A new variant of Mercer’s inequality [A.McD. Mercer, A variant of Jensen’s inequality, J. Inequal. Pure Appl. Math. 4(4) (2003) Article 73] of Jessen’s type is given. Moreover, versions of Chebyshev’s inequality and Hardy–Littlewood– Pólya inequality for some abstract nonnegative linear functionals are obtained
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