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Results tagged with inequalities
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user 49208
for questions involving inequalities, upper and lower bounds.
0
votes
Find the best constant to this bounded inequality
Start with $(\sum x_k)^2=\sum x_k^2 + 2\sum_{i>j}x_ix_j .$
Under given conditions and summing up progressions give
$$(\sum x_k)^2 \le \frac{5n}{3} -\frac{2}{9}(1-(\frac{1}{4})^n).
$$
So
$$\sum x_k \le …
- 1,560
0
votes
An upper bound for the difference between arithmetic and harmonic mean
Some results for differences $A_n-G_n, G_n-H_n$ and so after summing up for $A_n-H_n$ you may find in the book: Classical and new inequalities in analysis by D. S. Mitrinovic; J. E. Pecaric; A. M. …
- 1,560
1
vote
A "quadratic" triangular inequality
The first paper on characterization of inner product spaces was:
P. Jordan and J. Von Neumann, On inner products in linear, metric spaces, Ann. of Math. (2) 36 (1935), no. 3, 719–723.
There is a boo …
- 1,560
1
vote
Inequality for a gamma function
Standard inequalities gives not power but exponential growth
$$
|\frac{\Gamma(s)}{\Gamma(2-s)}|\le \frac{1}{\pi} \sinh(\pi |s|).
$$
Really the better estimate is true? …
- 1,560
4
votes
0
answers
204
views
Integral inequality of Polya
Find a direct proof of this inequality based on some classical inequalities, such as Cauchy-Bunyakovskii, Holder, Jensen, Hardy and so on.
Problem 2. …
- 1,560
2
votes
Accepted
Estimate of a ratio of two incomplete gamma functions
In fact it seems to be a consequence of the Cauchy theorem from calculus. Really, by it and a formula for derivative of incomplete gamma function (cf. Wiki for example) we evaluate
$\frac{f(x)-f(y)}{ …
- 1,560
0
votes
About Turan`s problem(inequality) in multivariable
For the case of a single variable an obvious condition on polynomials is that a ratio
$$
\frac{P_{n-1}(x)P_{n+1}(x)}{(P_n(x))^2}
$$
is monotone. Then sharp estimates hold true with limits via $P_n( …
- 1,560
0
votes
Derivatives of radial functions can be bounded by derivatives in terms of radial distance?
There are many estimates of Sobolev type of the form
$$
||D^{\alpha}f||_1 \le ||\Delta f||_2
$$
for different pairs of norms. For your case the r.h.s. will be one-dimensional in r
$||(D^2-\frac{n-1}{r …
- 1,560
3
votes
1
answer
425
views
Inverse Hadamard determinant inequality
As far as I remembered there is an inverse Hadamard inequality for the determinant of the form
$$
|D|>\prod_j \sqrt{(a_{jj}^2-\sum_{i\neq j}a_{ij}^2)}
$$
providing all values in $(\cdot)>0$.
Please h …
- 1,560
2
votes
Proof of the "Neo-classical Inequality", a fractional extension of the binomial theorem
As far as I know this inequality was first proved by Terry Lyons in 90-s by standard method of Lagrange multipliers.
- 1,560
5
votes
1
answer
295
views
Estimate of incomplete binomial integral
Let $0\le k \le n$. Prove that
$$
n\binom{n}{k}\int_{0}^{\frac{k}{n+1}}t^k(1-t)^{n-k}\,dt \le 1/2.
$$
As far as I know
1) it is proved for $\frac{k}{n+1}\le 1/2$ and
2) not proved for $1/2 <\frac{k}{ …
- 1,560