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Exploring Various Means: Arithmetic, Geometric, Harmonic, and Beyond

Delve into the world of mean functions including symmetrical properties, fixed points, homogeneity, monotony, continuity, and their relationships. Learn about the geometric mean, arithmetic mean, harmonic mean, power mean, Lehmer mean, and more. Discover connections, properties, and methods to construct different types of means. Explore mixed means, like arithmetic-geometric and geometric-harmonic, and their convergence sequences.

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Exploring Various Means: Arithmetic, Geometric, Harmonic, and Beyond

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  1. Mean M(x1,x2,…xN) (no weighted): 1. M is a function: 2. Simmetry: for all permutation 3. Fixed point: 4. Homogenity: 5. Monotony: 6. Continuity:

  2. Geometric Mean Arithmetic Mean So for the point 2, to make a mean, we have to use some operation with commutativeproperty; the simplest are: Moreover for the fixed pointproperty (3), if all number are iguals the mean is the same value, so we have to add something:

  3. Traditional Block ………… ………… Harmonic Mean First method to build anothers means: Media For example, if we choose f(x)=1/x and arithmetic mean as traditional block, we discover the Harmonic Mean:

  4. Power Mean if we choose f(x)=xs and arithmetic mean as traditional block, we find the Power Mean:

  5. Second method to build anothers means: for two numbers, but is possible to generalize

  6. Third form, to build anothers means: Lehmer mean

  7. First group Harmonic Mean Power Mean Power Mean of orden 2 (Root Mean Squares) Quadratic Mean

  8. Strange connection geometric-quadratic Power mean orden 1/2 Looks Heronian Second group Identric Mean Logarithmic Mean Stolarsky Mean

  9. All means with negative p have the the coniugate mean respect the arithmentic mean, with positive p. Clic there is a demostration Third group Contraharmonic Mean

  10. Equivalent Form of Lehmer Mean q=-p We can write the Lehmer Mean in this form too: Like weighted mean….

  11. Heronian Mean Heinz Mean 1 0 Logarithmic Mean Average of Means like Heronian and Heinz. Another type

  12. Mixed Means Arithmetic-Geometric Mean Arithmetic-Harmonic Mean Geometric-Harmonic Mean These two sequences converge to the same number.

  13. For the Arithmetic-Geometric Mean there is a closed for expression: Where K(x) is the complete elliptic integral of first kind.

  14. Demostration of an expression:

  15. An observation: Another observation:

  16. Moreover: • Elementary Symmetric Mean • And…. • 1. Median • 2. Mode

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