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

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.Comment: 11 page

On two new means of two arguments III

In this paper authors establish the two sided inequalities for the following two new means $$X=X(a,b)=Ae^{G/P-1},\quad Y=Y(a,b)=Ge^{L/A-1}.$$ As well as many other well known inequalities involving the identric mean $I$ and the logarithmic mean are refined from the literature as an application.Comment: 14. arXiv admin note: substantial text overlap with arXiv:1509.0197

On Seiffert-like means

We investigate the representation of homogeneous, symmetric means in the form M(x,y)=\frac{x-y}{2f((x-y)/(x+y))}. This allows for a new approach to comparing means. As an example, we provide optimal estimate of the form (1-\mu)min(x,y)+ \mu max(x,y)<= M(x,y)<= (1-\nu)min(x,y)+ \nu max(x,y) and M((x+y)/2-\mu(x-y)/2,(x+y)/2+\mu(x-y)/2)<= N(x,y)<= M((x+y)/2-\nu(x-y)/2,(x+y)/2+\nu(x-y)/2) for some known means

A new way to prove L'Hospital Monotone Rules with applications

Let $-\infty \leq a<b\leq \infty$. Let $f$ and $g$ be differentiable functions on $(a,b)$ and let $g^{\prime }\neq 0$ on $(a,b)$. By introducing an auxiliary function $H_{f,g}:=\left( f^{\prime }/g^{\prime }\right) g-f$, we easily prove L'Hoipital rules for monotonicity. This offer a natural and concise way so that those rules are easier to be understood. Using our L'Hospital Piecewise Monotone Rules (for short, LPMR), we establish three new sharp inequalities for hyperbolic and trigonometric functions as well as bivariate means, which supplement certain known results.Comment: 19 page