Bounds for the combination of Toader mean and the arithmetic mean in terms of the contraharmonic mean

Abstract

In the paper, the authors find the greatest value $\lambda$ and the least value $\mu $ such that the double inequality \begin{multline*} C(\lambda a+(1-\lambda)b,\lambda b+(1-\lambda )a)<\alpha A(a,b)+(1-\alpha)T(a,b)\\ < C(\mu a+(1-\mu)b,\mu b+(1-\mu )a) \end{multline*} holds for all $\alpha\in(0,1)$ and $a,b>0$ with $a\ne b$, where $$ C(a,b)=\frac{a^{2}+b^{2}}{a+b},\quad A(a,b)=\frac{a+b}2, $$ and $$ T(a,b)=\frac{2}{\pi}\int_{0}^{{\pi}/{2}}\sqrt{a^2\cos^2\theta+b^2\sin^2\theta}\,d\theta$$ denote respectively the contraharmonic, arithmetic, and Toader means of two positive numbers $a$ and $b$.