A generalized form of Grüss type inequality and other integral inequalities
© Minculete and Ciurdariu; licensee Springer. 2014
Received: 6 November 2013
Accepted: 28 February 2014
Published: 20 March 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:
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 .
called the h-integral arithmetic mean for a Riemann-integrable function f.
where k is a real constant.
the variance of f.
where k is a constant.
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].
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 .
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.
where is a Riemann-integrable function with .
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.
where a and b are real numbers.
Therefore, we deduce relationship (1). □
Proof If we take and , in relation (7), then we obtain equalities (8) and (9). □
where a, b, c, and d are real numbers.
We can prove an inequality for integrable functions similar to the inequality of Cauchy-Schwarz for random variables given by the following.
because . □
Proof Using Theorem 6 and Lemma 2, we deduce the statement. □
for every .
Taking into account that , because and dividing by , we obtain the inequality of the statement. □
Lemma 9 represents generalized integral variant of inequality of Cauchy. □
Now we compare inequalities (21) and (12) to see which is stronger.
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), .
imply for all ,
, i.e. if , , then .
, 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 .
Now if we consider this inequality and previous theorem we deduce as in Lemma 2 the following result. □
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).
4 Properties of h-variance
- 1.If , then we have the following inequality:(27)
- 2.If , then we have the following inequality:(28)
- 3.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.
This length function satisfies the required properties of a norm and is called the Euclidean norm on .
This distance function is called the Euclidean metric. This formula expresses a special case of the Pythagorean theorem.
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)
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.
- 1.Taking into account the integral arithmetic mean and h-integral arithmetic mean for a Riemann-integrable function we can rewrite the following inequalities.
- (a)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
- (c)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:
- 2.(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
- (b)If and if are two integrable functions on with , and , then we have
- 3.(i) Under the previous conditions we have
- (ii)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 .
- (i)If , , is a convex and continuous function and are two integrable functions on , and, in addition, if is integrable on , then
- (ii)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.
We would like to thank the anonymous reviewer for providing valuable comments to improve the manuscript.
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