On Generalizations of Grüss Inequality in Inner Product Spaces and Applications

Let be an orthonormal system of vectors in unitary space It is well known that for all , the following inequality holds [1, page 333]:

(11)

where is defined by

(12)

The equality in (1.1) holds if and only if is linearly dependent. Applying (1.1) on for by choosing , and , we immediately obtain the Pre-Grüss inequality as follows:

(13)

where and is the Chebyshev functional

(14)

Let , , , and be real numbers such that and for all Combining (1.3) with the following well-known inequality:

(15)

we obtain a premature Grüss inequality,

(16)

and the original Grüss inequality (see [2]),

(17)

Note that the discrete version of inequality (1.7) has the following form:

(18)

where are real numbers so that , for all , and , .

In [3, 4], Dragomir starting from inequality (1.2) proved the following Grüss type inequality in real or complex inner product spaces.

Let be a unit vector in If , , , and are complex numbers and are vectors in that satisfy the conditions,

(19)

then the following inequality holds:

(110)

The constant that appears at the right side of the inequality is optimal in the sense that it cannot be replaced by a smaller one.

Some generalizations and refinements of inequality (1.10) can be found in [1, 4–10] In this paper, we achieve an improvement of inequality (1.1) in the sense of subtracting a nonnegative quantity from the right part of (1.1). In this way, every result that stems from (1.1), such as the inequality (1.10) and its generalizations and refinements, can also be improved. Furthermore, we apply our improvements of inequalities (1.3), (1.6), and (1.7) to achieve refinements of some well-known trapezoid-Grüss and Ostrowki-Grüss type inequalities. Some refinements of inequality (1.8) as well as an additive reverse of the Schwarz inequality are also given.

Let be an inner product space over the real or complex number field . For our purpose, we need the following three lemmas.

Lemma 2.1 (see [2, page 599]).

For all , , one has

(21)

where is the Gramian of the vectors . The equality in (2.1) holds if and only if , or is linearly dependent, or is linearly dependent.

Lemma 2.2.

Let be an orthonormal system of vectors in , then, for any vectors , one has that

(22)

is linearly dependent, if and only if is linearly dependent.

Proof.

Let be linearly dependent, then clearly is also linearly dependent. Conversely, let be linearly dependent, then since is linearly independent, there exist and not all zero, such that

(23)

which for some can be rewritten as follows:

(24)

Hence, we get

(25)

Further, for , we have

(26)

Hence from (2.5), we get , Consequently, from (2.4) we obtain

(27)

and because at least one of is nonzero, it is derived that

(28)

is linearly dependent.

Lemma 2.3.

Let be an orthonormal system of vectors in and be any vectors in , then,

(29)

where

(210)

Proof.

Define the linear mapping

(211)

given by It is easy to verify that

(212)

which completes the proof.

Theorem 2.4.

Let be an orthonormal system of vectors in and let be linear independent, such that , then for all the following inequality holds:

(213)

where

(214)

and is given as in (1.2). The equality in (2.13) holds if and only if , or is linearly dependent.

Proof.

If we apply inequality (2.1) for by choosing , , , and and taking into consideration that for all , we have

(215)

then we get

(216)

Now, from the condition , the following is derived:

(217)

(218)

From the condition " is linear independent", it follows that . Now, if we set (2.17) and (2.18) in (2.16), we readily get the conclusion. Finally, according to Lemma 2.1 we have that the equality in (2.16) holds if and only if

(219)

or is linearly dependent, or is linearly dependent. Now, since , (2.19) can be rewritten as follows:

(220)

Moreover, according to Lemma 2.3, we have that

(221)

Combining (2.20) with (2.21), we conclude that (2.19) is equivalent to

(222)

From the conditions of this theorem, we clearly have that is linearly independent and since we obtain that is also linear independent. Finally, according to Lemma 2.2, we have that is linearly dependent if and only if is linearly dependent, which completes the proof.

Corollary 2.5.

Assuming that the conditions of Theorem 2.4 hold and then one has

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The equality in (2.23) holds if and only if is linearly dependent.

Proof.

From the condition , we obtain that . Hence, (2.23) can be rewritten as follows:

(224)

Furthermore, from (2.13) and (2.14), we have

(225)

Consequently, we can apply the elementary inequality

(226)

for

(227)

to obtain

(228)

Combining (2.24) with (2.28) we get the desired result.

Letting , we easily derive that and , hence the equality in (2.23) holds. Conversely, it is clear that the equality in (2.23) holds if and only if the equalities in (2.24) and (2.28) hold. That is

(229)

(230)

Now, from (2.29) it follows that for some ,

(231)

(232)

Putting (2.31) and (2.32) in (2.30), we get after some algebraic calculations,

(233)

and since , we conclude that .

Therefore, there is in such that,

(234)

Putting (2.34) in (2.32), dividing the result by and finally subtracting this result from (2.31), we conclude that is linearly dependent.

Remark 2.6.

The main result in [11] is an identity, which by setting can be equivalently written in the following form:

(235)

It is hard to find a positive lower bound of the term . Therefore, it is difficult to derive a refinement of inequality (1.1) like (2.13) for from the above identity.

The main result of this section is a refinement of Dragomir's inequality (1.10).

Theorem 3.1.

Let be a unit vector in If , , , and are real or complex numbers with , , and are vectors in satisfying the following conditions:

(31)

then for all nonproportional vectors such that the following inequality holds

(32)

where is as given in (2.14).

Proof.

We distinguish two cases.

Case 1.

Let either or . Without loss of generality, let us assume that Then from the condition , we have that . Thus, . So Consequently, (3.2) reduces to inequality (1.10).

Case 2.

Let , then we can apply inequality (2.23) by to obtain

(33)

Furthermore, from the conditions (3.1), we have the following (see [3]):

(34)

Finally, combining (3.3) with (3.4) leads to the asserted inequality (3.2).

Now, working similar as above we can show the following result, which we will use in the next section, to obtain refinements of some known integral inequalities.

Theorem 3.2.

Let be a unit vector in If are real or complex numbers with , and are vectors in satisfying the following condition:

(35)

then for all nonproportional vectors such that the following inequality holds:

(36)

Remark 3.3.

Based on inequality (2.23), one can improve, in a way similar to Theorem 3.1, all results related to Grüss inequality as in [1, 4–8] For example, in [1, pages 333-334], [7, page 2751], and [4, page 90], Dragomir obtained the following inequality by using Arcel's inequality:

(37)

which is used to derive the following refinement of (1.10)

(38)

Now, if we combine the inequalities (3.3), (3.7), and (3.4), we get the following improvement of (3.8):

(39)

First we will use the results of Section 3 to obtain improvements of inequalities (1.3), (1.6), and (1.7).

Theorem 4.1.

Let be bounded on and let be not proportional and so that Then, provided that are nonconstant functions, the following inequalities hold:

(41)

(42)

(43)

where

(44)

and is the Chebyshev functional defined by (1.4).

Proof.

Applying inequality (2.23) on for as well as the inequalities (3.6) and (3.2) by choosing , , , , , we easily get the required inequalities.

Note that all known trapezoid-Grüss, midpoint-Grüss, and Ostrowski-Grüss type inequalities, which are proved by applying the inequalities (1.3), (1.6), and (1.7), can be improved by using the inequalities of Theorem 4.1. For example, in [12, page 39] we can see the following trapezoid-Grüss type inequality:

(45)

where is a twice differentiable mapping on such that , for all

In the following, we apply inequality (4.2) to derive a refinement of inequality (4.5).

Proposition 4.2.

Let , , and be as above, then

(46)

Proof.

Let be functions defined by , , , and , then it is easy to verify that

(47)

Finally, since equalities , hold, we can apply inequality (4.2), using the above relations to get the required inequality.

In [13, page 167] we can see the following Ostrowski-Grüss type inequality for all :

(48)

where is a differentiable function on such that , for all , and is defined by

(49)

We propose improvement of this result as follows.

Proposition 4.3.

Let be as above, then,

(410)

Proof.

If we apply inequality (4.2) by choosing

(411)

(412)

(413)

and take into account that, after some calculations, we obtain

(414)

(415)

(416)

(417)

(418)

(419)

(420)

and we easily derive the asserted inequality.

According to [13, page 168], the following inequality holds:

(421)

If we apply inequality (4.10) for , we get the following refinement of the previous inequality.

Corollary 4.4.

Let be as in Proposition 4.3, then,

(422)

It can now be observed that (4.19) can be written as follows:

(423)

or equivalently

(424)

which, by using inequality (2.26), leads to the following result.

Corollary 4.5.

Let be as above, then,

(425)

Now, we will use again inequality (4.2) to give another improvement of inequality (4.8).

Proposition 4.6.

Let be as in Proposition 4.3, then for all ,

(426)

where

(427)

Proof.

Let us choose , , and as given in (4.11) and (4.12). We distinguish the following two cases.

Case 1 ( ).

Then, by choosing

(428)

it can be easily verified that

(429)

So, we can apply inequality (4.2), by using (4.12), (4.14), (4.15), and (4.16) as well as

(430)

to obtain the desired inequality (4.22).

Case 2 ( ).

Then, by choosing

(431)

we easily derive the following:

(432)

Finally, application of inequality (4.2) using the above relations leads to the claimed result.

In this section, some refinements of the discrete version of Grüss inequality (1.8), as well as, an additive reverse of the Schwarz inequality are provided.

Theorem 5.1.

Let , be real numbers so that , for where and . Let be real numbers such that the vectors are not proportional and

(51)

Then,

(52)

Proof.

Define the vectors , , , , in the Euclidian inner product space . Then, we have

(53)

and . Hence, we can apply inequality (3.2) of Theorem 3.1 for the vectors , , , , and , as given above, to complete the proof.

Corollary 5.2.

Let be as is in Theorem 5.1, then for all ,

(54)

where

(55)

Proof.

If or or , then inequality (5.4) reduces to (1.8). If , , and , then by choosing the vectors , such that

(56)

it follows that these vectors satisfy the conditions of Theorem 5.1. Hence, we can apply Theorem 5.1 to obtain the desired result.

Now, if we apply Corollary 5.2 by choosing and , we immediately obtain the following result.

Corollary 5.3.

Let be real numbers so that , for all , where and . Then for all , one has the following inequality:

(57)

Corollary 5.4.

Let and , then

(58)

Proof.

If we apply inequality (5.7) by choosing , , , and , , we directly get the desired inequality.

Now, we will use inequality (5.7) to derive an additive reverse of the Schwarz inequality.

Applying times the inequality (5.7), namely for all pairs of integers with , adding the resulting inequalities, using the Lagrange identity, and finally dividing the resulting inequality by , then,

(59)

Now, if we solve inequality (5.9) with respect to and then replace by , we obtain the following reverse of the Schwarz inequality.

Proposition 5.5.

Let , , be real numbers so that , for all where and . then,

(510)