scholarly journals Cauchy Mean Theorem

2014 ◽  
Vol 22 (2) ◽  
pp. 157-166
Author(s):  
Adam Grabowski
Keyword(s):  
Simple Proof ◽  
Geometric Mean ◽  
Jensen’S Inequality ◽  
Arithmetic Mean ◽  
Real Numbers ◽  
Jensen's Inequality ◽  
Mizar Mathematical Library ◽  
Proof By Induction

Summary The purpose of this paper was to prove formally, using the Mizar language, Arithmetic Mean/Geometric Mean theorem known maybe better under the name of AM-GM inequality or Cauchy mean theorem. It states that the arithmetic mean of a list of a non-negative real numbers is greater than or equal to the geometric mean of the same list. The formalization was tempting for at least two reasons: one of them, perhaps the strongest, was that the proof of this theorem seemed to be relatively easy to formalize (e.g. the weaker variant of this was proven in [13]). Also Jensen’s inequality is already present in the Mizar Mathematical Library. We were impressed by the beauty and elegance of the simple proof by induction and so we decided to follow this specific way. The proof follows similar lines as that written in Isabelle [18]; the comparison of both could be really interesting as it seems that in both systems the number of lines needed to prove this are really close. This theorem is item #38 from the “Formalizing 100 Theorems” list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/.

scholarly journals On limit problems associated with some inequalities

1973 ◽  
Vol 14 (2) ◽  
pp. 123-127
Author(s):  
P. H. Diananda
Keyword(s):  
Geometric Mean ◽  
Arithmetic Mean ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Real Numbers ◽  
Limit Problems ◽  
Generalized Mean ◽  
Image Position ◽  
Necessary And Sufficient

Let {an} be a sequence of non-negative real numbers. Suppose thatThen M1,n is the arithmetic mean, MO,n the geometric mean, and Mr,n the generalized mean of order r, of a1, a2, …, an. By a result of Everitt [1] and McLaughlin and Metcalf [5], {n(Mr,n–Ms,n)}, where r ≧ l ≧ s, is a monotonic increasing sequence. It follows that this sequence tends to a finite or an infinite limit as n → ∞. Everitt [2, 3] found a necessary and sufficient condition for the finiteness of this limit in the cases r, s = 1, 0 and r ≧ 1 > s > 0. His results are included in the following theorem.

scholarly journals THE THERMODYNAMIC LIMIT FOR FINITE DIMENSIONAL CLASSICAL AND QUANTUM DISORDERED SYSTEMS

2004 ◽  
Vol 16 (05) ◽  
pp. 629-637 ◽  
Author(s):  
PIERLUIGI CONTUCCI ◽  
CRISTIAN GIARDINÀ ◽  
JOSEPH PULÉ
Keyword(s):  
Thermodynamic Limit ◽  
Simple Proof ◽  
Spin Glasses ◽  
Disordered Systems ◽  
Jensen’S Inequality ◽  
Specific Pressure ◽  
Jensen's Inequality ◽  
Dimensional Lattice ◽  
Finite Dimensional ◽  
Quantum Disordered Systems

We provide a very simple proof for the existence of the thermodynamic limit for the quenched specific pressure for classical and quantum disordered systems on a d-dimensional lattice, including spin glasses. We develop a method which relies simply on Jensen's inequality and which works for any disorder distribution with the only condition (stability) that the quenched specific pressure is bounded.

scholarly journals Hardy Martingales and Jensen's inequality

1997 ◽  
Vol 55 (2) ◽  
pp. 185-195
Author(s):  
Nakhlé H. Asmar ◽  
Stephen J. Montgomery-Smith
Keyword(s):  
Abelian Group ◽  
Simple Proof ◽  
Compact Abelian Group ◽  
Analytic Functions ◽  
Arbitrary Order ◽  
Jensen’S Inequality ◽  
Lexicographic Order ◽  
Dual Group ◽  
Dimensional Torus ◽  
Jensen's Inequality

Hardy martingales were introduced by Garling and used to study analytic functions on the N-dimensional torus 𝕋N, where analyticity is defined using a lexicographic order on the dual group ℤN. We show how, by using basic properties of orders on ℤN, we can apply Garling's method in the study of analytic functions on an arbitrary compact Abelian group with an arbitrary order on its dual group. We illustrate our approach by giving a new and simple proof of a famous generalised Jensen's Inequality due to Helson and Lowdenslager[5].

scholarly journals Jensen's inequality for distributions possessing higher moments, with application to sharp bounds for Laplace-Stieltjes transforms

1998 ◽  
Vol 40 (1) ◽  
pp. 80-85 ◽  
Author(s):  
B. Guljaš ◽  
C. E. M. Pearce ◽  
J. Pečarić
Keyword(s):  
Probability Distributions ◽  
Jensen’S Inequality ◽  
Distribution Functions ◽  
Higher Moments ◽  
Real Numbers ◽  
Jensen's Inequality ◽  
Sharp Bounds ◽  
Stieltjes Transforms

AbstractA new version of Jensen's inequality is established for probability distributions on the non-negative real numbers which are characterized by moments higher than the first. We deduce some new sharp bounds for Laplace-Stieltjes transforms of such distribution functions.

On Statistics of Log-Ratio of Arithmetic Mean to Geometric Mean for Nakagami-m Fading Power

2012 ◽  
Vol E95-B (2) ◽  
pp. 647-650
Author(s):  
Ning WANG ◽  
Julian CHENG ◽  
Chintha TELLAMBURA
Keyword(s):  
Geometric Mean ◽  
Arithmetic Mean ◽  
Log Ratio
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