Sharp bounds for the Toader mean of order 3 in terms of arithmetic, quadratic and contraharmonic means

In the article, we present the best possible parameters α 1, β 1, α 2, β 2 ∈ ℝ and α 3, β 3 ∈ [1/2, 1] such that the double inequalities α 1 C ( a , b ) + ( 1 − α 1 ) A ( a , b ) < T 3 ( a , b ) < β 1 C ( a , b ) + ( 1 − β 1 ) A ( a , b ) , α 2 C ( a , b ) + ( 1 − α 2 ) Q ( a , b ) < T 3 ( a , b ) < β 2 C ( a , b ) + ( 1 − β 2 )