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A generalized form of Grüss type inequality and other integral inequalities
Journal of Inequalities and Applications volume 2014, Article number: 119 (2014)
The aim of this presentation is to show several integral inequalities. Among these inequalities we have the inequality , where denotes the h-variance of f, which is a bounded function defined on with , and , are two constants. This inequality is important because it proves a generalized form of the Grüss type inequality. This improvement is given by the inequality
Using the integral arithmetic mean and h-integral arithmetic mean for a Riemann-integrable function f we can also rewrite several integral inequalities. In addition, we will give a generalization of inequality of Grüss for normalized isotonic linear functionals.
In 1935, Grüss  proved the following integral inequality which gives an approximation for the integral of a product of two functions in terms of the product of integrals of the two functions:
Let f and g be two bounded functions defined on with and , where , , , are four constants. Then we have
and the inequality is sharp, in the sense that the constant cannot be replaced by a smaller one.
It is well known that an important resource for studying inequalities is [2–4]. In , Peng and Miao established a form of inequality of Grüss type for functions whose first and second derivatives are absolutely continuous and the third derivative is bound. Also, in  Dragomir presented several integral inequalities of Grüss type, and in  he showed some Grüss type inequalities in inner product spaces and applications for the integral. Another improvement of the Grüss inequality was obtained by Mercer in . Moreover, in , a Grüss type inequality was used in order to obtain some sharp Ostrowski-Grüss type inequalities by Liu. Kechriniotis and Delibasis showed in  several refinements of inequality of Grüss in inner product spaces using a Kurepa’s results for Gramians. New generalizations of the inequality of Gruss were presented in  using Riemann-Liouville fractional integrals. Cerone and Dragomir studied in  some refinements of Grüss’ inequality. Florea and Niculescu in  treated the problem of estimating the deviation of the values of a function from its mean value.
The estimation of the deviation of a function from its mean value is characterized in terms of random variables.
We denote by the space of Riemann-integrable functions on the interval , and by the space of real-valued continuous functions on the interval .
The integral arithmetic mean for a Riemann-integrable function is the number
If and , then a generalization for the integral arithmetic mean is the number
called the h-integral arithmetic mean for a Riemann-integrable function f.
We find the following property of the h-integral arithmetic mean for a Riemann-integrable function f:
where k is a real constant.
If the function f is a Riemann-integrable function, we denote by
the variance of f.
The expression for the variance can be expanded thus:
In the same way we defined the h-variance of a Riemann-integrable function f by
The expression for the h-variance can be expanded thus:
It is easy to see another form of the h-variance, given by the following:
We note the following property of the h-variance of an integrable function f:
where k is a constant.
In , Aldaz showed a refinement of the AM-GM inequality and used in the proof that
is a measure of the dispersion of about its mean value, which is, in fact, comparable to the variance,
The covariance is a measure of how much two Riemann-integrable functions change together and is defined as
and it is equivalent to the form
In fact the covariance is the Chebyshev functional attached to functions f and g. In  is written as . The properties of the Chebyshev functional have been studied by Elezović, Marangunić and Pec̆arić in their paper, . For other generalizations of the Grüss’ inequality, see [17, 18].
The h-covariance is a measure of how much two integrable functions change together and is defined as
and it is equivalent to the form
In , Pečarić used the generalization of the Chebyshev functional notion attached of functions f and g to the Chebyshev h-functional attached of functions f and g defined by . Here, Pečarić showed some generalizations of the inequality of Grüss by the Chebyshev h-functional. It is easy to see that, in terms of the covariance, this can be written as .
In terms of covariance the inequality of Grüss becomes
and in terms of Chebyshev functional the inequality of Grüss becomes
Let be real numbers, assume for all and the average , and X a discrete random variable given by
In 1935, Popoviciu  proved the following inequality:
Bhatia and Davis showed in  that the following inequality holds:
The inequality of Bhatia and Davis represents an improvement of Popoviciu’s inequality, because .
If there is additional information about the mean values of the two functions in the inequality of Grüss then Zitikis argued in his paper, , that the inequality can be sharpened and he gave also a probabilistic interpretation for it.
2 Main results
We will present in this paper the integral version of inequalities (3) and (4). Therefore we have the following inequalities.
Lemma 1 Let f be a Riemann-integrable function defined on with , where and are two constants. Then we have
where is a Riemann-integrable function with .
Proof Since , we obtain the following inequality:
But it is easy to see that
Lemma 2 Let f be a Riemann-integrable function defined on with , where and are two constants and a Riemann-integrable function with . Then we have the following relations:
Proof It is easy to see that can be rewritten thus:
Next, using the idea of Dragomir , we will make several calculations, namely
Since f is a bounded function defined on with , and the function , it follows that . Therefore, it is easy to see that . This inequality proved the inequality of the statement. □
Next we show several relations between h-variance and h-covariance.
Lemma 3 If , and a Riemann-integrable function with , then we have the following equality:
where a and b are real numbers.
Proof From the expression of the variance, we have
Therefore, we deduce relationship (1). □
Proposition 4 If , then we have the following equality:
Proof If we take and , in relation (7), then we obtain equalities (8) and (9). □
Remark 1 From relations (8) and (9), we find the parallelogram law in terms of h-variance, namely
Lemma 5 If , then we have the following equality:
where a, b, c, and d are real numbers.
Proof From the expression of the covariance, we have
We can prove an inequality for integrable functions similar to the inequality of Cauchy-Schwarz for random variables given by the following.
Theorem 6 If , then we have the inequality
Proof If , then relation (12) is true. If , then we calculate the h-variance for the function:
because . □
Proposition 7 Let f and g be two Riemann-integrable functions defined on with and , where , , , are four constants, and we have a Riemann-integrable h function, with . Then we have
Proof Using Theorem 6 and Lemma 2, we deduce the statement. □
Remark 2 Inequality (13) is a refinement of inequality of Grüss, because
Remark 3 In the proof of Lemma 2 we found the equality in terms of random variables, given by
Remark 4 (a) Let f be a Riemann-integrable function defined on with , where , are two constants. Then we have
(b) Let f be a Riemann-integrable function defined on with , where , are two constants. Then we have
(c) Let f and g be two Riemann-integrable functions defined on with and where , , , are four constants. Then we have
Theorem 8 If , with and , then we have the inequality
Proof For the integrable functions f, g and q, with , we take the following integrable function:
We calculate the variance of the function w, thus:
and applying Lemma 3, we have
Using Lemma 2, we deduce the following inequality:
Returning to calculation of the function w, we have
Therefore, we deduce the equality
Since , it follows that
for every .
This implies that
Taking into account that , because and dividing by , we obtain the inequality of the statement. □
Remark 5 Let f, g and q be three integrable functions, with and . If we take the following function:
then we have the inequality
Lemma 9 Let f, g be two Riemann-integrable functions defined on . Then we have
Proof If , then relation (21) is true. Now we consider that .
for all , so, we have
which means that
for all , which implies
Lemma 9 represents generalized integral variant of inequality of Cauchy. □
Now we compare inequalities (21) and (12) to see which is stronger.
Theorem 10 Let f, g be two Riemann-integrable functions defined on . Then we have
Proof We calculate the difference of the terms which appear in inequality (12) and (21), thus:
But, applying the inequality between the arithmetic mean and the geometric mean and Lemma 9, we deduce the relation
From this, we obtain the inequality
Consequently the statement is true. □
3 A refinement of Grüss’ inequality for normalized isotonic linear functionals
There are many directions in which the inequality of Grüss  has been generalized.
Using the notion of normalized isotonic linear functional which appears in the paper , we will give a generalization of inequality of Grüss which is related to a theorem of Andrica and Badea (1988), .
Let E be a nonempty set, L a linear class of real-valued functions and having the properties:
imply for all ,
, i.e. if , , then .
An isotonic linear functional (in  is called positive definite functional) is a functional satisfying:
, for all and .
If and , then .
The mapping A is said to be normalized if .
Theorem 11 Let be such that and assume that there exist real numbers and so that .
Then for any normalized isotonic linear functional one has the inequality
Proof Taking into account the hypothesis we have by using (L1) and (L2) that and thus by (A2), we see that
Using also (A1) and (A3) we have,
and from this we obtain
From the inequality of Cauchy-Schwarz for a normalized isotonic linear functional, , a counterpart of the CBS inequality, we obtain for where and is any normalized isotonic linear functional
Now if we consider this inequality and previous theorem we deduce as in Lemma 2 the following result. □
Theorem 12 Let such that and , where , , , are given real numbers. Then for any normalized linear isotonic functional one has the inequality
Proof Because we can write
we can apply the CBS-inequality for a normalized linear isotonic functional, if , by Theorem 11 we have
Remark 6 If we take in the first theorem , (the Lebesque space of integrable functions on ) and g satisfying the condition on the interval , then we obtain inequality (17).
Theorem 13 Let be such that , and there exist real constants , so that . Then for any an isotonic linear functional so that one has the inequality
Proof We use the normalized isotonic linear functional A on L defined by
4 Properties of h-variance
If , then we have the following inequality:(27)
Proof From equality (8), we have
Applying the inequality of Cauchy-Schwarz for integrable functions given by
it follows that
which implies the inequality of the statement. □
If , then we have the following inequality:(28)
Proof From relation (9), we have
Applying the inequality of Cauchy-Schwarz for integrable functions, we obtain
which implies the inequality of the statement. □
The natural way to obtain these quantities is by introducing and using the standard inner product (also known as the dot product) on . The inner product of any two continuous f and g functions is defined by
The result is always a real number. Therefore the set can be organized as an Euclidean space.
The inner product of with itself is always non-negative. This product allows us to define the ‘length’ of an integrable function f through
This length function satisfies the required properties of a norm and is called the Euclidean norm on .
Finally, one can use the norm to define a metric (or distance function) on by
This distance function is called the Euclidean metric. This formula expresses a special case of the Pythagorean theorem.
From relation (27), we have
so we obtain the triangle inequality,
Remark 7 From , the analog of the arithmetic mean in the context of finite measure spaces is the integral arithmetic mean, which, for a μ-integrable function is the number
where represents in probability theory the conditional expectation of the random variable f.
A result which represents another estimate of Jensen’s inequality, in the sense of generalizations of the integral arithmetic mean and variance, can also be found in , page 53.
Lemma 14 (, 1.8) (Another estimate of Jensen’s inequality)
Let be a finite measure space and let be a μ-integrable function (or ). If f is a twice differentiable function given on an I interval that includes the image of g; and , then
where denotes the variance of g.
Izumio et al. showed in  some extensions of Grüss’ inequality and they also studied the integral-type Grüss’ inequalities for Lebesque space for a finite positive measure μ on Ω.
Taking into account the integral arithmetic mean and h-integral arithmetic mean for a Riemann-integrable function we can rewrite the following inequalities.
In the case when the integral form of the inequality from Theorem 2.4 (see ) was given by Theorem 2.5. Under the conditions of Theorem 2.5, the inequality becomes
In , Mortici gave a new refinement of Radon’s inequality. Using the integral form of the reverse of inequality from Theorem 2.5 (see ) we obtain, for , and , if are two integrable functions on with , a continuous function on , the inequality
From the integral form of the inequality from Consequence 1 (see ) we deduce if are two integrable functions, g a continuous function on , , and , the following inequality:
(a) Starting from Theorem 2.3, the inequalities (2.5) and (2.6), Theorem 2.7 and Theorem 2.9, given in , by using Theorem 12 (see ) and Theorem 8 (see ) we can obtain the next properties for certain Riemann-integrable functions. If and if are two integrable functions on with , and , then
If and if are two integrable functions on with , and , then we have
Using the integral form of the inequality (2.5) and (2.6) from Theorem 2.3 (see ), under the conditions of Theorem 5 (see ) we find that, for every , , and if are two continuous functions on with , , we have
If are two integrable functions on then
(i) Under the previous conditions we have
If are two integrable functions on with , , and , , then we have
The following two inequalities are rewritten here and have as a starting point an inequality from .
If , , is a convex and continuous function and are two integrable functions on , and, in addition, if is integrable on , then
If , , is a convex and continuous function, are two integrable functions on with , if is integrable on and are such that , , then
Starting from the inequality of Halliwell and Mercer (see ), we can establish the following result.
If , , are two continuous and strict positive functions on and , then the following inequality holds:
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We would like to thank the anonymous reviewer for providing valuable comments to improve the manuscript.
The authors declare that they have no competing interests.
The work presented here was carried out in collaboration between all authors. The study was initiated by NM. The author NM also played the role of the corresponding author. All authors contributed equally and significantly in writing this article. All authors have contributed to, seen and approved the manuscript.
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Minculete, N., Ciurdariu, L. A generalized form of Grüss type inequality and other integral inequalities. J Inequal Appl 2014, 119 (2014). https://doi.org/10.1186/1029-242X-2014-119
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