Bounds for Quotients of Inverse Trigonometric and Inverse Hyperbolic Functions

We establish new simple bounds for the quotients of inverse trigonometric and inverse hyperbolic functions such as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfrac><mrow><msup><mo form="prefix">sin</mo><mrow><mo>−</mo><mn>1</mn></mrow></msup><mi>x</mi></mrow><mrow><msup><mo form="prefix">sinh</mo><mrow><mo>−</mo><mn>1</mn></mrow></msup><mi>x</mi></mrow></mfrac></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfrac><mrow><msup><mo form="prefix">tanh</mo><mrow><mo>−</mo><mn>1</mn></mrow></msup><mi>x</mi></mrow><mrow><msup><mo form="prefix">tan</mo><mrow><mo>−</mo><mn>1</mn></mrow></msup><mi>x</mi></mrow></mfrac></semantics></math></inline-formula>. The main results provide polynomial bounds using even quadratic functions and exponential bounds under the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>e</mi><mrow><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup></mrow></msup><mo>.</mo></mrow></semantics></math></inline-formula> Graph validation is also performed.