} ( ∑_{j=1}^{n} |(y_{i}, y_{j}|z)|)^{δ})^{\frac{1}{δq}} , if γ > 1, \frac{1}{γ} + \frac{1}{δ} = 1; \\ (∑_{i=1}^{n} |c_{i}|^{p})^{\frac{1}{p}} (∑_{i=1}^{n} |c_{i}|^{q})^{\frac{1}{q}} max_{1≤i≤n} ( ∑_{j=1}^{n} |(y_{i}, y_{j}|z)|); \end{matrix}

where p > 1, \frac{1}{p} + \frac{1}{q} = 1. Proof. We note that

∑_{i=1}^{n} c_{i} (x, y_{i}|z) = (x, ∑_{i=1}^{n} \overline{c_{i}}y_{i}|z) .

Using Schwarz’s inequality in 2-inner product spaces, we have

(4.2) |∑_{i=1}^{n} c_{i} (x, y_{i}|z)|^{2} ≤ ∥x|z∥^{2} ∥∑_{i=1}^{n} \overline{c_{i}}y_{i}|z∥^{2} .

Finally, using Lemma 1 with α_{i} = \overline{c_{i}}, z_{i} = y_{i} (i = 1, …, n) , we deduce the desired inequality (4.1).

PEČARIĆ’S TYPE INEQUALITIES 11

Remark 1. If in (4.1) we choose p = q = 2, we obtain amongst others, the following result

(4.3)  |∑_{i=1}^{n} c_{i} (x, y_{i}|z)|^{2}
≤ ∥x|z∥^{2} (∑_{i=1}^{n} |c_{i}|^{2} (∑_{j=1}^{n} |(y_{i}, y_{j}|z)|))
≤ ∥x|z∥^{2} × \{ \begin{matrix}M; \\ N; \\ P; \end{matrix}

where

M := \{ \begin{matrix}max_{1≤i≤n} |c_{i}|^{2} ∑_{i,j=1}^{n} |(y_{i}, y_{j}|z)|; \\ max_{1≤i≤n} |c_{i}| (∑_{i=1}^{n} |c_{i}|^{2γ})^{\frac{1}{2γ}} ( ∑_{i,j=1}^{n} |(y_{i}, y_{j}|z)|)^{\frac{1}{2}} (∑_{i=1}^{n} ( ∑_{j=1}^{n} |(y_{i}, y_{j}|z)|)^{δ})^{\frac{1}{2δ}} , \\ if γ > 1, \frac{1}{γ} + \frac{1}{δ} = 1; \\ max_{1≤i≤n} |c_{i}| (∑_{i=1}^{n} |c_{i}|^{2})^{\frac{1}{2}} ( ∑_{i,j=1}^{n} |(y_{i}, y_{j}|z)|)^{\frac{1}{2}} max_{1≤i≤n} ( ∑_{j=1}^{n} |(y_{i}, y_{j}|z)|)^{\frac{1}{2}} ; \end{matrix}

N := \{ \begin{matrix}max_{1≤i≤n} \{|c_{i}|\} (∑_{i=1}^{n} |c_{i}|^{2α})^{\frac{1}{2α}} ( ∑_{i,j=1}^{n} |(y_{i}, y_{j}|z)|)^{\frac{1}{2}} (∑_{i=1}^{n} ( ∑_{j=1}^{n} |(y_{i}, y_{j}|z)|)^{β})^{\frac{1}{2β}} , \\ if α > 1, \frac{1}{α} + \frac{1}{β} = 1; \\ (∑_{i=1}^{n} |c_{i}|^{2α}) & \frac{1}{2α} & (∑_{i=1}^{n} |c_{i}|^{2γ}) & \frac{1}{2γ} & (∑_{i=1}^{n} ( ∑_{j=1}^{n} |(y_{i}, y_{j}|z)|)^{β}) & \frac{1}{2p} \\ × (∑_{i=1}^{n} ( ∑_{j=1}^{n} |(y_{i}, y_{j}|z)|)^{δ})^{\frac{1}{2δ}} , if α > 1, \frac{1}{α} + \frac{1}{β} = 1 \\ and γ > 1, \frac{1}{γ} + \frac{1}{δ} = 1; \\ (∑_{i=1}^{n} |c_{i}|^{2}) & \frac{1}{2} & (∑_{i=1}^{n} |c_{i}|^{2α}) & \frac{1}{2α} & max_{1≤i≤n} & ( ∑_{j=1}^{n} |(y_{i}, y_{j}|z)|) & \frac{1}{2} \\ × (∑_{i=1}^{n} ( ∑_{j=1}^{n} |(y_{i}, y_{j}|z)|)^{β})^{\frac{1}{2β}} , if α > 1, \frac{1}{α} + \frac{1}{β} = 1; \end{matrix}

12 Y.J. CHO^{⋆}, S.S. DRAGOMIR, C.-S. LIN, S.S. KIM^{♦}, AND Y.-H. KIM

and

P := \{ \begin{matrix}max_{1≤i≤n} \{|c_{i}|\} (∑_{i=1}^{n} |c_{i}|^{2})^{\frac{1}{2}} max_{1≤i≤n} ( ∑_{j=1}^{n} |(y_{i}, y_{j}|z)|)^{\frac{1}{2}} ( ∑_{i,j=1}^{n} |(y_{i}, y_{j}|z)|)^{\frac{1}{2}} ; \\ (∑_{i=1}^{n} |c_{i}|^{2}) & \frac{1}{2} & (∑_{i=1}^{n} |c_{i}|^{2γ}) & \frac{1}{2γ} & max_{1≤i≤n} & ( ∑_{j=1}^{n} |(y_{i}, y_{j}|z)|) & \frac{1}{2} \\ × (∑_{i=1}^{n} ( ∑_{j=1}^{n} |(y_{i}, y_{j}|z)|)^{δ})^{\frac{1}{2δ}} , if γ > 1, \frac{1}{γ} + \frac{1}{δ} = 1; \\ (∑_{i=1}^{n} |c_{i}|^{2}) max_{1≤i≤n} ( ∑_{j=1}^{n} |(y_{i}, y_{j}|z)|); \end{matrix}

which contains the version of Pečarić’s inequality for 2-inner products, i.e., the inequality

(4.4)  |∑_{i=1}^{n} c_{i} (x, y_{i}|z)|^{2} ≤ ∥x|z∥^{2} (∑_{i=1}^{n} |c_{i}|^{2} (∑_{j=1}^{n} |(y_{i}, y_{j}|z)|))
≤ (∑_{i=1}^{n} |c_{i}|^{2}) max_{1≤i≤n} (∑_{j=1}^{n} |(y_{i}, y_{j}|z)|) .

5. Some Results of Bombieri Type for 2-Inner Products The following results of Bombieri type hold.

Theorem 3. Let x, y_{1}, …, y_{n}, z ∈ X. Then one has the inequalities:

∑_{i=1}^{n} |(x, y_{i}|z)|^{2} ≤ ∥x|z∥ [∑_{i=1}^{n} |(x, y_{i}|z)|^{p} (∑_{j=1}^{n} |(y_{i}, y_{j}|z)|)]^{\frac{1}{2p}} (5.1)

× [∑_{i=1}^{n} |(x, y_{i}|z)|^{q} (∑_{j=1}^{n} |(y_{i}, y_{j}|z)|)]^{\frac{1}{2q}}

≤ ∥x|z∥ × \{ \begin{matrix}Q; \\ R; \\ S; \end{matrix}

Q := \{ \begin{matrix}max_{1≤i≤n} |(x, y_{i}|z)| ( ∑_{i,j=1}^{n} |(y_{i}, y_{j}|z)|)^{\frac{1}{2}} ; \\ max_{1≤i≤n} & |(x, y_{i}|z)| & \frac{1}{2} & (∑_{i=1}^{n} |(x, y_{i}|z)|^{γq}) & \frac{1}{2γq} & ( ∑_{i,j=1}^{n} |(y_{i}, y_{j}|z)|) & \frac{1}{2p} \\ × (∑_{i=1}^{n} ( ∑_{j=1}^{n} |(y_{i}, y_{j}|z)|)^{δ})^{\frac{1}{2δq}} , if γ > 1, \frac{1}{γ} + \frac{1}{δ} = 1; \\ max_{1≤i≤n} & |(x, y_{i}|z)| & \frac{1}{2} & (∑_{i=1}^{n} |(x, y_{i}|z)|^{q}) & \frac{1}{2q} & ( ∑_{i,j=1}^{n} |(y_{i}, y_{j}|z)|) & \frac{1}{2p} \\ × max_{1≤i≤n} ( ∑_{j=1}^{n} |(y_{i}, y_{j}|z)|)^{\frac{1}{2q}} ; \end{matrix}

R := \{ \begin{matrix}max_{1≤i≤n} & |(x, y_{i}|z)| & \frac{1}{2} & (∑_{i=1}^{n} |(x, y_{i}|z)|^{αp}) & \frac{1}{2αβ} & ( ∑_{i,j=1}^{n} |(y_{i}, y_{j}|z)|) & \frac{1}{2q} \\ × (∑_{i=1}^{n} ( ∑_{j=1}^{n} |(y_{i}, y_{j}|z)|)^{β})^{\frac{1}{pβ}} , if α > 1, \frac{1}{α} + \frac{1}{β} = 1; \\ (∑_{i=1}^{n} |(x, y_{i}|z)|^{αp}) & \frac{1}{2αp} & (∑_{i=1}^{n} |(x, y_{i}|z)|^{γq}) & \frac{1}{2γq} & (∑_{i=1}^{n} ( ∑_{j=1}^{n} |(y_{i}, y_{j}|z)|)^{β}) & \frac{1}{2pβ} \\ × (∑_{i=1}^{n} ( ∑_{j=1}^{n} |(y_{i}, y_{j}|z)|)^{δ})^{\frac{1}{2δq}} , if α > 1, \frac{1}{α} + \frac{1}{β} = 1 \\ and γ > 1, \frac{1}{γ} + \frac{1}{δ} = 1; \\ (∑_{i=1}^{n} |(x, y_{i}|z)|^{q}) & \frac{1}{2q} & (∑_{i=1}^{n} |(x, y_{i}|z)|^{αp}) & \frac{1}{2αp} & max_{1≤i≤n} & ( ∑_{j=1}^{n} |(y_{i}, y_{j}|z)|) & \frac{1}{2p} \\ × (∑_{i=1}^{n} ( ∑_{j=1}^{n} |(y_{i}, y_{j}|z)|)^{β})^{\frac{1}{2pβ}} , if α > 1, \frac{1}{α} + \frac{1}{β} = 1; \end{matrix}

and

S := \{ \begin{matrix}max_{1≤i≤n} & |(x, y_{i}|z)| & \frac{1}{2} & (∑_{i=1}^{n} |(x, y_{i}|z)|^{p}) & \frac{1}{2p} & max_{1≤i≤n} & ( ∑_{j=1}^{n} |(y_{i}, y_{j}|z)|) & \frac{1}{2p} \\ × ( ∑_{i,j=1}^{n} |(y_{i}, y_{j}|z)|)^{\frac{1}{2q}} ; \\ (∑_{i=1}^{n} |(x, y_{i}|z)|^{p}) & \frac{1}{2p} & (∑_{i=1}^{n} |(x, y_{i}|z)|^{γq}) & \frac{1}{2γq} & max_{1≤i≤n} & ( ∑_{j=1}^{n} |(y_{i}, y_{j}|z)|) & \frac{1}{2p} \\ × (∑_{i=1}^{n} ( ∑_{j=1}^{n} |(y_{i}, y_{j}|z)|)^{δ})^{\frac{1}{2δq}} , if γ > 1, \frac{1}{γ} + \frac{1}{δ} = 1; \\ (∑_{i=1}^{n} |(x, y_{i}|z)|^{p})^{\frac{1}{2p}} (∑_{i=1}^{n} |(x, y_{i}|z)|^{q})^{\frac{1}{2q}} max_{1≤i≤n} ( ∑_{j=1}^{n} |(y_{i}, y_{j}|z)|)^{\frac{1}{2}} , \end{matrix}

where p > 1, \frac{1}{p} + \frac{1}{q} = 1.

Proof. The proof follows by Theorem 2 on choosing c_{i} = \overline{(x, y_{i}|z)}, for i ∈ \{1, ..., n\} and taking the square root in both sides of the inequalities involved. We omit the details.

Remark 2. We observe, by the last inequality in (5.1), we get

(5.2) \begin{matrix}(∑_{i=1}^{n} |(x, y_{i}|z)|^{2}) & 2 \\ (∑_{i=1}^{n} |(x, y_{i}|z)|^{p}) & \frac{1}{p} & (∑_{i=1}^{n} |(x, y_{i}|z)|^{q}) & \frac{1}{q} \end{matrix} ≤ ∥x|z∥^{2} max_{1≤i≤n} (∑_{j=1}^{n} |(y_{i}, y_{j})|) ,

where p > 1, \frac{1}{p} + \frac{1}{q} = 1, and provided that not all (x, y_{i}|z) , for i ∈ \{1, ..., n\} are zero.

Remark 3. If in this inequality we choose p = q = 2, then we obtain the following Bombieri’s type result for 2-inner products

(5.3)  ∑_{i=1}^{n} |(x, y_{i}|z)|^{2} ≤ ∥x|z∥^{2} max_{1≤i≤n} (∑_{j=1}^{n} |(y_{i}, y_{j}|z)|) .

6. Applications for Determinantal Integral Inequalities

Let (Ω, Σ, μ) be a measure space consisting of a set Ω, a σ−algebra Σ of subsets of Ω and a countably additive and positive measure μ 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

(6.1) (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} ] ,

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

(6.2)  ∥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

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

and

(6.4)  ∥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.

Using the representations (6.3), (6.4) and the inequalities for 2-inner products and 2-norms established in the previous sections, we can get some interesting determinantal integral inequalities.

We give here only two examples.

Proposition 1. Let f, g_{1}, ..., g_{n}, h ∈ L_{ρ}^{2} (Ω) , where ρ : Ω → [0, ∞) is a measurable function on Ω, then we have the inequality

(∑_{i=1}^{n} | \begin{matrix}∫_{Ω} ρfg_{i}dμ & ∫_{Ω} ρfhdμ \\ ∫_{Ω} ρg_{i}hdμ & ∫_{Ω} ρh^{2}dμ \end{matrix} |^{2})^{2}

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

× max_{1≤i≤n} \{∑_{j=1}^{n} | det [ \begin{matrix}∫_{Ω} ρg_{j}g_{i}dμ & ∫_{Ω} ρg_{j}hdμ \\ ∫_{Ω} ρg_{i}hdμ & ∫_{Ω} ρh^{2}dμ \end{matrix} ] |\}

× (∑_{i=1}^{n} |det [ \begin{matrix}∫_{Ω} ρfg_{i}dμ & ∫_{Ω} ρfhdμ \\ ∫_{Ω} ρg_{i}hdμ & ∫_{Ω} ρh^{2}dμ \end{matrix} ]|^{p})^{1/p}

× (∑_{i=1}^{n} |det [ \begin{matrix}∫_{Ω} ρfg_{i}dμ & ∫_{Ω} ρfhdμ \\ ∫_{Ω} ρg_{i}hdμ & ∫_{Ω} ρh^{2}dμ \end{matrix} ]|^{q})^{1/q} ,

where p > 1, \frac{1}{p} + \frac{1}{q} = 1.

The proof follows by the inequality for 2-inner products incorporated in (5.2).

Proposition 2. Let f, g_{1}, ..., g_{n}, h ∈ L_{ρ}^{2} (Ω) , where ρ : Ω → [0, ∞) is a measurable function on Ω, then we have the inequality

∑_{i=1}^{n} | \begin{matrix}∫_{Ω} ρfg_{i}dμ & ∫_{Ω} ρfhdμ \\ ∫_{Ω} ρg_{i}hdμ & ∫_{Ω} ρh^{2}dμ \end{matrix} |^{2}

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

× max_{1≤i≤n} \{∑_{j=1}^{n} | det [ \begin{matrix}∫_{Ω} ρg_{j}g_{i}dμ & ∫_{Ω} ρg_{j}hdμ \\ ∫_{Ω} ρg_{i}hdμ & ∫_{Ω} ρh^{2}dμ \end{matrix} ] |\} .

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] E. BOMBIERI, A note on the large sieve, Acta Arith., 18(1971), 401-404.

[2] 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

[3] 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.

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

[5] J.E. PEČARIĆ, On some classical inequalities in unitary spaces, Mat. Bilten (Scopje), 16(1992), 63-72.

Department of Mathematics, College of Education, Gyeongsang National University, Chinju 660-701, Korea.

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

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, Bishop’s University Lennoxville, Quebec J1M 1Z7, Canada

E-mail address: plin@ubishops.ca

Department of Matematics, Dongeui University, Pusan 614-714, Korea.

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

Department of Applied Mathematics, Changwon National University, Changwon 641-773, Korea

E-mail address: yhkim@sarim.changwon.ac.kr