engleski

Bounds for the Normalized Jensen-Mercer Functional

We introduce the normalized Jensen-Mercer functional M_{; ; ; n}; ; ; (f, x, p)=f(a)+f(b)-∑ _{; ; ; i=1}; ; ; ⁿ p_{; ; ; i}; ; ; f(x_{; ; ; i}; ; ; )-f(a+b-∑ _{; ; ; i=1}; ; ; ⁿ p_{; ; ; i}; ; ; x_{; ; ; i}; ; ; ) and establish the inequalities of type MM_{; ; ; n}; ; ; (f, x, q)≥ M_{; ; ; n}; ; ; (f, x, p)≥ mM_{; ; ; n}; ; ; (f, x, q), where f is a convex function, x=(x₁ , … .x_{; ; ; n}; ; ; ) and m and M are real numbers satisfying certain conditions. We prove them for the cases when p and q are nonnegative n-tuples and when p and q satisfy the conditions as for the Jensen-Steffensen inequality. We also give their integral versions in both cases.

Jensen-Mercer functional; Jensen-Mercer inequality; convex functions; bounds

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