We present the best possible parameters α 1 , α 2 , β 1 , β 2 ∈ R and α 3 , β 3 ∈ ( 1 / 2,1 ) such that the double inequalities α 1 A ( a , b ) + ( 1 - α 1 ) C ( a , b ) < N Q A ( a , b ) < β 1 A ( a , b ) + ( 1 - β 1 ) C ( a , b ) , A α 2 ( a , b ) C 1 - α 2 ( a , b ) < N Q A ( a , b ) < A β 2 ( a , b ) C 1 - β 2 ( a , b ) , and C [ α 3 a + ( 1 - α 3 ) b , α 3 b + ( 1 - α 3 ) a ] < N Q A ( a , b ) < C [ β 3 a + ( 1 - β 3 ) b , β 3 b + ( 1 - β 3 ) a ] hold for all a , b > 0 with a ≠ b and give several sharp inequalities...