A double inequality for tanhx

Abstract

Abstract In this paper, we prove that, for x>0 $x>0$, 1−exp(−x2x2+1)<tanhx<1−exp(−x3x3+1)3. $$ \sqrt{1-\exp \biggl(-\frac{x^{2}}{\sqrt{x^{2}+1}} \biggr)}< \tanh x< \sqrt[3]{1-\exp \biggl(- \frac{x^{3}}{\sqrt{x^{3}+1}} \biggr)}. $$This solves an open problem proposed by Ivády.

Keywords

• Inequality
• Exponential function
• Hyperbolic function