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In the article, we prove that the double inequalities αNAG(a, b) + (1 - α) Q(a, b) < T D (a, b) < βNAG(a, b) + (1 - β) Q(a, b), λNAG(a, b) + (1 - λ) C(a, b) < T D(a, b) < µNAG(a, b) + (1 - µ) C(a, b) hold for all a, b > 0 with a 6= b if and only if α ≥ 3/10, β ≤ 2 (√ 2π - 4)/h(2√2 - 1)πi=0.1542 ···, λ ≥ 9/16 and µ ≤ 4 (π - 2)/(3π) = 0.4845 ···, where Q(a, b),C(a, b), NAG (a, b) and T D(a, b) are the quadratic, contra-harmonic, secondNeuman and Toader means of a and b, respectively. [ABSTRACT FROM AUTHOR]