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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A generalization of two inequalities involving means

Authors: Scott Lawrence and Daniel Segalman
Journal: Proc. Amer. Math. Soc. 35 (1972), 96-100
MSC: Primary 26A86
MathSciNet review: 0304586
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Abstract: Fan has proven an inequality relating the arithmetic and geometric means of $ ({x_1}, \cdots ,{x_n})$ and $ (1 - {x_1}, \cdots ,1 - {x_n})$, where $ 0 < {x_i} \leqq \tfrac{1}{2},i = 1, \cdots ,n$. Levinson has generalized Fan's inequality; his result involves functions with positive third derivatives on (0, 1). In this paper, the above condition that requires $ 0 < {x_i} \leqq \tfrac{1}{2}$ has been replaced by a condition which only weights the $ {x_i}$ to the left side of (0, 1) in pairs, and Levinson's differentiability requirement has been replaced by the analogous condition on third differences.

References [Enhancements On Off] (What's this?)

  • [1] N. Levinson, Generalization of an inequality of Ky Fan, J. Math. Anal. Appl. 8 (1964), 133–134. MR 0156928
  • [2] Tiberiu Popoviciu, Remarques sur une formule de la moyenne des différences divisées généralisées, Mathematica (Cluj) 2 (25) (1960), 323–324 (French). MR 0131705

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