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A(a,b)]+(1-\\lambda)C^{1/6}(a,b)A^{5/6}(a,b)<M(a,b)<\\mu [1/6 C(a,b)+5/6 A(a,b)]++(1-\\mu)C^{1/6}(a,b)A^{5/6}(a,b)$ hold for all $a,b>0$ with $a\\neq b$ if and only if $\\alpha\\leq (3-3\\sqrt[6]{2}\\log(1+\\sqrt{2}))/[(2+\\sqrt{2}-3\\sqrt[6]{2})\\log(1+\\sqrt{2})]=0.777...$, $\\beta\\geq 4/5$, $\\lambda\\leq (6-6\\sqrt[6]{2}\\log(1+\\sqrt{2}))/(7-6\\sqrt[6]{2}\\log(1+\\sqrt{2}))=0.274...$, and $\\mu\\geq 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