Treffer: On the generalized power-type Toader mean

Summary: This paper deals with the so-called generalized power-type Toader mean which is defined by \[ \boldsymbol{T}_n(a, b) = \left(\frac{2}{\pi}\int_0^{\pi/2}\sqrt{a^n\cos^2\theta+b^2\sin^2\theta}d\theta\right)^{2/n} \] for \(a, b > 0\) with non-zero integer \(n\). In this study, we establish the following chain of inequalities \[ \begin{aligned} \boldsymbol{H}(a, b) < \boldsymbol{T}_{-1}(a, b) &< \boldsymbol{G}(a, b) < \boldsymbol{T}_1(a, b) < \boldsymbol{A}(a, b)\\ &< \boldsymbol{T}_2(a, b) < \boldsymbol{Q}(a, b) < \boldsymbol{T}_3(a, b)< \boldsymbol{T}_4(a, b) < \boldsymbol{C}(a, b) \end{aligned} \] for all \(a, b > 0\) with \(a \neq b\), where \(\boldsymbol{H}(a, b) = 2ab/(a+b)\), \(\boldsymbol{G}(a, b) = \sqrt{ab}\), \(\boldsymbol{A}(a, b) = (a+b)/2\), \(\boldsymbol{Q}(a, b) = [(a^2+b^2)/2]^{1/2}\) and \(\boldsymbol{C}(a, b) = (a^2+b^2)/(a+b)\) are the harmonic, geometric, arithmetic, quadratic and contra-harmonic means, respectively. Further, we provide sharp bounds for \(\boldsymbol{T}_{-1}(a, b)\) and \(\boldsymbol{T}_4(a, b)\) in terms of bivariate means mentioned above. As applications, new bounds for complete elliptic integral of the second kind are established.