for questions involving inequalities, upper and lower bounds.
For a list of mathematical inequalities, see Wikipedia
Questions tagged [inequalities]
1655 questions
51
votes
24 answers
Most elementary proof showing that exponential growth wins against polynomial growth
This question is motivated by teaching : I would like to see a completely elementary proof showing for example that for all natural integers $k$ we have eventually $2^n>n^k$.
All proofs I know rely somehow on properties of the logarithm.
(I have…
Roland Bacher
- 17,432
29
votes
8 answers
Is there a good reason why $a^{2b} + b^{2a} \le 1$ when $a+b=1$?
The following problem is not from me, yet I find it a big challenge to give a nice (in contrast to 'heavy computation') proof. The motivation for me to post it lies in its concise content.
If $a$ and $b$ are nonnegative real numbers such that…
Sunni
- 1,858
27
votes
5 answers
Is this inequality on sums of powers of two sequences correct?
Let $a_1,\dots,a_n$ and $b_1,\dots,b_n$ be two sequences of non negative numbers such that for every positive integer $k$,
$$ a_1^k+\cdots+a_n^k \leq b_1^k+\cdots+b_n^k,$$
and
$$a_1+\cdots+a_n = b_1+\cdots+b_n.$$
Can we…
Mostafa
- 4,454
24
votes
4 answers
This inequality why can't solve it by now (Only four variables inequality)?
I asked a question at Math.SE last year and later offered a bounty for it, only johannesvalks give Part of the answer; A few months ago, I asked the author(Pham kim Hung) in Facebook, he said that now there is no proof by hand.and use of software…
math110
- 4,230
20
votes
6 answers
subadditive implies concave
Let $f:R_+\to R_+$ be smooth on $(0,\infty)$, increasing, $f(0)=0$ and
$\lim_{x\to\infty}=\infty$. Assume also that $f$ is subadditive:
$f(x+y)\le f(x)+f(y)$ for all $x,y\ge 0$. Must $f$ be concave? The converse is obvious.
alex
- 213
17
votes
4 answers
How to prove a known inequality from a book
The following inequality is from page 125 of D.S. Mitrinovic, J. Pecaric, A.M. Fink, Classical and new inequalities in analysis, Kluwer
Academic Publishers, Dordrecht/Boston/London, 1993.
If $a_i>0$, $b_i>0$ for $i=1,\cdots, n$ and $A=\frac{\max…
Sunni
- 1,858
16
votes
4 answers
An inequality concerning Lagrange's identity
we know Lagrange's identity
$$(a^2_{1}+a^2_{2}+a^2_{3})(b^2_{1}+b^2_{2}+b^2_{3})=(a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3})^2+\sum_{i=1}^{2}\sum_{j=i+1}^{3}(a_{i}b_{j}-a_{j}b_{i})^2$$
then we have Cauchy-Schwarz…
math110
- 4,230
14
votes
0 answers
Nice proof of inequality $(1-x^p)^{1/p}(1-x^q)^{1/q}\ge (1-x)(1+x^c)^{1/c}$ where $2^{1/c} = p^{1/p} q^{1/q}$?
Let $0\leq x < 1$, $1 \leq p < \infty$ and $q$ be the conjugate exponent defined by
$$1/p + 1/q = 1.$$
I am looking for a nice proof that
$$ \frac{(1-x^p)^{1/p}(1-x^q)^{1/q}}{(1-x)(1+x^c)^{1/c}} \geq 1,$$
where $c$ is defined via
$$2^{1/c} =…
George Shakan
- 2,283
14
votes
4 answers
An inequality improvement on AMM 11145
I have asked the same question in math.stackexchange, I am reposting it here, looking for answers:
How to show that for $a_1,a_2,\cdots,a_n >0$ real numbers and for $n \ge 3$:…
r9m
- 810
- 6
- 18
14
votes
2 answers
When is the product of a set of numbers greater than the sum of them?
This could well be too general a question, but I'd be interested in solutions to special cases too. Say you have some finite set of positive real numbers $x_i$, when is it the case that $\sum_i x_i > \prod_i x_i$? And when are they equal?
The…
Seamus
- 367
13
votes
2 answers
One specific inequality
How to prove this inequality $$\left(a+\frac{1}{2} \left(a b-\sqrt{a^2-1} \sqrt{b^2-1}\right)\right)^{3/4}-\frac{\sqrt{3} \cos\left[\frac{3 (\pi -t)}{4}\right]}{2 \left(\frac{1}{2}+b\right)^{1/4}}-(a+\cos t)^{3/4}\ge 0$$ for $a,b\ge 1$ and…
MathArt
- 333
13
votes
3 answers
I conjecture inequalities $\sum_{k=1}^{n}\{kx\}\le\frac{n}{2}x$
I conjecture the following inequality:
For $x > 1$, and $n$ a positive integer,
$$\sum_{k=1}^{n}\{kx\}\le\dfrac{n}{2}x.$$
For $n=1$, the inequality becomes
$$\{x\}\le\dfrac{x}{2}\Longleftrightarrow \{x\}\le [x];$$
and, for $n = 2$, it…
math110
- 4,230
13
votes
2 answers
When is alternating sum $\sum_{i}f(a_i)-\sum_{i
Let $a_1,a_2,\ldots,a_n\geq 1$, and let $f:\mathbb{R}^+\rightarrow\mathbb{R}^+$. Consider the sum
$$S(f)=\sum_{i}f(a_i)-\sum_{i
nan
- 363
13
votes
1 answer
A delicate elementary inequality
The following "piecewise-quadratic" inequality emerged in a joint work of Rom
Pinchasi and myself. The inequality is surprisingly delicate, and all our
attempts to simplify it made it false. By the end of the day, we were able to
prove the…
Seva
- 22,827
12
votes
1 answer
Maximize product of sums
Let $n,k\geq 2$ be positive integers. For each $1\leq i\leq n$, let $I_i$ be a nonempty subset of $\{1,2,\dots,k\}$. Let $P_i=\sum_{j\in I_i}x_j$, and let $P=P_1\cdot P_2\cdot\dots\cdot P_n$. (For example, $P=x_1(x_1+x_2)(x_1+x_3)$.)
We want to…
nan
- 363