Sharp Bounds for Trigonometric and Hyperbolic Functions with Application to Fractional Calculus

Abstract

Sharp bounds for cosh(x)/x, sinh(x)/x, and sin(x)/x were obtained, as well as one new bound for e(x)+arctan(x)/root x. A new situation to note about the obtained boundaries is the symmetry in the upper and lower boundary, where the upper boundary differs by a constant from the lower boundary. New consequences of the inequalities were obtained in terms of the Riemann-Liovuille fractional integral and in terms of the standard integral.