Elementary Functions - 4.32 Inequalities

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
4.32.E1	${\displaystyle{\displaystyle\cosh x\leq\left(\frac{\sinh x}{x}\right)^{3}}}$ \cosh@@{x} \leq \left(\frac{\sinh@@{x}}{x}\right)^{3}		cosh(x) <= ((sinh(x))/(x))^(3)	Cosh[x] <= (Divide[Sinh[x],x])^(3)	Failure	Failure	Successful [Tested: 3]	Successful [Tested: 3]
4.32.E2	${\displaystyle{\displaystyle\sin x\cos x<\tanh x}}$ \sin@@{x}\cos@@{x} < \tanh@@{x}	${\displaystyle{\displaystyle x>0}}$	sin(x)*cos(x) < tanh(x)	Sin[x]*Cos[x] < Tanh[x]	Failure	Failure	Successful [Tested: 3]	Successful [Tested: 3]
4.32.E2	${\displaystyle{\displaystyle\tanh x<x}}$ \tanh@@{x} < x	${\displaystyle{\displaystyle x>0}}$	tanh(x) < x	Tanh[x] < x	Failure	Failure	Successful [Tested: 3]	Successful [Tested: 3]
4.32.E3	${\displaystyle{\displaystyle|\cosh x-\cosh y|\geq|x-y|\sqrt{\sinh x\sinh y}}}$ |\cosh@@{x}-\cosh@@{y}| \geq |x-y|\sqrt{\sinh@@{x}\sinh@@{y}}	${\displaystyle{\displaystyle x>0,y>0}}$	abs(cosh(x)- cosh(y)) >= abs(x - y)*sqrt(sinh(x)*sinh(y))	Abs[Cosh[x]- Cosh[y]] >= Abs[x - y]*Sqrt[Sinh[x]*Sinh[y]]	Failure	Failure	Successful [Tested: 9]	Successful [Tested: 9]
4.32.E4	${\displaystyle{\displaystyle\operatorname{arctan}x\leq\tfrac{1}{2}\pi\tanh x}}$ \atan@@{x} \leq \tfrac{1}{2}\pi\tanh@@{x}	${\displaystyle{\displaystyle x\geq 0}}$	arctan(x) <= (1)/(2)*Pi*tanh(x)	ArcTan[x] <= Divide[1,2]*Pi*Tanh[x]	Failure	Failure	Successful [Tested: 3]	Successful [Tested: 3]