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this paper, we prove that the inequalities $\\alpha [1/3 Q(a,b)+2/3 A(a,b)]+(1-\\alpha)Q^{1/3}(a,b)A^{2/3}(a,b)<M(a,b) <\\beta [1/3 Q(a,b)+2/3 A(a,b)]+(1-\\beta)Q^{1/3}(a,b)A^{2/3}(a,b)$ and $\\lambda [1/6 C(a,b)+5/6 A(a,b)]+(1-\\lambda)C^{1/6}(a,b)A^{5/6}(a,b)<M(a,b)<\\mu [1/6 C(a,b)+5/6 A(a,b)]++(1-\\mu)C^{1/6}(a,b)A^{5/6}(a,b)$ 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 8","authors_text":"Miao-Kun Wang, Yu-Ming Chu","cross_cats":["math.OC"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-09-13T15:05:37Z","title":"Refinements of the inequalities between Neuman-Sandor, arithmetic, contra-harmonic and quadratic 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