More results on a functional generalization of the Cauchy-Schwarz inequality
© Masjed-Jamei and Hussain; licensee Springer 2012
Received: 17 July 2012
Accepted: 1 October 2012
Published: 17 October 2012
By using a specific functional property, some more results on a functional generalization of the Cauchy-Schwarz inequality, such as an extension of the pre-Grüss inequality and a refinement of the Cauchy-Schwarz inequality via the generalized Wagner inequality, are given for both discrete and continuous cases.
play an important role in different branches of modern mathematics such as Hilbert space theory, classical real and complex analysis, numerical analysis, probability and statistics, qualitative theory of differential equations and their applications. To date, a large number of generalizations and refinements of the inequalities (1) and (2) have been investigated in the literature, e.g., [2–7].
Recently in , we have presented a functional generalization of the Cauchy-Bunyakovsky-Schwarz inequality for both discrete and continuous cases as follows.
Thus, inequalities (3) and (4) are respectively generalizations of the discrete and continuous Cauchy-Bunyakovsky-Schwarz inequalities for , and in (3) and and in (4).
Also, the equality holds if in (3) where r is constant and in (4) where R is constant.
The aim of this paper is to extend the results of the above-mentioned theorem by using a specific functional property.
2 On the generalized Cauchy-Schwarz inequality
By using the aforesaid functional property, many classical inequalities such as Chebyshev, Stefensen and Aczel inequalities have been generalized in . Here we wish to apply the property (8) to extend the results of Theorem 1. For this purpose, we first express the following lemma.
for any , and then notes that the discriminant Δ of P must be negative. □
in which and are two sequences of real numbers, and and are two arbitrary functions of m and k variables. Moreover, the equality holds if in (10) and for the constant .
The equality holds in (11) if and for the constant .
In this section we study two special cases of inequalities (10) and (11) which are remarkable.
Example 3 (An extension of the pre-Grüss inequality)
Before deriving the main result, let us recall some initial comments.
For the above inequality gives the same result as the pre-Grüss inequality (12) while for (or ) the Cauchy-Schwarz inequality is obtained. Also, for and , the Wagner inequality  is derived. An interesting case of the inequality (13) is when (i.e., ), which reveals its importance in numerical integration formulas.
where . In particular, replacing in (16) gives the same as the pre-Grüss inequality.
Example 4 (A refinement of the Cauchy-Schwarz inequality via the generalized Wagner inequality)
Clearly, for and in (18), the inequality (17) is derived. Moreover, (18) is also a special case of the inequality (10).
in (19) then it is directly concluded that in (21) and conversely. Therefore, the solution of (20) would be either or . This means that the refinement (19) is valid for any or provided that the condition (22) holds.
Similarly, the latter result holds for the continuous case and we have
The work of the first author is supported by the grant from ‘Iran National Science Foundation’ No. 91002576. The second author gratefully acknowledges the support from the Deanship of Scientific Research (DSR) at King Abdulaziz University during this research.
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