Jordanian inequality

{\ displaystyle {\ frac {2} {\ pi}} x \ leq \ sin (x) \ leq x {\ text {for}} x \ in \ left [0, {\ frac {\ pi} {2} } \ right]}

Unit circle with acute angle x. A second circle with a radius is drawn around E.

{\ textstyle | EG | = \ sin (x)}

{\ displaystyle {\ begin {aligned} & | DE | \ leq | {\ widehat {DC}} | \ leq | {\ widehat {DG}} | \\\ Leftrightarrow & \ sin (x) \ leq x \ leq {\ tfrac {\ pi} {2}} \ sin (x) \\\ Rightarrow & {\ tfrac {2} {\ pi}} x \ leq \ sin (x) \ leq x \ end {aligned}}}

The Jordan inequality or Jordan inequality provides a linear upper and lower estimate of the sine function for acute angles . It is named after Camille Jordan .

Inequality

The Jordan inequality is:

{\ displaystyle {\ frac {2} {\ pi}} x \ leq \ sin (x) \ leq x {\ text {for}} x \ in \ left [0, {\ frac {\ pi} {2} } \ right].}

It is used, among other things, in function theory . It can be proven analytically with the aid of the mean value theorem of integral calculus . Geometrically, their correctness can be recognized directly from the unit circle with the help of a second circle (see drawing).

Corollaries, Extensions, and Related Inequalities

As a consequence of the Jordan inequality we get that for a real number with : ${\ displaystyle x}$ ${\ displaystyle 0 <| x | \ leq {\ frac {\ pi} {2}}}$

{\ displaystyle {\ frac {2} {\ pi}} \ leq {\ frac {\ sin (x)} {x}} <1}

.

The related Redheffer-Williams inequality is named after US mathematicians Raymond Redheffer and JP Williams: For a real number is always ${\ displaystyle x \; (x \ neq 0)}$

{\ displaystyle {\ frac {\ sin (x)} {x}} \ geq {\ frac {{\ pi} ^ {2} -x ^ {2}} {{\ pi} ^ {2} + x ^ {2}}}}

.

literature

Serge Colombo: Holomorphic Functions of One Variable . Taylor & Francis 1983, ISBN 0-677-05950-7 , pp. 167-168
Feng Qi, Da-Wei Niu, Jian Cao: Refinements, Generalizations, and Applications of Jordan's Inequality and Related Problems . In: Journal of Inequalities and Applications , Volume 2009 (52 pages), doi: 10.1155 / 2009/271923
Meng-Kuang Kuo: Refinements of Jordan's inequality . Journal of Inequalities and Applications 2011, 2011: 130, doi: 10.1186 / 1029-242X-2011-130
Dragoslav Mitrinović : Analytic Inequalities . Springer Verlag (The Basic Teachings of Mathematical Sciences in Individual Representations. Volume 165), Berlin 1970, ISBN 3-540-62903-3 , p. 33

Web links

Jordan Inequality in the Proof Wiki
Eric W. Weisstein : Jordan's inequality . In: MathWorld (English).
Jordan and Kober inequalities on cut-the-knot.org

Individual evidence

↑ Eric W. Weisstein : Jordan's inequality . In: MathWorld (English).
^ Serge Colombo: Holomorphic Functions of One Variable . Taylor & Francis 1983, ISBN 0-677-05950-7 , pp. 167-168
↑ After Feng Yuefeng: Proof without words: Jordan's inequality . In: Mathematics Magazine , Volume 69, No. 2, 1996, p. 126
↑ Feng Qi, Da-Wei Niu, Jian Cao: Refinements, Generalizations, and Applications of Jordan's Inequality and Related Problems . In: Journal of Inequalities and Applications , Volume 2009 (52 pages), doi: 10.1155 / 2009/271923
↑ Dragoslav Mitrinović : Analytic Inequalities . Springer Verlag (The Basic Teachings of Mathematical Sciences in Individual Representations. Volume 165), Berlin 1970, ISBN 3-540-62903-3 , p. 33

[1] Eric W. Weisstein : Jordan's inequality . In: MathWorld (English).

[2] Serge Colombo: Holomorphic Functions of One Variable . Taylor & Francis 1983, ISBN 0-677-05950-7 , pp. 167-168

[3] After Feng Yuefeng: Proof without words: Jordan's inequality . In: Mathematics Magazine , Volume 69, No. 2, 1996, p. 126

[Feng-4] Feng Qi, Da-Wei Niu, Jian Cao: Refinements, Generalizations, and Applications of Jordan's Inequality and Related Problems . In: Journal of Inequalities and Applications , Volume 2009 (52 pages), doi: 10.1155 / 2009/271923

[5] Dragoslav Mitrinović : Analytic Inequalities . Springer Verlag (The Basic Teachings of Mathematical Sciences in Individual Representations. Volume 165), Berlin 1970, ISBN 3-540-62903-3 , p. 33