Questions tagged [inequality]

Questions on proving, manipulating and applying inequalities. Do not use this tag just because an inequality appears somewhere in your question.

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A Nice Problem In Additive Number Theory

$\color{red}{\mathbf{Problem\!:}}$ $n\geq3$ is a given positive integer, and $a_1 ,a_2, a_3, \ldots ,a_n$ are all given integers that aren't multiples of $n$ and $a_1 + \cdots + a_n$ is also not a ...
ShBh's user avatar
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26 votes
4 answers
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Prove $\left(\frac{a+1}{a+b} \right)^{\frac25}+\left(\frac{b+1}{b+c} \right)^{\frac25}+\left(\frac{c+1}{c+a} \right)^{\frac25} \geqslant 3$

$a,b,c >0$ and $a+b+c=3$, prove $$\left(\frac{a+1}{a+b} \right)^{\frac25}+\left(\frac{b+1}{b+c} \right)^{\frac25}+\left(\frac{c+1}{c+a} \right)^{\frac25} \geqslant 3$$ I try to apply AM-GM $$\left(...
HN_NH's user avatar
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25 votes
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741 views

How large must $f(f(z))-z^2$ be in the unit disk?

This is a follow-up question to $|f(f(z))-z^2|$ must be large somewhere in the disc $\mathbb{D}$?, where the following was proven: Let $f$ be holomorphic in the unit disk $\Bbb D$ with $f(0) = 0$ and ...
Martin R's user avatar
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24 votes
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Gradient Estimate - Question about Inequality vs. Equality sign in one part

$\DeclareMathOperator{\diam}{diam}\newcommand{\norm}[1]{\lVert#1\rVert}\newcommand{\abs}[1]{\lvert#1\rvert}$For $u \in C^{1}(\overline{\Omega})$, for $\Omega\subset \subset \mathbb{R^{n}}$ a bounded ...
user avatar
22 votes
0 answers
1k views

Refinement of a famous inequality

I refine a famous inequality this is the following : Let $x,y>0$ then we have : $$x^n+y^n\leq \Big(\frac{x^n+y^n}{x^{n-1}+y^{n-1}} \Big)^n+\Big(\frac{x+y}{2}\Big)^n$$ It's equivalent to : $$x^n+1\...
user avatar
18 votes
0 answers
487 views

Smallest $c$ such that $f'<cf$ holds for all $f$ such that $f,f',f'',f'''>0$ and $f''' \le f.$

Let $f: \mathbb{R} \to \mathbb{R}$ be a $C^3$ function such that $f,f',f'',f'''>0$ and $f''' \le f.$ What is the smallest $c$ such that we can guarantee $f'<cf$? Since $f(x)=e^x$ works, we must ...
Display name's user avatar
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18 votes
0 answers
662 views

Bounding a polynomial from below

Let $\sigma >0$ be fixed. For even $k \in \mathbb{N} \cup \{0\}$, we consider the polynomial \begin{equation} \varphi_k(x) = \sum_{j=0}^{k} (-1)^j {k \choose j} b_j \, x^{2j} \quad x \in (-1,1), \...
char's user avatar
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18 votes
0 answers
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A very general method for solving inequalities repaired

Yesterday, I asked a question about a very general method for solving equations I had found here. As it turned out, there were quite some problems with my method and I got a lot of good feedback. ...
Mastrem's user avatar
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17 votes
1 answer
339 views

Ratio of product from one point and minimum distance

Given points $A_0,A_1,\ldots,A_n$ in the plane, let $m$ denote the minimum distance among any two points. What is the minimum value of $$\dfrac{|A_0A_1|\cdot|A_0A_2|\cdot\ldots\cdot|A_0A_n|}{m^n}?$$ ...
simmons's user avatar
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15 votes
1 answer
666 views

$\sum\limits_i a_i^2\sum\limits_i b_i^2+\left(\sum\limits_ia_i b_i\right)^2\geq \sqrt{\sum\limits_i a_i^4\sum\limits_i b_i^4}+\sum\limits_ia_i^2b_i^2$

I have no idea about how to prove (or disprove) the following inequality: $$ \left(\sum_{i=1}^n a_i^2\right)\left(\sum_{i=1}^n b_i^2\right)+\left(\sum_{i=1}^na_i b_i\right)^2\geq \sqrt{\left(\sum_{i=1}...
Ludwig's user avatar
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15 votes
0 answers
990 views

How to prove this polynomial inequality?

How can we prove the following? If $\frac{dP_{n}}{dz}|_{z=z_{0}}=0$ then $|P_{n}(z_{0})|<2$ for all $n>1$, where $P_{n}(z)\equiv P_{n-1}^{2}+z$ and $P_{1}\equiv z$ $z$ is in the complex plane. ...
Jerry Guern's user avatar
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14 votes
1 answer
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About the inequality : $x^{x^{x^{x^{x^x}}}}\geq x^{x^{x^{((e-2)(1+e))x\left(1+\sqrt{x}\left((\sqrt{x})^3-1\right)\right)}}}\geq x^{x^{\frac{16}{27}}}$

This inequality is due to user RiverLi : Let $0<x\leq 1$ then we have : $$x^{x^{x^{x^{x^x}}}}\geq x^{x^{\frac{16}{27}}} \geq 0.5x^2+0.5$$ I propose another one wich states : Let $0<x\leq 1$ ...
Miss and Mister cassoulet char's user avatar
14 votes
0 answers
565 views

prove or disprove an inequality on bounds of derivatives for radial functions

Suppose $f$ is a radial function, i.e., $f(x)=f(|x|)$, and $f \in C^\infty(\bar{B})$, where $\bar{B}$ is the closure of the unit ball in $\mathbb{R}^n$. Prove or disprove the following. Given any ...
booksee's user avatar
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14 votes
0 answers
306 views

Question about an upper bound

Let each of the positive integers $a_1 ,\dots, a_n$ be less than $m$ such that the least common multiple of any two of the positive integers $a_1 ,\dots, a_n$ is greater than the integer $m$. Then ...
Souvik Dey's user avatar
  • 8,327
13 votes
1 answer
291 views

Inequality with summation of cosine terms $\left|1 + 2\sum_{j=1}^k \cos (\frac{2\pi n}{q}j) \right| \leq 1 + 2\sum_{j=1}^k \cos (\frac{2\pi }{q}j)$

I got stuck on the following problem: Let $q\in \mathbb{N}$ be a fixed odd number and $k,n \in \{ 1,…,\frac{q-1}{2}\}$. I want to show that $$ \left|1 + 2\sum_{j=1}^k \cos (\frac{2\pi n}{q}j) \right| ...
Philipp's user avatar
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Inequality problem with factorials

I am not sure if this kind of "question" is welcome on MSE. Here is an olympiad-like problem that I would like to share with you: Let $a,b,c$ be nonnegative integers. Prove that $$ \frac{(a+b)!^2}...
Puzzle's user avatar
  • 156
13 votes
2 answers
169 views

Reference Request for inequalities without variables

Prove that $\dfrac{1}{2} < \dfrac{1}{101} + \dfrac{1}{102} + \dots + \dfrac {1}{200} < 1$ I started a math club at my school in the hopes of promoting math interest, and I want to interest my ...
Airdish's user avatar
  • 2,481
12 votes
0 answers
430 views

Hilbert's Inequality - improved???

Assume for convenience that $a_n\ge0$ (this also clarifies why certain inequalities below are in fact stronger than certain other inequalities below). Of the various inequalities Hilbert proved, I'm ...
David C. Ullrich's user avatar
12 votes
0 answers
3k views

Bounds for the exponential integral

In Abramowitz and Stegun: Handbook of Mathematical Functions (on page 229, property 5.1.20) it is stated that $$ \frac{1}{2} \log \left(1 + \frac{2}{x} \right) < \exp(x) E_1(x) < \log \left(1 + ...
user7064's user avatar
  • 313
11 votes
0 answers
593 views

For positive $a$, $b$, $c$, $d$, if $\sum_{cyc}\frac1{1+a}=2$, (dis)prove $\frac{4(3\sqrt2-4)}{a+b+c+d}+\sum_{cyc}\frac1{\sqrt a}\geq3\sqrt2$

An open problem from Art of Problem Solving (AoPS): If $a,b,c,d$ are positive real numbers such that $$\frac{1}{1+a}+\frac 1{1+b}+\frac 1{1+c}+\frac 1{1+d}=2,$$then prove or disprove $$\frac1{\sqrt a}...
Will's user avatar
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11 votes
0 answers
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An inequality about quasi-linear function

Let $\gamma$ be a positive, nondecreasing, continuous, function defined on $[0,\infty]$. Suppose that $\gamma(x+y)\le C(\gamma(x)+\gamma(y))$. In addition, suppose $$ \int_{2}^{\infty}\frac{dr}{\gamma(...
Mr.xue's user avatar
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11 votes
0 answers
492 views

On the Abstract Concreteness Method (bka $ABC-$Method).

I was reading Zdravko Cvetkovski's excellent book Inequalities: Theorems, Techniques, and selected problems, when I arrived at the $16$th chapter: the $ABC-$Method. I had some questions related to ...
Dr. Mathva's user avatar
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11 votes
0 answers
478 views

How to determine the minimal constant $\lambda = \lambda(n,k)$

Given fixed positive integers $n,k$, determine the minimal constant $\lambda = \lambda(n,k)$ for which the following inequality holds for any $a_1,a_2,...,a_n>0$ (taking indices mod $n$ if required)...
math110's user avatar
  • 93.4k
11 votes
1 answer
463 views

Very hard inequality: $\frac{a}{\sqrt{a+pb}}+\frac{b}{\sqrt{b+pc}}+\frac{c}{\sqrt{c+pa}} \le k_p \sqrt{a+b+c}.$

Given $p>0$. Find the smallest real number $k_p$ such that the following inequality holds for any non-negative reals $a,b,c$: $$\frac{a}{\sqrt{a+pb}}+\frac{b}{\sqrt{b+pc}}+\frac{c}{\sqrt{c+pa}} \le ...
f10w's user avatar
  • 4,519
10 votes
3 answers
472 views

show that $\sum_{k=1}^{n}(1-a_{k})<\frac{2}{3}$

Let $a_{1}=\dfrac{1}{2}$, and such $a_{n+1}=a_{n}-a_{n}\ln{a_{n}}$,show that $$\sum_{k=1}^{n}(1-a_{k})<\dfrac{2}{3}$$ My attemp: let $1-a_{n}=b_{n}$,then we have $$b_{n+1}=b_{n}+(1-b_{n})\ln{(1-b_{...
math110's user avatar
  • 93.4k
9 votes
1 answer
296 views

Bound a function with parameter involving logarithm

In my master project I encounter the following function $$f_\varepsilon(x) = \ln\left(\frac{x^{1 + a} + \varepsilon^\beta}{\lambda(x)^2 + \varepsilon^2}\right)$$ for $a$ close to zero, $\beta \in (1, ...
Falcon's user avatar
  • 3,990
9 votes
0 answers
192 views

Prove or disprove that the power of positive term polynomial will be eventually single peak

This is a question that a classmate asked me three years ago. Let $P(x)=\sum_{i=0}^n a_ix^i$ be a polynomial such that each $a_i>0$. Prove or disprove that there exists a positive integer $r$ such ...
JetfiRex's user avatar
  • 2,451
9 votes
0 answers
588 views

Proof of one-side version of Bennett-Bernstein inequality

I'm going to prove the following: For independent random variables $X_i$, $i \in [m]$ satisfying $X_i-E[X_i] \le b$ for some constant $b > 0$. Let $\bar{X} = \dfrac{1}{m}\sum_{i=1}^m X_i$, we have ...
Andrews's user avatar
  • 3,971
9 votes
0 answers
234 views

Is there a combinatorial proof to this inequality?

I verified that this inequality: $$ \sum_{i=0}^{k-1} \sum_{j=0}^{k+1} {3 k-3\choose i} {3 k+3\choose j} \geq \sum_{i=0}^{k} \sum_{j=0}^{k} {3 k\choose i} {3 k\choose j} $$ holds for all $k$ between 1 ...
Erel Segal-Halevi's user avatar
9 votes
0 answers
181 views

Inequality for hook numbers in Young diagrams

Consider a Young diagram $\lambda = (\lambda_1,\ldots,\lambda_\ell)$. For a square $(i,j) \in \lambda$ define hook numbers $h_{ij} = \lambda_i + \lambda_j' -i - j +1$ and complementary hook numbers $...
Igor Pak's user avatar
  • 1,356
9 votes
1 answer
354 views

Any similar Lagrange's identity inequality

Following problem I have post MO ,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_{...
math110's user avatar
  • 93.4k
9 votes
0 answers
396 views

Proof of Binet-Cauchy identity through the polarization transformation

In Steele's "The Cauchy-Schwarz Master Class" exercise 3.7 asks to prove the Cauchy-Binet identity: $$\langle a,s \rangle \langle b,t \rangle - \langle a,t \rangle \langle s,b \rangle = \...
Giovanni N's user avatar
9 votes
1 answer
160 views

How prove this inequality $(\sum_{k=1}^{3n}a_{k})^3\ge 27n^2\sum_{k=1}^{n}a_{k}a_{n+k}a_{2n+k}$

let $$0\le a_{1}\le a_{2}\le \cdots\le a_{3n}$$ show that $$\left(a_{1}+a_{2}+a_{3}+\cdots+a_{3n-1}+a_{3n}\right)^3\ge 27n^2\sum_{k=1}^{n}a_{k}a_{n+k}a_{2n+k}$$ we know when $n=1$,this is $$(a+...
math110's user avatar
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9 votes
1 answer
3k views

Relation betweeen Hoeffding inequality and Chernoff bound?

If I am correct, both Hoeffding inequality and Chernoff bound are about bounds on the probability of sample mean deviates from the true mean. Besides that, I wonder how Hoeffding inequality and ...
Tim's user avatar
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9 votes
1 answer
638 views

Using Jensen's inequality to prove another inequality?

Suppose $u(\cdot)$ and $v(\cdot)$ are two differentiable, strictly increasing, and strictly concave real functions. Specifically, $v(\cdot)$ is "more concave" than $u(\cdot)$ in the sense that there ...
Tuzi's user avatar
  • 101
8 votes
0 answers
195 views

Prove that $\Phi_{420}(69) > \Phi_{69}(420)$

Let $\Phi_n(x)$ denote the nth cyclotomic polynomial. Prove that $\Phi_{420}(69) > \Phi_{69}(420)$. Observe that if $\phi$ denotes the Euler-phi function, \begin{align} \phi(420) &= (2^2-2) \...
user3472's user avatar
  • 1,195
8 votes
0 answers
157 views

Does the inequality $c_a \le xy^{y^a/x^a} + yx^{x^a/y^a} - x^a - y^a \le C_a$ hold?

Update: Posted in MO since it is unanswered in MSE Let $0 \le x,y \le 1$ and $a$ be a real. Consider the function $$ f(x,y,a) = xy^{y^a/x^a} + yx^{x^a/y^a} - x^a - y^a \tag 1 $$ For a fixed $a$, the ...
Nilotpal Sinha's user avatar
8 votes
0 answers
255 views

$|𝐸[𝑋]|+𝜌(𝑋)≥1$?

Suppose $X_1, X_2, ... \sim X$ are i.i.d. random variables on $\mathbb{Z}$. Then the sequence $\{P(\sum_{i=1}^{d(X)n} X_i = 0)^{\frac{1}{d(X)n}}\}_{n=1}^\infty$ converges to some constant $\rho(X) \in ...
Chain Markov's user avatar
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8 votes
0 answers
241 views

show that $\left(\sum_{k=1}^{n}x_{k}\tan{k}\right)\left(\sum_{k=1}^{n}x_{k}\cot{k}\right)\le n^3\sum_{k=1}^{n}x^2_{k}$

let $x_{k}>0(k=1,2,3,\cdots,n)$ show that $$\left(\sum_{k=1}^{n}x_{k}\tan{k}\right)\left(\sum_{k=1}^{n}x_{k}\cot{k}\right)\le n^3\sum_{k=1}^{n}x^2_{k}$$ My try use AM-GM and Cauchy-Schwarz ...
math110's user avatar
  • 93.4k
8 votes
0 answers
269 views

Is it true that $\phi(\mu)=\mu F(\mu)^2-\int_{\mu}^{\overline{v}}F(v)[1-F(v)]dv\geq 0$?

Consider a random variable $V$ with distribution function $F$ and density function $f$ with support $[\underline{v},\overline{v}]$, where $0\leq\underline{v}<\overline{v}$. The mean is $\mu$. Here $...
smcc's user avatar
  • 5,704
8 votes
0 answers
319 views

Number of simultaneously solvable linear equations with nonnegative variables

Let $N$ variables $x_i \ge 0$ but not all of them be zero. One may fix $\sum_{i=1}^N x_i = 1$. There are $P$ equations which need to be solved, with coefficients $a^k_i$ indexed with superscripts $k =...
Andreas's user avatar
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8 votes
0 answers
3k views

Sum of minima is at most minimum of sum

I'd like to show the conditions under which the two are equal? $$\min_{x_i,y_j}\sum_{ij} |f(x_i,y_j)|^2 \ge \sum_{ij} \min_{x_i,y_j}|f(x_i,y_j)|^2$$ I believe the inequality can be proved as follows:...
raegakj's user avatar
  • 163
8 votes
0 answers
727 views

Von Neumann's Trace Inequality for Multiple Matrices

Von Neumann's trace inequality states that $|tr(AB)| \le \sum_{i=1}^{n} \sigma_i(A) \sigma_i(B)$ where $A, B$ are general square matrices with singular values $(\sigma_i(A)), (\sigma_i(B))$, ...
Zuza's user avatar
  • 247
8 votes
0 answers
357 views

I conjecture $3(a^ab^bc^c)^{\frac{1}{a+b+c}}\ge (a^ab^b)^{\frac{1}{a+b}}+(b^bc^c)^{\frac{1}{b+c}}+(c^ca^a)^{\frac{1}{c+a}}$

I Conjecture, for $a, b, c > 0$, $$3\left(a^ab^bc^c\right)^{\dfrac{1}{a+b+c}}\ge \left(a^ab^b\right)^{\frac{1}{a+b}}+\left(b^bc^c\right)^{\frac{1}{b+c}}+\left(c^ca^a\right)^{\frac{1}{c+a}}.$$ This ...
math110's user avatar
  • 93.4k
8 votes
0 answers
327 views

A Matrix Norm Inequality $\|A^{1/2}B^{1/2}(A+B)^{-1/2}\|_F \geq \|A^{1/2}(A+B)^{-1/2}B^{1/2}\|_F$

Let $\|X\|_F:= \sqrt{\text{Tr} \left(XX^\dagger\right) }$ denote the Frobenius norm. Does anyone know how to show the norm inequality: $\left\|A^{\frac12}B^{\frac12}(A+B)^{-\frac12}\right\|_F \geq ...
Hao-Chung Cheng's user avatar
8 votes
0 answers
490 views

How prove this inequality $\sum\limits_{cyc}\frac{1}{a^5_{1}(a_{2}+2a_{3})^2}\ge\frac{n}{9}$

Question: let $a_{1},a_{2},\cdots,a_{n}>0$,and such $$a_{1}a_{2}\cdot \cdots a_{n}=1$$ prove or disprove: $$\dfrac{1}{a^5_{1}(a_{2}+2a_{3})^2}+\dfrac{1}{a^5_{2}(a_{3}+2a_{4})^2}+\cdots+\dfrac{...
user avatar
8 votes
1 answer
262 views

Prove $\sup_{0\le x\le 1}|f(x)|\le\int_0^1(|f(t)|+|f'(t)|)dt$

Let $f\in C^1([0,1])$. Prove the following: $$\sup_{0\le x\le 1}|f(x)|\le\int_0^1(|f(t)|+|f'(t)|)dt$$ and $$|f(1/2)|\le\int_0^1(|f(t)|+\frac12|f'(t)|)dt$$ Note that $e^{-x}(e^xf(x))'=f(x)+f'(x)$. ...
Christmas Bunny's user avatar
8 votes
0 answers
335 views

Close power towers

In another question I gave some ways to determine which of two power towers is larger, but my answer there is incomplete because it doesn't handle the case where two towers are very close at each ...
Zander's user avatar
  • 10.9k
8 votes
0 answers
286 views

Multivariate polynomial with all coefficients positive

Let $n\geq 3$ be an integer. Consider the following polynomials : $$ f(x_1,x_2, \ldots ,x_n)=\bigg(\frac{1}{n}\sum_{k=1}^n x_k^n\bigg)^{2n-2}- \bigg(\prod_{k=1}^n \frac{x_k^{2n-2}+\big(\prod_{j\neq k}...
Ewan Delanoy's user avatar
  • 61.6k
7 votes
0 answers
91 views

Inequality involving minors of a hermitian matrix

Let $A$ be an $n \times n$ hermitian matrix with $n \geq 3$. I am trying to prove the following inequality involving its minors $$\left| \sum_{k=3}^n A_{3k} A_{[12k],[123]} \right| \leq \sqrt{\sum_{i &...
meler's user avatar
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