204,031 research outputs found
A Best Possible Double Inequality for Power Mean
We answer the question: for any p,q∈ℝ with p≠q and p≠-q, what are the greatest value λ=λ(p,q) and the least value μ=μ(p,q), such that the double inequality Mλ(a,b)0 with a≠b? Where Mp(a,b) is the pth power mean of two positive numbers a and b
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
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
A new way to prove L'Hospital Monotone Rules with applications
Let . Let and be differentiable
functions on and let on . By introducing an
auxiliary function , 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
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