109 research outputs found
Some sharp inequalities involving Seiffert and other means and their concise proofs
In the paper, by establishing the monotonicity of some functions involving
the sine and cosine functions, the authors provide concise proofs of some known
inequalities and find some new sharp inequalities involving the Seiffert,
contra-harmonic, centroidal, arithmetic, geometric, harmonic, and root-square
means of two positive real numbers and with .Comment: 10 page
The monotonicity results and sharp inequalities for some power-type means of two arguments
For with , we define
M_{p}=M^{1/p}(a^{p},b^{p})\text{if}p\neq 0 \text{and} M_{0}=\sqrt{ab}, where
and 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 is a mean of
and , then is also, and call "power-type mean". We prove the
power-type means , , , are increasing in on
and establish sharp inequalities among power-type means ,
, , , , , , % . 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 As well as
many other well known inequalities involving the identric mean 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
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