Parameter Dependent Refinements for the Sinc Hyperbolic Function and Applications
Abstract
In this paper, we prove for \ $x \in (0,\infty)$ the double inequality for the Sinc hyperbolic function
$$\left( -2\, \left( {\frac {\tanh x }{x}} \right) ^{-\frac{ r}{2}}+3\, \left( \cosh x \right) ^{\frac{ r}{3}} \right) ^{\frac{1}{r}}
< \ {\frac {\sinh x }{x}}\ <\ \cosh x \left( \frac{2}{3} \left( {\frac {\tanh x }{x}} \right) ^{\frac{3}{2}
\,r}+\frac{1}{3} \right) ^{\frac{1}{r}}.$$ Both bounds converge to \ ${\frac {\sinh x }{x}}$ \ as $r$ approaches zero from the right.
Similar double inequalities are provided for the Huygens and the Wilker functions as well as for the Lazarewi\'c function.
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References
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