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

Abstract

Abstract 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 ) Q ( a , b ) , C ( α 3 ; a , b ) < T 3 ( a , b ) < C ( β 3 ; a , b ) $$\begin{array}{} \begin{split} \displaystyle \alpha_{1}C(a, b)+(1-\alpha_{1})A(a, b) & \lt T_{3}(a, b) \lt \beta_{1}C(a, b)+(1-\beta_{1})A(a, b), \\ \alpha_{2}C(a, b)+(1-\alpha_{2})Q(a, b) & \lt T_{3}(a, b) \lt \beta_{2}C(a, b)+(1-\beta_{2})Q(a, b), \\ C(\alpha_{3}; a, b) & \lt T_{3}(a, b) \lt C(\beta_{3}; a, b) \end{split} \end{array}$$ hold for a, b > 0 with a ≠ b, and provide new bounds for the complete elliptic integral of the second kind, where A(a, b) = (a + b)/2 is the arithmetic mean, Q ( a , b ) = a 2 + b 2 / 2 $\begin{array}{} \displaystyle Q(a, b)=\sqrt{\left(a^{2}+b^{2}\right)/2} \end{array}$ is the quadratic mean, C(a, b) = (a 2 + b 2)/(a + b) is the contra-harmonic mean, C(p; a, b) = C[pa + (1 – p)b, pb + (1 – p)a] is the one-parameter contra-harmonic mean and T 3 ( a , b ) = ( 2 π ∫ 0 π / 2 a 3 cos 2 ⁡ θ + b 3 sin 2 ⁡ θ d θ ) 2 / 3 $\begin{array}{} T_{3}(a,b)=\Big(\frac{2}{\pi}\int\limits_{0}^{\pi/2}\sqrt{a^{3}\cos^{2}\theta+b^{3}\sin^{2}\theta}\text{d}\theta\Big)^{2/3} \end{array}$ is the Toader mean of order 3.