MMN-2334
Bounding the convex combination of arithmetic and integral means in terms of one-parameter harmonic and geometric means
Wei-Mao Qian; Wen Zhang; Yu-Ming Chu;Abstract
In the article, we find the best possible parameters $\lambda_{1}$, $\mu_{1}$, $\lambda_{2}$
and $\mu_{2}$ on the interval $[0, 1/2]$ such that the double inequalities
\begin{equation*}
H(a, b; \lambda_{1})<\alpha A(a,b)+(1-\alpha)T(a,b)<H(a, b; \mu_{1}),
\end{equation*}
\begin{equation*}
G(a, b; \lambda_{2})<\alpha A(a,b)+(1-\alpha)T(a,b)<G(a, b; \mu_{2})
\end{equation*}
hold for all $\alpha\in [0, 1]$ and $a, b>0$ with $a\neq b$, where $A(a,b)=(a+b)/2$, $T(a,b)=2\int_{0}^{\pi/2}a^{\cos^{2}\theta}b^{\sin^{2}\theta}d\theta/\pi$, $H(a, b; \lambda)=2[\lambda a+(1-\lambda)b][\lambda b+(1-\lambda)a]/(a+b)$, $G(a, b; \mu)=\sqrt{[\mu a +(1-\mu)b][\mu b+(1-\mu)a]}$ are the arithmetic, integral, one-parameter harmonic and one-parameter geometric means of $a$ and $b$, respectively.
Vol. 20 (2019), No. 2, pp. 1157-1166
DOI: 10.18514/MMN.2019.2334