Chebyshev and Grüss Type Inequalities Involving Two Linear Functionals and Applications
نویسندگان
چکیده
In the present paper we prove the Chebyshev inequality involving two isotonic linear functionals. Namely, if A and B are isotonic linear functionals, then A(p f g)B(q)+A(p)B(q f g) A(p f )B(qg) + A(pg)B(q f ) , where p,q are non-negative weights and f ,g are similarly ordered functions such that the above-mentioned terms are well-defined. If functionals are equal, i.e. A = B and if p = q , then the above inequality becomes the Chebyshev inequality involving one isotonic linear functional: A(p)A(p f g) A(p f )A(pg) in which we recognize a generalization of the well-known classical integral and discrete Chebyshev inequalites as special cases. We derive various properties of functionals related to the difference of the right-hand and the left-hand sides of the above-mentioned inequalities. The most remarkable results are the Grüss type inequalities for two functionals. Inequalities involving some fractional integral operators are also given. Mathematics subject classification (2010): 26D20, 26A33.
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