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Inequalities, which we can prove by the Karamata's inequal;ity

  • Let $(a_1,a_2,...,a_n)\succ(b_1,b_2,...b_2)$ and $f$ is a convex function. Prove that: $$f(a_1)+f(a_2)+...+f(a_n)\geq f(b_1)+f(b_1)+...+f(b_n).$$

  • Let $(a_1,a_2,...,a_n)\succ(b_1,b_2,...b_2)$ and $f$ is a concave function. Prove that: $$f(a_1)+f(a_2)+...+f(a_n)\leq f(b_1)+f(b_1)+...+f(b_n).$$

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