Bessel Functions - 10.14 Inequalities; Monotonicity

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
10.14#Ex1 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |\BesselJ{\nu}@{x}| \leq 1}
|\BesselJ{\nu}@{x}| \leq 1		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \nu \geq 0, \realpart@@{(\nu+k+1)} > 0} 
abs(BesselJ(nu, x)) <= 1

Abs[BesselJ[\[Nu], x]] <= 1
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
10.14#Ex2 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |\BesselJ{\nu}@{x}| \leq 2^{-\frac{1}{2}}}
|\BesselJ{\nu}@{x}| \leq 2^{-\frac{1}{2}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \nu \geq 1, \realpart@@{(\nu+k+1)} > 0} 
abs(BesselJ(nu, x)) <= (2)^(-(1)/(2))

Abs[BesselJ[\[Nu], x]] <= (2)^(-Divide[1,2])
Failure Failure Successful [Tested: 2] Successful [Tested: 2]
10.14.E2 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 0 < \BesselJ{\nu}@{\nu}}
0 < \BesselJ{\nu}@{\nu}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \nu > 0, \realpart@@{(\nu+k+1)} > 0} 
0 < BesselJ(nu, nu)

0 < BesselJ[\[Nu], \[Nu]]
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
10.14.E2 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \BesselJ{\nu}@{\nu} < \frac{2^{\frac{1}{3}}}{3^{\frac{2}{3}}\EulerGamma@{\tfrac{2}{3}}\nu^{\frac{1}{3}}}}
\BesselJ{\nu}@{\nu} < \frac{2^{\frac{1}{3}}}{3^{\frac{2}{3}}\EulerGamma@{\tfrac{2}{3}}\nu^{\frac{1}{3}}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \nu > 0, \realpart@@{(\nu+k+1)} > 0} 
BesselJ(nu, nu) < ((2)^((1)/(3)))/((3)^((2)/(3))* GAMMA((2)/(3))*(nu)^((1)/(3)))

BesselJ[\[Nu], \[Nu]] < Divide[(2)^(Divide[1,3]),(3)^(Divide[2,3])* Gamma[Divide[2,3]]*\[Nu]^(Divide[1,3])]
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
10.14.E3 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |\BesselJ{n}@{z}| \leq e^{|\imagpart@@{z}|}}
|\BesselJ{n}@{z}| \leq e^{|\imagpart@@{z}|}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{(n+k+1)} > 0} 
abs(BesselJ(n, z)) <= exp(abs(Im(z)))

Abs[BesselJ[n, z]] <= Exp[Abs[Im[z]]]
Failure Failure Successful [Tested: 7] Successful [Tested: 7]
10.14.E4 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |\BesselJ{\nu}@{z}| \leq \frac{|\tfrac{1}{2}z|^{\nu}e^{|\imagpart@@{z}|}}{\EulerGamma@{\nu+1}}}
|\BesselJ{\nu}@{z}| \leq \frac{|\tfrac{1}{2}z|^{\nu}e^{|\imagpart@@{z}|}}{\EulerGamma@{\nu+1}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{(\nu+k+1)} > 0, \realpart@@{(\nu+1)} > 0} 
abs(BesselJ(nu, z)) <= ((abs((1)/(2)*z))^(nu)* exp(abs(Im(z))))/(GAMMA(nu + 1))

Abs[BesselJ[\[Nu], z]] <= Divide[(Abs[Divide[1,2]*z])^\[Nu]* Exp[Abs[Im[z]]],Gamma[\[Nu]+ 1]]
Failure Failure Successful [Tested: 7] Successful [Tested: 7]
10.14.E5 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |\BesselJ{\nu}@{\nu x}| \leq \frac{x^{\nu}\exp@{\nu(1-x^{2})^{\frac{1}{2}}}}{\left(1+(1-x^{2})^{\frac{1}{2}}\right)^{\nu}}}
|\BesselJ{\nu}@{\nu x}| \leq \frac{x^{\nu}\exp@{\nu(1-x^{2})^{\frac{1}{2}}}}{\left(1+(1-x^{2})^{\frac{1}{2}}\right)^{\nu}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \nu \geq 0, 0 < x, x \leq 1, \realpart@@{(\nu+k+1)} > 0} 
abs(BesselJ(nu, nu*x)) <= ((x)^(nu)* exp(nu*(1 - (x)^(2))^((1)/(2))))/((1 +(1 - (x)^(2))^((1)/(2)))^(nu))

Abs[BesselJ[\[Nu], \[Nu]*x]] <= Divide[(x)^\[Nu]* Exp[\[Nu]*(1 - (x)^(2))^(Divide[1,2])],(1 +(1 - (x)^(2))^(Divide[1,2]))^\[Nu]]
Failure Failure Successful [Tested: 3] Skip - No test values generated
10.14.E7 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 1 \leq \frac{\BesselJ{\nu}@{\nu x}}{x^{\nu}\BesselJ{\nu}@{\nu}}}
1 \leq \frac{\BesselJ{\nu}@{\nu x}}{x^{\nu}\BesselJ{\nu}@{\nu}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \nu \geq 0, 0 < x, x \leq 1, \realpart@@{(\nu+k+1)} > 0} 
1 <= (BesselJ(nu, nu*x))/((x)^(nu)* BesselJ(nu, nu))

1 <= Divide[BesselJ[\[Nu], \[Nu]*x],(x)^\[Nu]* BesselJ[\[Nu], \[Nu]]]
Failure Failure Successful [Tested: 3] Skip - No test values generated
10.14.E7 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{\BesselJ{\nu}@{\nu x}}{x^{\nu}\BesselJ{\nu}@{\nu}} \leq e^{\nu(1-x)}}
\frac{\BesselJ{\nu}@{\nu x}}{x^{\nu}\BesselJ{\nu}@{\nu}} \leq e^{\nu(1-x)}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \nu \geq 0, 0 < x, x \leq 1, \realpart@@{(\nu+k+1)} > 0} 
(BesselJ(nu, nu*x))/((x)^(nu)* BesselJ(nu, nu)) <= exp(nu*(1 - x))

Divide[BesselJ[\[Nu], \[Nu]*x],(x)^\[Nu]* BesselJ[\[Nu], \[Nu]]] <= Exp[\[Nu]*(1 - x)]
Failure Failure Successful [Tested: 3] Skip - No test values generated
10.14.E8 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |\BesselJ{n}@{nz}| \leq \frac{\left|z^{n}\exp@{n(1-z^{2})^{\frac{1}{2}}}\right|}{\left|1+(1-z^{2})^{\frac{1}{2}}\right|^{n}}}
|\BesselJ{n}@{nz}| \leq \frac{\left|z^{n}\exp@{n(1-z^{2})^{\frac{1}{2}}}\right|}{\left|1+(1-z^{2})^{\frac{1}{2}}\right|^{n}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{(n+k+1)} > 0} 
abs(BesselJ(n, n*z)) <= (abs((z)^(n)* exp(n*(1 - (z)^(2))^((1)/(2)))))/((abs(1 +(1 - (z)^(2))^((1)/(2))))^(n))

Abs[BesselJ[n, n*z]] <= Divide[Abs[(z)^(n)* Exp[n*(1 - (z)^(2))^(Divide[1,2])]],(Abs[1 +(1 - (z)^(2))^(Divide[1,2])])^(n)]
Failure Failure Successful [Tested: 7] Successful [Tested: 7]
10.14.E9 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |\BesselJ{n}@{nz}| \leq 1}
|\BesselJ{n}@{nz}| \leq 1		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle n = 0, \realpart@@{(n+k+1)} > 0} 
abs(BesselJ(n, n*z)) <= 1

Abs[BesselJ[n, n*z]] <= 1
Failure Failure Error Successful [Tested: 21]
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