The monotonicity of ratios involving arc tangent function with applications

In this paper, we investigate the monotonicity of the functions x bar right arrow 1/x (1 - a + root 2/3ax(2) + a(2)) arctan x, x bar right arrow 1/x (4/pi(2)+ root 4/pi(2) x(2)+ a) arctan x, on (0, infinity) for a > 0, which not only gives relative errors of known bounds with quadratic for arctan x, but also yields some new accurate bounds. Moreover, the known bounds are extended and a more accurate estimate for arctan x is presented.