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Showing content from https://dlmf.nist.gov/4.18 below:

§4.18 Inequalities ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions

Keywords:

inequalities, trigonometric functions

Notes:

For (4.18.1) see Copson (1935, p. 136). (4.18.3) follows by the same method and (4.18.2) is a consequence. (4.18.5) to (4.18.9) are straightforward and (4.18.10) is obtained from the Maclaurin expansions of cos ⁡ z and sin ⁡ z . For the second inequality in (4.18.4), it is sufficient to show f ⁡ ( x ) ≡ 4 ⁢ x ⁢ ( 1 − x ) − sin ⁡ ( π ⁢ x ) ≥ 0 for 0 ≤ x ≤ 1 / 2 . The function f ⁡ ( x ) is zero at x = 0 and x = 1 / 2 and it has no zeros in ( 0 , 1 / 2 ) , because f′ ⁡ ( x ) = 4 ⁢ ( 1 − 2 ⁢ x ) − π ⁢ cos ⁡ ( π ⁢ x ) can have only one zero in ( 0 , 1 / 2 ) where y = 4 ⁢ ( 1 − 2 ⁢ x ) intersects y = π ⁢ cos ⁡ ( π ⁢ x ) . The first inequality is proved similarly.

Referenced by:

Erratum (V1.0.19) for Notation

Permalink:

http://dlmf.nist.gov/4.18

See also:

Annotations for Ch.4

For more inequalities see Mitrinović (1964, pp. 101–111), Mitrinović (1970, pp. 235–265), and Bullen (1998, pp. 250–254).

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