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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3 votes
3 answers
192 views

Inequality $\{a,b,c,d\}\subset[0,1]\Rightarrow\sum\limits_{cyc}(a^4+a^2b^2)+8\prod\limits_{cyc}(1-a)\geq1$

Let $\{a,b,c,d\}\subset[0,1]$. Prove that: $$a^4+b^4+c^4+d^4+a^2b^2+b^2c^2+c^2d^2+d^2a^2+8(1-a)(1-b)(1-c)(1-d)\geq1$$ I tried convexity, the substitution $a=\frac{x}{x+1}...$ and more, but without ...
2 votes
1 answer
65 views

Prove $\sqrt{\frac{a+3}{a+3b}}+\sqrt{\frac{b+3}{b+3c}}+\sqrt{\frac{c+3}{c+3a}} \ge 3$

Let $a,b,c$ are positives such that $ab+bc+ca=3$. Prove that: $$\sqrt{\frac{a+3}{a+3b}}+\sqrt{\frac{b+3}{b+3c}}+\sqrt{\frac{c+3}{c+3a}} \ge 3$$ Once, I see this problem in AoPS:https://...
2 votes
1 answer
79 views

How to prove $\sum\limits_{cyc} \frac{1}{x+yz} \le \frac{9}{2(xy+yz+zx)}$ for all $x,y,z >0:x+y+z=3.$?

When I entered a test at my school, I stuck this problem (it is also posted here) Let $x,y,z$ be positive real numbers such that $x+y+z=3$, prove that $$\frac{1}{x+yz}+\frac{1}{y+zx}+\frac{1}{z+xy} \...
0 votes
0 answers
52 views

What are all possible set of $x,y,z$ such that $|x|=|y|=|z|=1$ and $x^3+y^3+z^3=-xyz$?

In this question, I asked for all the possible values of $|x+y+z|$ such that $x,y,z$ are complex numbers, $|x|=|y|=|z|=1$, and $x^3+y^3+z^3=-xyz$, and the answer was only $1$ and $2$. I found these ...
0 votes
0 answers
23 views

Searching for a more concise solution to $|x - 1| + |x + 1| < 2$ [duplicate]

I came up with what I think is the solution to exercise 11. (v) on chapter 1 of the third edition of book Calculus by Michael Spivak. Find all numbers $x$ for which $|x - 1| + |x + 1| < 2$. ...
0 votes
0 answers
33 views

A modified version of log-sum inequality.

Suppose $a_i>0, b_i>0, c_i>0 \; \forall i = 1, 2, \dots, n$, and $$\sum_{i} a_i b_i \ln(\frac{c_i}{b_i}) \geq 0, $$ where $a_i, b_i, c_i$ are not constant over $i$. Moreover, $$\sum_{i}a_ib_i\...
0 votes
3 answers
74 views

$\frac{1}{2a^2+3}+\frac{1}{2b^2+3}+\frac{1}{2c^2+3}\geq\frac{3}{5}.$

Let $a, b, c \in\mathbb{R}, a, b,c\geq \frac{1}{4}$ and $a+b+c=3$. Show that $\frac{1}{2a^2+3}+\frac{1}{2b^2+3}+\frac{1}{2c^2+3}\geq\frac{3}{5}.$ My idea: $(4a-1)(a-1)^2\geq 0$. Then I tried to divide ...
0 votes
1 answer
77 views

Prove that $f(u+v)\le f(u) +f(v)$ being $f(u)= Au\cdot u\in\mathbb R$

Let $A\in\mathbb R^{n\times n}$ be a symmetric and positive definite matrix. Consider the function $$f: u\in\mathbb R^n\mapsto f(u)= (Au\cdot u)^{1/2}\in\mathbb R.$$ Let $\|\cdot\|_1$ denote a norm ...
1 vote
0 answers
113 views

Factorial equation and no solution existing :A conjecture

Conjecture : Let $x<0,n>2$ ,$n$ an integer such that $|x|<(n!+1)(n^2+1)$ then it seems that : $$1/(x/(n!+1))!-1/(x/(n^2+1))!=0$$ Have no integer solution in $x$ where : $$x!=\Gamma(x+1)$$ Or :...
4 votes
1 answer
95 views

Proof of Doob's Maximal Inequality for Positive Supermartingales

I am working through a course on Stochastic Processes and am looking for a proof verification for an alternative to the one that I have been presented. My proof, as noted below, currently contains a ...
3 votes
4 answers
951 views

shorter proof of generalized mediant inequality?

Show $\frac{a_{1}+...+a_{n}}{b_{1}+...+b_{n}}$ is between the smallest and largest fraction $\frac{a_{i}}{b_{i}}$, where $b_{i}>0$. Attempt Assume the largest is $\frac{a_{n}}{b_{n}}\Rightarrow$ ...
3 votes
0 answers
89 views
+100

Maximum or tight upper bound of $f_n g_n$

Denote $f_n=f_n(x_1,x_2,w_1,w_2,w_3)$, $g_n=g_n(x_1,x_2,w_1,w_2,w_3)$ and define, $$f_n:=x_1^n(1-x_1)^n x_2^n(1-x_2)^n \sum_{i+j+k=n}(1-w_1)^i(1-w_2)^j(1-w_3)^k$$ and $$g_n:=g_n(x_1,x_2,w_1,w_2,w_3)=\...
0 votes
0 answers
75 views

How to prove $\frac{a}{a’}<\frac{a+b+c+d}{a’+b’+c’+d’}<\frac{d}{d’}$

Given $a, b, c, d, a', b', c', d'>0$ and $\displaystyle\frac{a}{a’}<\frac{b}{b’}<\frac{c}{c’}<\frac{d}{d’}$ , prove that $\displaystyle\frac{a}{a’}<\frac{a+b+c+d}{a’+b’+c’+d’}<\frac{...
0 votes
0 answers
50 views

A functional inequality over integers

Given positive integers $N$ and $n$, we define a function $f(x)$ as follows: $$f(x):=\frac{x^2-3x}{2}~\text{ when } x\leq N,$$ and $$f(x):=\frac{1}{2n}x^2+\frac{n-4}{2}x~\text{ when } x>N.$$ ...
42 votes
8 answers
11k views

Elementary central binomial coefficient estimates

How to prove that $\quad\displaystyle\frac{4^{n}}{\sqrt{4n}}<\binom{2n}{n}<\frac{4^{n}}{\sqrt{3n+1}}\quad$ for all $n$ > 1 ? Does anyone know any better elementary estimates? Attempt. We ...
0 votes
0 answers
33 views

Asymptotic Analysis question

Let $x>0$ be some variable that I am interested in analyzing in the limit $x\to\infty$. Let $y = x+c$ for some constant $c$. Is $\sqrt{y} - \sqrt{x} \geq f(c) > 0$ for some function $f$ of only ...
8 votes
4 answers
382 views

Prove that $\int_{0}^{1}f(x)^2dx\geq 4$

Let $f:[0,1]\to \mathbb{R} $ be an integrable function with $\int_{0}^{1}f(x)dx=\int_{0}^{1}xf(x)dx=1$. Prove that $\int_{0}^{1}f(x)^2dx\geq 4$. I got that $\int_{0}^{1}F(x)dx=F(0)$, but I don't think ...
-1 votes
0 answers
44 views

Find the maximum value of $a$ under the condition $2e\ln x\leq ax+b\leq \frac{1}{2}x^2+e$,where $a,b\in\mathbb{R}$,$x>0$

Assume there exists $a,b\in \mathbb{R}$,such that $$2e\ln x\leq ax+b\leq \frac{1}{2}x^2+e$$ hold for $\forall x>0$.find the maximum value of $a$. I guess the critical situation is $y=ax+b$ it the ...
0 votes
1 answer
10 views

Inequation of the product of ascending and random integers

We have three sequences of positive integers $l$, $p$ and $q$ such that: $$ p_1 \geq p_2 \geq \cdots \geq p_k\ \ and\ q_1 \geq q_2 \geq \cdots \geq q_k \geq \cdots \geq q_h \ \ where:\ \ k < h $$ ...
6 votes
3 answers
383 views

challenging inequality with complex numbers

The statement of the problem: Let $n \in \mathbb N $ \ {0} and $z_1,z_2, ... z_n \in \mathbb C $. Prove that $$\sum_{i=1}^n |z_i||z-z_i| \ge \sum_{i=1}^n |z_i|^2$$ holds for any $z \in \mathbb C $ $\...
0 votes
0 answers
30 views

Case of equality in Jensen's Inequality

I haven't really understood parts of the proof given for $\textrm{Problem 6.2}$ of Steele's Cauchy-Schwarz Master Class. Suppose that $f : [a, b] → \mathbb{R} $ is strictly convex and show that if \...
1 vote
1 answer
40 views

Determine all $(p,q)$ where Lorentz quasi-norm $\|\cdot\|_{L^{p,q}}$ is norm

On a measure space $(X,\mu)$, for $0<p,q<\infty$ the Lorentz space $L^{p,q}(\mu)$ is defined by $$\|f\|_{L^{p,q}(\mu)}:=p^\frac1q\|t\cdot\mu(|f|>t)^\frac1p\|_{L^q(\mathbb R_+,\frac{dt}t)}=p^\...
-2 votes
0 answers
54 views

The sum of five natural numbers, two of which are greater than 40, is 127. Accordingly, what is the largest of these numbers?

What I think the question wanted to convey is that we should find the largest possible value in $a,b,c,d,e$ so that their sum is $127$, but so that two of them are greater than $40$. The given (...
0 votes
2 answers
65 views

Inequality with logarithms and radicals of order 4

The statement of the problem : If $x,y \in (1, \infty)$ , prove that $\sqrt[4]{x} \bullet y^{\log_x y} \ge y$ My approach : If x = y , we have that $y^{\log_x y} = y$ , and with the fact that $x > ...
2 votes
1 answer
135 views

Looking for an existing proof for a property of triangles

In my paper, I need the following lemma. I can prove it, but it is a little lengthy to be put inside the paper. I am wondering is there any existing proof that I can quote. Lemma 1: Let the nodes of ...
7 votes
2 answers
300 views

If the roots of $z^4+az^3+bz^2+z$ are distinct and concyclic in the complex plane, does $ab\in\mathbb R$ imply $1<ab<9$?

HMMT February 2022, Team Round, Problem 6 (proposed by Akash Das) is: Let $\operatorname{\it P\!}{\left(x\right)}=x^4+ax^3+bx^2+x$ be a polynomial with four distinct roots that lie on a circle in the ...
2 votes
2 answers
94 views

Is there a spelling mistake or am I missing something

Here, $[ \cdot]$ is $\lfloor \cdot \rfloor $ floor function. N $ \in N$. Where did $\frac{[Nx]} N + \frac{1}{2N}$ came from and how does $x$ differs by $\frac{1}{2N}$. Shouldn't it be $\frac{1}N$ if ...
0 votes
1 answer
56 views

Demonstrate that: $\frac{1-a^2+c^2}{5c(3a+2b\sqrt{2})}+\frac{1-b^2+a^2}{5a(3b+2c\sqrt{2})}+\frac{1-c^2+b^2}{5b(3c+2a\sqrt{2})} \geq \frac{6}{5}$

The question Let $a,b,c \in (0,\infty)$ with $a^2+b^2+c^2=\frac{1}{4}$. Demonstrate that: $$\frac{1-a^2+c^2}{5c(3a+2b\sqrt{2})}+\frac{1-b^2+a^2}{5a(3b+2c\sqrt{2})}+\frac{1-c^2+b^2}{5b(3c+2a\sqrt{2})} \...
2 votes
1 answer
101 views

Prove: $\sum_{i=1}^{n-k+1} \frac{x_{n-k+1}}{X_{i}} > \sum_{i=1}^{k} \frac{x_{k}}{X_{i}}$

Let $n$ and $k$ two natural numbers such that $ k < \frac{n}{2}$ and $n$ real numbers such that $x_{1} > x_2 > \ldots > x_{n} > 0$ I'm not 100% sure whether this inequality always holds....
1 vote
2 answers
50 views

How to prove this (geometrically correct) inequality?

Let $x_1, x_1^*\in \left[0,a\right]$ and $x_2, x_2^*\in \left[0,b\right]$ where $a, b \in \left(0,1\right)$ and $a<b$. Suppose \begin{equation*} \frac{x_1}{x_2}>\frac{a}{b} \end{equation*} and \...
4 votes
1 answer
175 views

For $x,y,z \in \mathbb{R} $, prove that $ x(x+y)^5+y(y+z)^5+z(z+x)^5 \geq \frac{32}{243}(x+y+z)^6 $

For real numbers $x,y,z$ ,prove that $$x(x+y)^5+y(y+z)^5+z(z+x)^5 \geq \frac{32}{243}(x+y+z)^6$$ For $x,y,z>0$ I have a simple solution: By Hölder inequality: $$\sum x(x+y)^5 \geq \frac{(\sum x(x+...
0 votes
4 answers
153 views

Need help showing $(a^p + b^p) \le (a^2 + b^2)^{p/2}$, where $p \ge 2$, and $a,b \ge 0$.

I've been so far trying to show: $(\frac{a^2}{a^2 + b^2})^{p/2} + (\frac{b^2}{a^2 + b^2})^{p/2} \le 1.$ Also, it holds true that $\frac{a^2}{a^2 + b^2} \le 1$ and $\frac{b^2}{a^2+b^2}\le 1.$ I'm ...
4 votes
5 answers
136 views

The triangle $ABC$ is right-angled in $A$. Prove that the inequality $(1-\sin B)(1-\sin C)\leq \frac{{(\sqrt{2}-1)}^2}{2}$ holds.

The question The triangle $ABC$ is right-angled in $A$. Prove that the inequality $(1-\sin B)(1-\sin C)\leq \frac{{(\sqrt{2}-1)}^2}{2}$ holds The idea Let's note the sides of the triangle as $a,b,c$ ...
5 votes
2 answers
136 views

Show $\root{-e}\of{e}<\ln2$ without a calculator

I tried manipulating the expression to come up with inequalities such as $1<e^{\frac{1}{e}}\ln2$. One idea I have is to show that $\lim_{x\to\infty}\left(\frac{1}{x}+\ln2\right)\left(\frac{1}{x}+1\...
3 votes
0 answers
79 views

On a step on a complex inequality with summation.

To make it short I have a doubt in the last step of a complex number inequality, the problem is the next one Let complex numbers $z_1,z_2,z_3,...,z_n$ all modulus $1$ and $z_1+z_2+z_3+...+z_{2012}=0$ ...
0 votes
0 answers
39 views

inequality about sequences satisfied$S_n\le \lambda n$

A nonnegative sequence $\{a_n\}$ satisfies that at least $n$ of the first $n^2$ terms of the sequence $\{a_n\}$ are greater than $n$, and the sum of the first $n$ terms $S_n\le \lambda n$ holds for ...
9 votes
2 answers
276 views

Prove that $\mathbb{E}\exp{\lambda\xi} \le \exp\left(\lambda^2 \Vert\xi\Vert_{\psi_2}^2\right)$

Problem: Let $\xi$ be a real random variable. We say that $\xi$ is $\psi_2$ when $\exists \lambda >0$ such that $\mathbb{E}\exp(\xi^2/\lambda^2) \le e$. We denote by $\Vert \xi \Vert_{\psi_2}$ the ...
3 votes
1 answer
173 views

Prove that $\mathbb{P}\left(\Vert A \Vert \ge cK(\sqrt{N} + \sqrt{K} + t)\right) \le 2\exp(-t^2)$

Problem: Let $A = (a_{ij}): l_2^k \to l_2^N$ be a random matrix whose coefficients $a_{ij}$ are independent, centered, $\psi_2$ (the definition is in this post) and satisfy. \begin{align*} \...
2 votes
1 answer
43 views

An inequality about the $BMO$ space from Garfako's Modern Fourier Analysis

In Garfako's Modern Fourier Analysis I came across the following inequality: $$ \left| \left| f \right|-\underset{Q}{\mathrm{Avg}}\left| f \right| \right|\le \left| f-\underset{Q}{\mathrm{Avg}}f \...
0 votes
1 answer
31 views

$e^{xL}(x+x^3)<\frac{3\pi^2x}{L^2}$ for $x>0$ small enough and $L>0$ small enough

I'm trying to prove that $e^{xL}(x+x^3)<\frac{3\pi^2x}{L^2}$ for $x>0$ small enough and $L>0$ small enough and fixed, that is, there exists $\delta>0$ such that $e^{xL}(x+x^3)<\frac{3\...
3 votes
2 answers
71 views

Information-theoretic Inequality

If we have two discrete RVs, X, and Y. How can we show: $$\sum_{x,y} p(x|y)p(y|x) \geq 1.$$ The question goes further with finding a sufficient and necessary condition for equality. My attempt: For ...
2 votes
0 answers
51 views

How to prove this inequality based on several inequalities?

Consider $x$, $y$, $z$, $w$, all in $\left(0,1\right)$. Suppose $\frac{1-z}{w}<\frac{1-x}{y}<1<\frac{z}{1-w}<\frac{x}{1-y}$. I want to prove $\frac{w\left(1-x\right)-y\left(1-z\right)}{x+y-...
26 votes
7 answers
1k views

Show by hand : $e^{e^2}>1000\phi$

Problem: Show by hand without any computer assistance: $$e^{e^2}>1000\phi,$$ where $\phi$ denotes the golden ratio $\frac{1+\sqrt{5}}{2} \approx 1.618034$. I come across this limit showing: $$\...
1 vote
0 answers
30 views

Does the solution exist for the linear matrix inequality $A^TP+PA-Q<0$, where $P=P^T>0$, $Q=Q^T>0$? [closed]

I am studying the consensus problem of a linear multi-agent system, and a control gain needs to be designed to reach the consensus property. But the design procedure depends on the solution of the ...
16 votes
3 answers
1k views

Hardy's Inequality: Problems $3.14$ and $3.15$ in Rudin's RCA

In Problem $3.14$, we prove (a) Hardy's inequality, (b) the condition for equality, and I shall talk about (c), (d) below. Problem $3.15$ is the discrete case of Hardy's inequality. I have asked three ...
-1 votes
2 answers
87 views

For $a,b,c>0$ and $a+b+c=1$ prove $\frac{1}{ab+2c^{2}+2c}+\frac{1}{bc+2a^{2}+2a}+\frac{1}{ca+2b^{2}+2b}\ge \frac{1}{ab+bc+ca}$ [closed]

For $a,b,c>0$ and $a+b+c=1.$ Prove$:$ $$\frac{1}{ab+2c^{2}+2c}+\frac{1}{bc+2a^{2}+2a}+\frac{1}{ca+2b^{2}+2b}\geqq \frac{1}{ab+bc+ca}$$ well i try to use AM-GM but turns out: $$\frac{9}{4(a^{2}+b^{...
0 votes
1 answer
20 views

Young's inequality for scalar multiplied with absolute value of function

Young's inequality states the following: For scalar time functions $x(t)\in\mathbb{R}$ and $y(t)\in\mathbb{R}$, the following holds: $$ x\cdot y \leq \frac{1}{2\cdot \epsilon}\cdot x^2 + \frac{\...
7 votes
2 answers
646 views

Relation between Gaussian width and its squared version

I'm currently reading through Roman Vershynin's High Dimensional Probability and working through one of the exercises (7.6.1). Consider a set $T \subseteq \mathbf{R}^n$ and define its Gaussian width $...
2 votes
2 answers
48 views

Prove that $a^2+b^2+c^2 \geq ab(a+b+\sqrt{ab})+cb(c+b+\sqrt{cb})+ ac(a+c+\sqrt{ac} )$

the question Let $a,b,c$ be positive numbers such that $a+b+c=1$. Prove that $a^2+b^2+c^2 \geq ab(a+b+\sqrt{ab})+cb(c+b+\sqrt{cb})+ ac(a+c+\sqrt{ac} )$. the idea After i put some values to $a$, $b$, ...
1 vote
2 answers
99 views

How to solve the following mixed exponential inequality: $3^x + x < 4$? (from Spivak's Calculus)

I have encountered this inequality in Spivak's Calculus (first chapter exercises), which I'm not sure how to solve. $$ 3^x + x < 4 $$ I might be wrong but my gut feeling says the inequality holds ...

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