Abstract: In the article, we prove that the double inequalities

$$\begin{array}{l}{G^p}[{{\rm{\lambda }}_1}a + (1 - {{\rm{\lambda }}_1})b,{{\rm{\lambda }}_1}b + (1 - {{\rm{\lambda }}_1})a]{A^{1 - p}}(a,b) < [A(a,b),G(a,b)]\\ < {G^p}[{\mu _1}a + (1 - {\mu _1})b,{\mu _1}b + (1 - {\mu _1})a]{A^{1 - p}}(a,b),\\{C^s}[{\rm{\lambda }}2a + (1 - {{\rm{\lambda }}_2})b,{{\rm{\lambda }}_2}b + (1 - {{\rm{\lambda }}_2})a]{A^{1 - p}}(a,b) < [A(a,b),Q(a,b)]\\ < {C^s}[\mu 2a + (1 - \mu 2)b,\mu 2b + (1 - {\mu _2})a]{A^{1 - p}}(a,b)\end{array}$$ hold for all a, b > 0 with $a\neq b$ if and only if $\lambda_{1}\leq 1/2-\sqrt{1-(2/\pi)^{2/p}}/2$, $\mu_{1}\geq 1/2-\sqrt{2p}/(4p)$, $\lambda_{2}\leq1/2+\sqrt{2^{3/(2s)}(\mathcal{E}(\sqrt{2}/2)/\pi)^{1/s}-1}/2$ and $\mu_{2}\geq 1/2+\sqrt{s}/(4s)$ if $\lambda_{1}, \mu_{1}\in (0, 1/2)$, $\lambda_{2}, \mu_{2}\in (1/2, 1)$, $p\geq 1$ and $s\geq 1/2$, where $G(a, b)=\sqrt{ab}$, $A(a,b)=(a+b)/2$, $T(a,b)=2\int_{0}^{\pi/2}\sqrt{a^{2}\cos^{2}t+b^{2}\sin^{2}t}{\rm d}t/\pi$, $Q(a,b)=\sqrt{\left(a^{2}+b^{2}\right)/2}$, $C(a, b)=(a^{2}+b^{2})/(a+b)$ and $\mathcal{E}(r)=\int_{0}^{\pi/2}\sqrt{1-r^{2}\sin^{2}t}{\rm d}t$.

Key words: Geometric mean, arithmetic mean, Toader mean, ontraharmonic mean, complete elliptic integral

CLC Number: 

• 26E60

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