A double inequality for bounding Toader mean by the centroidal mean

Abstract

In the paper, the authors find the best numbers $\alpha$ and $\beta$ such that $$\overline{C}(\alpha a+(1-\alpha )b,\alpha b+(1-\alpha )a)<T(a,b)<\overline{C}(\beta a+(1-\beta )b,\beta b+(1-\beta )a)$$ for all $a,b>0$ with $a\ne b$, where $\overline{C}(a,b)=2(a^2+ab+b^2)3(a+b)$ 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 centroidal mean and Toader mean of two positive numbers $a$ and $b$.

5 pages

References in corpus (3)

• Bounds for the combination of Toader mean and the arithmetic mean in terms of the contraharmonic mean
• Sharp bounds in terms of the power of the contra-harmonic mean for Neuman-Sandor mean
• Two sharp inequalities for bounding the Seiffert mean by the arithmetic, centroidal, and contra-harmonic means

Cited by in corpus (3)

• Very accurate approximations for the elliptic integrals of the second kind in terms of Stolarsky means
• Some sharp inequalities for the Toader-Qi mean
• Lehmer Transform and its Theoretical Properties