Refined quadratic estimations of Shafer’s inequality

Abstract

Abstract We establish an inequality by quadratic estimations; the double inequality π 2 x 4 + ( π 2 − 4 ) 2 + ( 2 π x ) 2 0 $x>0$ , where the constants ( π 2 − 4 ) 2 $(\pi^{2} -4)^{2}$ and 32 are the best possible.

Keywords

• Shafer’s inequality
• an upper bound for arctangent
• a lower bound for arctangent