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(λa+(1-λ)b,λb+(1-λ)a)<αA(a,b)+(1-α)T(a,b)\\ < C(μa+(1-μ)b,μb+(1-μ)a) \end{multline*} holds for all $\alpha \in (0,1)$ and $a,b>0$ with $a\ne b$, where $$C(a,b)=\frac{a^2+b^2}{a+b},A(a,b)=\frac{a+b}{2},$$ 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 $b$.

6 pages

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