Sharp bounds for the Toader mean in terms of arithmetic and geometric means

Abstract

For $$a,b>0$$ with $$a\ne b$$ , the Toader mean of a and b is defined by $$\begin{aligned} T\left( a,b\right) =\frac{2}{\pi }\int _{0}^{\pi /2}\sqrt{a^{2}\cos ^{2}t+b^{2}\sin ^{2}t}dt. \end{aligned}$$ In this paper, we prove that the double inequality $$\begin{aligned} \left( \frac{3}{2}\left( \frac{a+b}{2}\right) ^{q}-\frac{1}{2}\left( \sqrt{ab }\right) ^{q}\right) ^{1/q}<T\left( a,b\right) <\left( \frac{3}{2}\left( \frac{a+b}{2}\right) ^{p}-\frac{1}{2}\left( \sqrt{ab}\right) ^{p}\right) ^{1/p} \end{aligned}$$ holds if and only if $$0<p\le 3/2$$ and $$q\ge \ln \left( 3/2\right) /\ln \left( 4/\pi \right) $$ . This gives new sharp lower and upper bounds for the Toader mean, and improves several known results.