The monotonicity of ratios involving arc tangent function with applications

Abstract

Abstract In this paper, we investigate the monotonicity of the functions $$\begin{array}{} \begin{split}{} \displaystyle x &\mapsto &\frac{1}{x}\left( 1-a+\sqrt{\frac{2}{3}ax^{2}+a^{2}}\right) \arctan x, \\ x &\mapsto &\frac{1}{x}\left( \frac{4}{\pi ^{2}}+\sqrt{\frac{4}{\pi ^{2}}% x^{2}+a}\right) \arctan x \end{split} \end{array}$$ on (0, ∞) 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.