Sharp bounds for the Toader mean of order 3 in terms of arithmetic, quadratic and contraharmonic means
Let \(n\in \mathbb{Z} \) and \(a,b>0.\) Then the Toader mean \(T_{n}(a,b)\) of order \(n\) is defined by \[ T_{n}(a,b)=\left\{\begin{array}{cc} \left( \frac{2}{\pi}\int\nolimits_{0}^{\frac{\pi }{2}}\sqrt{a^{n}\cos ^{2}\theta +b^{n}\sin ^{2}\theta }d\theta \right) ^{\frac{2}{n}}, & n\neq 0 \\ \sqrt{ab} & n=0. \end{array} \right. \] In this paper, the authors study the bounds for the Toader mean of order \(3 \) in terms of the arithmetic, quadratic and contraharmonic means. They derive and prove in Theorems 3.1, 3.2 and 3.3 the best possible parameters \(\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}\in \mathbb{R}\) and \(\alpha _{3},\beta _{3}\in \lbrack \frac{1}{2},1]\) for which the double inequalities \[ \alpha _{1}C(a,b)+(1-\alpha _{1})A(a,b)<T_{3}(a,b)<\beta _{1}C(a,b)+(1-\beta _{1})A(a,b), \] \[ \alpha _{2}C(a,b)+(1-\alpha _{2})Q(a,b)<T_{3}(a,b)<\beta _{2}C(a,b)+(1-\beta _{2})Q(a,b), \] and \[ C(\alpha _{3};a;b)<T_{3}(a;b)<C(\alpha _{3};a;b) \] hold for \(a, b>0\) with \(a\neq b.\) Furthermore, new sharp bounds for the complete elliptic integral of the second kind were provided and proved, where \(A(a;b)=\frac{(a+b)}{2}\) is the arithmetic mean, \(Q(a;b)=\sqrt{\frac{ (a^{2}+b^{2})}{2}}\) is the quadratic mean, \(C(a;b)=\frac{(a^{2}+b^{2})}{(a+b) }\) 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 \(T_{3}(a;b)=\left( \frac{2}{\pi } \int\nolimits_{0}^{\frac{\pi }{2}}\sqrt{a^{3}\cos ^{2}\theta +b^{3}\sin ^{2}\theta }d\theta \right) ^{\frac{2}{3}}\) is the Toader mean of order \(3.\)
scientific article; zbMATH DE number 7439831
scientific article