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

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 $a\mathrm{\ne }b$, where $$C(a,b)=\frac{a^2+b^2}{a+b},A(a,b)=\frac{a+b}{2},$$ and $$T(a,b)=\frac{2}{\pi }{\int }_0^{\pi \mathrm{/}2}\sqrt{a^2{\cos }^2\theta +b^2{\sin }^2\theta }d\theta$$ denote respectively the contraharmonic, arithmetic, and Toader means of two positive numbers $a$ and $b$.