Sharp bounds for Toader mean in terms of arithmetic and second contraharmonic means
Summary: We present the best possible parameters \(\lambda_1,\mu_1\in \mathbb{R}\) and \(\lambda_2,\mu_2\in(1/2,1)\) such that double inequalities \(\lambda_1C(a,b)+(1-\lambda_1)A(a,b)<T(a,b)<\mu_1C(a,b)+(1-\mu_1)A(a,b)\), \(C[\lambda_2a+(1-\lambda_2)b,\lambda_2b+(1-\lambda_2)a]<T(a,b)<C[\mu_2a+(1-\mu_2)b,\mu_2b+(1-\mu_2)a]\) hold for all \(a,b>0\) with \(a\neq b\), where \(A(a,b)=(a+b)/2\), \(C(a,b)=(a^3+b^3)/(a^2+b^2)\) and \(T(a,b)=2\int_0^{n/2}\sqrt{a^2\cos^2\theta+b^2\sin^2\theta}d\theta/\pi\) are the arithmetic, second contraharmonic, and Toader means of \(a\) and \(b\), respectively.
scientific article; zbMATH DE number 6499106
scientific article