JacobiSN[x, m]/(Sqrt[m] JacobiCN[x, m] + JacobiDN[x, m])^2 <= (2 x)/(Pi (1 - m)^2) /; Element[x, Reals] && 0 <= x <= EllipticK[m] && 0 <= m <= 1

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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mfrac> <mrow> <mi> sn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> x </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <msqrt> <mi> m </mi> </msqrt> <mo> &#8290; </mo> <mrow> <mi> cn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> x </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> dn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> x </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> <mo> &#8804; </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> x </mi> </mrow> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> </mfrac> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> x </mi> <mo> &#8712; </mo> <mi> &#8477; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mn> 0 </mn> <mo> &#8804; </mo> <mi> x </mi> <mo> &#8804; </mo> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mn> 0 </mn> <mo> &#8804; </mo> <mi> m </mi> <mo> &#8804; </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <leq /> <apply> <times /> <apply> <ci> JacobiSN </ci> <ci> x </ci> <ci> m </ci> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> m </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> JacobiCN </ci> <ci> x </ci> <ci> m </ci> </apply> </apply> <apply> <ci> JacobiDN </ci> <ci> x </ci> <ci> m </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> x </ci> <apply> <power /> <apply> <times /> <pi /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> x </ci> <ci> &#8477; </ci> </apply> <apply> <leq /> <cn type='integer'> 0 </cn> <ci> x </ci> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> </apply> <apply> <leq /> <cn type='integer'> 0 </cn> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2002-12-18