The monotonicity results and sharp inequalities for some power-type means of two arguments

Abstract

For $a,b>0$ with $a\neq b$, we define M_{p}=M^{1/p}(a^{p},b^{p})\text{if}p\neq 0 \text{and} M_{0}=\sqrt{ab}, where $M=A,He,L,I,P,T,N,Z$ and $Y$ stand for the arithmetic mean, Heronian mean, logarithmic mean, identric (exponential) mean, the first Seiffert mean, the second Seiffert mean, Neuman-S\'{a}ndor mean, power-exponential mean and exponential-geometric mean, respectively. Generally, if $M$ is a mean of $a$ and $b$, then $M_{p}$ is also, and call "power-type mean". We prove the power-type means $P_{p}$, $T_{p}$, $N_{p}$, $Z_{p}$ are increasing in $p$ on $\mathbb{R}$ and establish sharp inequalities among power-type means $A_{p}$, $He_{p}$, $L_{p}$, $I_{p}$, $P_{p}$, $N_{p}$, $Z_{p}$, $Y_{p}$% . From this a very nice chain of inequalities for these means L_{2}<P<N_{1/2}<He<A_{2/3}<I<Z_{1/3}<Y_{1/2} follows. Lastly, a conjecture is proposed.