7 research outputs found
A Nice Separation of Some Seiffert-Type Means by Power Means
Seiffert has defined two well-known trigonometric means denoted by and . In a similar way it was defined by Carlson the logarithmic mean ℒ as a hyperbolic mean. Neuman and Sándor completed the list of such means by another hyperbolic mean ℳ. There are more known inequalities between the means ,, and ℒ and some power means . We add to these inequalities two new results obtaining the following nice chain of inequalities 0<ℒ<1/2<<1<ℳ<3/2<<2, where the power means are evenly spaced with respect to their order
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
Monotonicity of the Ratio of the Power and Second Seiffert Means with Applications
We present the necessary and sufficient condition for the monotonicity of the ratio of the power and second Seiffert means. As applications, we get the sharp upper and lower bounds for the second Seiffert mean in terms of the power mean
Unification and refinements of Jordan, Adamovi\'c-Mitrinovi\'cand and Cusa's inequalities
In this paper, we find some new sharp bounds for ,
which unify and refine Jordan, Adamovi\'{c}-Mitrinovi\'{c}and and Cusa's
inequalities. As applications of main results, some new Shafer-Fink type
inequalities for arc sine function and ones for certain bivariate means are
established, and a simpler but more accurate estimate for sine integral is
derived.Comment: 17 page
A Note on Jordan, Adamović-Mitrinović, and Cusa Inequalities
We improve the Jordan, Adamović-Mitrinović, and Cusa inequalities. As applications, several new Shafer-Fink type inequalities for inverse sine function and bivariate means inequalities are established, and a new estimate for sine integral is given