Algebra

Inequalities

330 𝑎₁+𝑎₂+⋯+𝑎_𝑛𝑏₁+𝑏₂+⋯+𝑏_𝑛 lies between the greatest and least of the fractions 𝑎₁𝑏₁, 𝑎₂𝑏₂, ⋯, 𝑎_𝑛𝑏_𝑛, the denominators being all of the same sign.

Proof

Let 𝑘 be the greatest of the fractions, and 𝑎_𝑟𝑏_𝑟 any other; then 𝑎_𝑟<𝑘𝑏_𝑟. Substitute in this way for each 𝑎. Similarly if 𝑘 be the least fraction. 331 𝑎+𝑏2>𝑎𝑏 332 𝑎₁+𝑎₂+⋯+𝑎_𝑛𝑛>^𝑛𝑎₁𝑎₂⋯𝑎_𝑛 or, Arithmetic mean > Geometric mean.

Proof

Substitute both for the greatest and least factors their Arithmetic mean. The product is thus increased in value. Repeat the process indefinitely. The limiting value of the G.M. is the A.M. of the quantities. 333 𝑎^𝑚+𝑏^𝑚2>𝑎+𝑏2^𝑚 excepting when 𝑚 is a positive proper fraction.

Proof

𝑎^𝑚+𝑏^𝑚=𝑎+𝑏2^𝑚{(1+𝑥)^𝑚+(1−𝑥)^𝑚} where 𝑥=𝑎−𝑏𝑎+𝑏. Employ Bin. Th. 334 𝑎^m₁+𝑎^m₂+⋯+𝑎^m_𝑛𝑛>𝑎₁+𝑎₂+⋯+𝑎_𝑛𝑛^𝑚 excepting when 𝑚 is a positive proper fraction.
Otherwise: The Arithmetic mean of the 𝑚^th powers is greater than the 𝑚^th power of the Arithmetic mean, excepting when m is a positive proper fraction.

Proof

Similar to (332). Substitute for the greatest and least on the left side, employing (333). 336 If 𝑥 and 𝑚 are positive, and 𝑥 and 𝑚𝑥 less than unity; then (1+𝑥)^−𝑚>1−𝑚𝑥 125, 240 337 If 𝑥, 𝑚, and 𝑛 are positive, and 𝑛 greater than 𝑚; then, by taking 𝑥 small enough, we can make 1+𝑛𝑥>(1+𝑥)^𝑚 For 𝑥 may be diminished until 1+𝑛𝑥 is >(1−𝑚𝑥)⁻¹, and this is >(1+𝑥)^𝑚, by last. 338 If 𝑥 be positive; log(1+𝑥)<𝑥 150 If 𝑥 be positive and >1, log(1+𝑥)>𝑥−𝑥²2155, 240 If 𝑥 be positive and <1, log11−𝑥>𝑥156 339 When 𝑛 becomes infinite in the two expressions 1⋅3⋅5⋅⋯⋅(2𝑛−1)2⋅4⋅6⋅⋯⋅2𝑛 and 3⋅5⋅7⋅⋯⋅(2𝑛+1)2⋅4⋅6⋅⋯⋅2𝑛 the first vanishes, the second becomes infinite, and their product lies between 12 and 1.
Shewn by adding 1 to each factor (see 73), and multiplying the result by the original fraction. 340 If 𝑚 be > 𝑛, and 𝑛 > 𝑎, 𝑚+𝑎𝑚−𝑎^𝑚 is < 𝑛+𝑎𝑛−𝑎^𝑛 341 If 𝑎, 𝑏 be positive quantities, 𝑎^𝑎𝑏^𝑏 is > 𝑎+𝑏2^𝑎+𝑏 Similarly 𝑎^𝑎𝑏^𝑏𝑐^𝑐 > 𝑎+𝑏+𝑐3^{𝑎+𝑏+𝑐} These and similar theorems may be proved by taking logarithms of each side, and employing the Expon. Th (158), ⋯

Sources and References

https://archive.org/details/synopsis-of-elementary-results-in-pure-and-applied-mathematics-pdfdrive