EXACT UPPER AND LOWER BOUNDS ON THE DIFFERENCE BETWEEN THE ARITHMETIC AND GEOMETRIC MEANS

@article{Pinelis2015EXACTUA,
  title={EXACT UPPER AND LOWER BOUNDS ON THE DIFFERENCE BETWEEN THE ARITHMETIC AND GEOMETRIC MEANS},
  author={Iosif Pinelis},
  journal={Bulletin of the Australian Mathematical Society},
  year={2015},
  volume={92},
  pages={149 - 158}
}
  • I. Pinelis
  • Published 1 March 2015
  • Mathematics
  • Bulletin of the Australian Mathematical Society
Exact upper and lower bounds on the difference between the arithmetic and geometric means are obtained. The inequalities providing these bounds may be viewed, respectively, as a reverse Jensen inequality and an improvement of the direct Jensen inequality, in the case when the convex function is the exponential. 
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References

SHOWING 1-9 OF 9 REFERENCES
Upper bounds for the variances of certain random variables
Jacobson (1969) proved that the variance of a unimodal distribution on the unit interval is bounded above by 1/9. This paper generalizes Jacobson's result by deriving sharp upper bounds for the
An asymptotically Gaussian bound on the Rademacher tails
An explicit upper bound on the tail probabilities for the normalized Rademacher sums is given. This bound, which is best possible in a certain sense, is asymptotically equivalent to the corresponding
SOME REVERSES OF THE JENSEN INEQUALITY WITH APPLICATIONS
  • S. Dragomir
  • Mathematics
    Bulletin of the Australian Mathematical Society
  • 2013
Abstract Two new reverses of the celebrated Jensen’s inequality for convex functions in the general setting of the Lebesgue integral, with applications to means, Hölder’s inequality and
A Better Bound on the Variance
TLDR
It is shown as only a commutative special case of a far more general result; and then it is shown for a new result analogous to it, in which the place of the square function is taken by the inverse function.
Tchebycheff systems and extremal problems for generalized moments: a brief survey
A brief presentation of basics of the theory of Tchebycheff and Markov systems of functions and its applications to extremal problems for integrals of such functions is given. The results, as well as
Problem 11800
  • Amer. Math. Monthly ,
  • 2014
The Markov Moment Problem and Extremal Problems