For \(a,b>0\) with \(a\ne b\) , let \( L=L(a,b),I=I(a,b),A=A(a,b),G=G(a,b),A_{p}=A_{p}(a,b)\) denote the logarithmic mean, identric mean, arithmetic mean, geometric mean and p-order power mean, respectively. In 2003, Alzer and Qiu asserted “there does not exist a real number p such that \(\sqrt{LI}<A_{p}(A,G)<(L+I)/2\) is valid for all \(a,b>0\) ” , which inspired us to prove the assertion and to find new sharp bounds for the product and sum of logarithmic and identric means. Precisely, we proved that the inequalities \(\begin{aligned} \sqrt{LI}< & \sqrt{A_{1/3}A_{2/3}}<A_{8/9}\left( A,G\right)<A_{1/2}, \\ A_{q_{0}}< & A_{\ln 2}\left( A,G\right)<\frac{L+I}{2}, \\ \frac{L+I}{2}< & \frac{A_{1/3}+A_{2/3}}{2}<A_{1/2} \end{aligned}\) hold for \(a,b>0\) with \(a\ne b\) , where \(q_{0}=\left( \ln 2\right) /\left( 1+\ln 2\right) \) . These yield several nice chains of inequalities for means. Finally, a conjecture is proposed.
