Questions on proving and manipulating inequalities.

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4
votes
1answer
53 views

Prove that $\frac{1}{\sqrt{1+a^2}}+\frac{1}{\sqrt{1+b^2}}+\frac{1}{\sqrt{1+c^2}}\le\frac{3}{2}$

For positive real numbers with $a+b+c=abc$ prove that $$\frac{1}{\sqrt{1+a^2}}+\frac{1}{\sqrt{1+b^2}}+\frac{1}{\sqrt{1+c^2}}\le\frac{3}{2}$$ I made the substitution $a=\tan(\alpha), b = ...
5
votes
4answers
73 views

For all reals $x$, prove $2^x > x$

How can I prove that for all reals $x$, $2^x > x$? I can prove this for integers with induction, but I can't figure out how to prove it for reals. Perhaps you could say that since $2^x$ is strictly ...
0
votes
1answer
16 views

Norm and Matrice Proof

I'm trying to show that the following statement is true: If $||\mathbf a - \theta \mathbf b||^2 - ||\mathbf a||^2 \geq 0$ for all $\theta \in [0,1]$, then $\mathbf a^T \mathbf b \le 0$. Is this ...
1
vote
0answers
35 views

How prove this $|S_{1}|-|S_{2}|\le 2^{2n}\binom{2n}{n}$

let $n\in N^{+}$,and define set $S=\{1,2,\cdots,4n\}$, for any$ a<b\in R^{+}$,defind $$S_{1}=\{X|X\subseteq S,a\le S(X)\le b,S(X)\equiv 1\pmod 2\}$$ $$S_{2}=\{X|X\subseteq S,a\le S(X)\le ...
-4
votes
0answers
21 views

Quadratic Equation Inequality [on hold]

a,b,c,p,q,r are real numbers such that ax^2+bx+c>=0,px^2+qx+r>=0 For all real numbers prove that apx^2+bqx+cr>=0
1
vote
0answers
29 views

Prove or disprove $\sum_{j=1}^{n}\sum_{i=1}^{n}|a_{i}-a_{j}|^2|b_{ij}|^2\le \cdots$

let $a_{i},b_{ij}\in C,i=1,2,\cdots,n,j=1,2,\cdots,n$,prove or disprove $$\sum_{j=1}^{n}\sum_{i=1}^{n}|a_{i}-a_{j}|^2|b_{ij}|^2\le ...
2
votes
3answers
41 views

Proof by induction of a recursive sequence

I am studying CIE A levels Further Maths and I am stuck at a question from June 2002: Q The sequence of positive numbers $u_1,u_2,u_3,...$ is such that $u_1<4$ and ...
2
votes
0answers
20 views

Inequality similar to Hoeffding

I have a coin with heads probability $p_1$. I toss it $n_1$ times. Let $\hat{p}_1$ be the empirical heads probability. Then we know from Hoeffding that $$P\left( \left|\hat{p}_1-p_1 \right| \geq ...
0
votes
2answers
20 views

Inequality: $\tan(x) > 1$

So far, I've not come very... far. It ends up with me trying to solve it more intuitively than mathematically. I figured, first I'll find the place of equality, which is at $x = \arctan 1 = ...
2
votes
4answers
46 views

If $a^2+b^2=1$ where $a,b>0$ then find the minimum value of $(a+b+{1\over{ab}})$

If $a^2+b^2=1$ where $a,b>0$ then find the minimum value of $(a+b+{1\over{ab}})$ This can be easily done by calculas but is there any way to do do this by algebra
1
vote
1answer
12 views

Determine the values of x for which the linear approximation is accurate to within 0.1.

So I've got a function: $$\frac{1}{(1+2x)^4}$$ with its linear approximation: $$1-8x$$ For all values of $x$ where the linear approximation is accurate within $0.1$, then surely we subtract the ...
5
votes
1answer
111 views

determinant inequality $ \det(A^2+B^2+(A-B)^2)\ge 3\det(AB-BA) $

A and B are two $2\times2$ reals matrices. then $$ \det \Big(A^2+B^2+(A-B)^2\Big)\ge 3\det(AB-BA) $$ well, it is seems interesting, but it is really hard to get started Thank you very much!
0
votes
0answers
44 views

what is the difference between $\log_ax^2$ and $2\log_ax$

when are we allowed to make the use of the formular $\log_ab^c=c\log_ab$ for example solving the logarithmic inequality $\log _2(-t)+\log_2(t)^2<3$ is solveable if $t <0$ but $\log ...
0
votes
1answer
34 views

Solve a system of inequalities

$$\begin{cases} \log_{2}^{2}(-\log_{2}x) + \log_{2}\log_{2}^{2}x \leq 3 & \\-4 |x^2-1|-3\geq \frac{1}{x^2-1}& \end{cases}$$ What I've tried: Make substitution $t=x^2-1$ and solve second ...
5
votes
7answers
111 views

Find value range of $2^x+2^y$

Assume $x,y \in \Bbb{R}$ satisfy $$4^x+4^y = 2^{x+1} + 2^{y+1}$$, Find the value range of $$2^x+2^y$$ I know $x=y=1$ is a solution of $4^x+4^y = 2^{x+1} + 2^{y+1}$ , but I can't go further more. I ...
0
votes
1answer
15 views

Increasing rate of a continuous function

Consider $f: X \rightarrow X$ continuous, with $X \subset \mathbb{R}^n$ compact convex. I am wondering on conditions on $f$ so that there exists $\epsilon > 0$ such that $$ (x-y)^\top \left( f(x) ...
0
votes
1answer
12 views

How to prove that the spectral radius of a linear operator is the infimum over all subordinate norms of the corresponding norm of the operator.

I am trying to understand a proof I have seen of the following theorem: $$\rho(A)=\inf_{\|\cdot\|}\|A\|.$$ I understand that to do this, the idea is to show that 1) $\rho(A)\leq\|A\|$ for any norm, ...
1
vote
1answer
44 views

Strong induction inequality proof

Use strong induction to prove that $$\frac{1}{2^3}+\frac{1}{3^3}+\cdots+\frac{1}{n^3}\leq\frac{5}{8}-\frac{1}{n}$$ $$n\geq2$$ I'm not sure how to go about this. I used base cases n=2, and n=3 but ...
-2
votes
1answer
28 views
0
votes
1answer
20 views

Redefine a discrete compact set

I need to find twice continuously differentiable functions $g_i: \mathbb{R}^2 \rightarrow \mathbb{R}$ $i=1,\ldots,I$ such that the set $\{ 0,1\}^2=\{(0,0), (1,0), (0,1), (1,1) \}$ can be written as ...
1
vote
0answers
34 views

Show that $\liminf\limits_{n\rightarrow\infty}a_{n}/a_{n-k}\leq \liminf\limits_{n\rightarrow\infty}a_{n}^{k/n}$

Here is an exercise: Let $\{a_{n}\}$ be a positive increasing sequence, can we prove that: $\liminf_{n\rightarrow\infty}\frac{a_{n}}{a_{n-k}}\leq \liminf_{n\rightarrow\infty}a_{n}^{k/n}$? Could ...
0
votes
0answers
14 views

Inequality/ Convergence for two operators with functional calculus

Given a sequence of functions $f_n \to f$ in $L^\infty(\mathbb{R}^2)$ and two self-adjoint, unbounded operators $A, B$ is it true that $\|f_n(A,B) - f(A,B)\| \to 0$? With only one operator I can ...
2
votes
1answer
27 views

How Find this maximum $P=\frac{4}{\sqrt{a^2+b^2+c^2+4}}-\frac{9}{(a+b)\sqrt{(a+2c)(b+2c)}}$

let $a,b,c>0$, find the maximum $$P=\dfrac{4}{\sqrt{a^2+b^2+c^2+4}}-\dfrac{9}{(a+b)\sqrt{(a+2c)(b+2c)}}$$ I think this inequality we can use AM-GM inequality to solve it,and Now first we ...
0
votes
1answer
22 views

Define a compact and convex set through inequality constraints

I need to find twice continuously differentiable functions $g_i: \mathbb{R}^2 \rightarrow \mathbb{R}$ $i=1,...,I$ such that the set $\{ 0,1\}^2=\{(0,0), (1,0), (0,1), (1,1) \}$ can be written as $\{x ...
4
votes
3answers
49 views

Help verifying my proof that for any $\epsilon>0$, there exists rational $\frac{j}{k}$ such that $0<\frac{j}{k}<\epsilon$

I'm trying to prove that that for any $\epsilon>0$, there exists rational $\frac{j}{k}$ such that $0<\frac{j}{k}<\epsilon$. Obviously $j,k\in \mathbb{N}$. This is not for homework, it's a ...
1
vote
1answer
14 views

2 Linear equation problems [on hold]

Write objective, constraints and graph for the following two problems: 1.A test offers 2 types of problems. Type A takes 3 Min to solve and B takes 2. You have 20 min to take the test and can only ...
0
votes
3answers
37 views

a simple inequality

Is it true that for any real numbers (a,b): $(a - b)^{2} \leq 3a^{2} + 3b^{2}$ Also, if this is true, is there a way to sharpen this bound say $(a - b)^{2} \leq K(a^{2} + b^{2})$, for some $K < ...
2
votes
2answers
40 views

Prove the inequality.Let a, b and c be nonnegative real numbers.

Let $a$, $b$ and $c$ be nonnegative real numbers. Prove that $a^4+b^4+c^2\ge 8^{½}abc$
1
vote
2answers
49 views

Proving that the line integral $\int_{\gamma_{2}} e^{ix^2}\:\mathrm{d}x$ tends to zero

Let $f(z) = e^{iz^2}$ and $\gamma_2 = \{ z : z = Re^{i\theta}, 0 \leq \theta \leq \frac{\pi}{4} \} $. All the sources I have found online, says that the line integral $$ \left| \int_{\gamma_2} ...
4
votes
3answers
84 views

Prove the inequality $({1+\frac{a}b})^n$ + $(1+\frac{b}a)^n$ $\geq$ $2^{n+1}$

Let $a$ and $b$ be positive real numbers and let $n$ be a natural number prove that $$\left({1+\frac ab}\right)^n+\left(1+\frac ba\right)^n\ge2^{n+1}.$$
-4
votes
0answers
40 views

Which of the following is correct? [on hold]

Let $$X = \frac{1}{1001} + \frac{1}{1002} + \frac{1}{1003} + \cdots + \frac{1}{3001}$$ Then find the correct option. (A) $X < 1$ (B) $X > \frac{3}{2}$ (C) $1 < X < \frac{3}{2}$ (D) ...
2
votes
0answers
32 views

Integral inequality related to derivation

While trying to understand a proof, i have stumbled upon the following statement: Let $f \in L^p(a,b)$ be a $p$-integrable function. Then the inequality $$\liminf_{s \rightarrow t} \frac{1}{t-s} ...
3
votes
0answers
54 views
+50

How prove this general inequality $a\left(\frac{\sin{x}}{x}\right)^m+b\left(\frac{\tan{x}}{x}\right)^n>a+b$

if $$m,n<0,a,b>0,a\left[\left(\dfrac{2}{\pi}\right)^m-1\right]\ge b,am\le 2bn$$ show that $$a\left(\dfrac{\sin{x}}{x}\right)^m+b\left(\dfrac{\tan{x}}{x}\right)^n>a+b,\forall ...
2
votes
3answers
51 views

Set of solutions for a binomial inequality

I bumped into the following inequality: $${a-b\choose c}{a\choose c}^{-1} \le \exp\left(-\frac{bc}{a}\right)$$ Playing with it a little bit, trying to bound it asymptotically for large $a$'s, using ...
1
vote
1answer
27 views

Prove trace inequality $\mathrm{tr}\{ABCBAD-ABCD-ADCB+CD\} \geq 0$

Let $\mathbf{A}$, $\mathbf{B}$, $\mathbf{C}$, $\mathbf{D}$ be four (generally non-commuting) positive semidefinite matrices of same size. I want to show that (or find a counterexample to) $$ ...
1
vote
1answer
20 views

For $1-r<|\theta|<1/2$, $|\frac{2r\sin{2\pi\theta}}{1-2r\cos{2\pi\theta}+r^2}-\frac{\cos\pi\theta}{\sin\pi\theta}|<C\frac{(1-r)^2}{|\theta|^3}$

For $1-r<|\theta|<1/2$ show that $$|\frac{2r\sin{2\pi\theta}}{1-2r\cos{2\pi\theta}+r^2}-\frac{\cos\pi\theta}{\sin\pi\theta}|<C\frac{(1-r)^2}{|\theta|^3}$$ This inequality shows that ...
3
votes
1answer
43 views

Very loose bound on sum of first binomials

Let $n\geq k\geq 2$. Is it always true that $$\binom{n}{0}+\binom{n}{1}+\cdots+\binom{n}{k}\leq n^k?$$ The left-hand side is dominated by the term $\dfrac{n^k}{k!}$, so the statement should be true. ...
0
votes
1answer
27 views

How to use induction on this type of inequality?

Given $a_1,a_2,\ldots,a_n>0$ and $a_1+a_2+\ldots+a_n<\frac{1}{2}$, prove that $(1+a_1)(1+a_2)\ldots(1+a_n)<2$. Some of you may have already seen this inequality. I was the one who asked ...
6
votes
3answers
151 views

An Inequality Problem with not nice conditions

How to show that $\dfrac{a^3}{a^2+b^2} + \dfrac{b^3}{b^2+c^2} + \dfrac{c^3}{c^2+a^2} \ge \dfrac32$, where $a^2+b^2+c^2=3$, and $a,b,c > 0$ ?
-1
votes
0answers
12 views

Inequality about ceiling function. How to prove it?

Proposition: Set $n,m\in{\mathbb{Z}}$ with $n\geq{2}$, then $$\lceil{\frac{n}{3}}\rceil\lceil{\frac{n}{3}}\rceil\leq\lceil{\frac{nm}{5}}\rceil$$ I've verified it for small cases (computationally), ...
0
votes
3answers
65 views

Is there any book about inequality? [closed]

I heard there is a book name 'inequality'. But I couldn't find the book. Is there any site or book about inequalities? What i want is collection of inequalities.
2
votes
0answers
56 views

Rising Sun Inequality (Dunford-Schwartz maximal inequality)

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be an absolutely integrable function, and let $f^*:\mathbb{R} \rightarrow \mathbb{R}$ be the one-sided signed Hardy-Littlewood maximal function $$f^*(x) := ...
0
votes
1answer
41 views

Induction proof that $4^n > 3^n+2^n$ for $n\ge2$

This is a problem with induction and proofs but I'm not sure how to start with proving this one. $$\text{Show that for any $n \geq 2$, $4^n > 3^n+2^n$}$$
0
votes
4answers
81 views

Proof that $(1 + x)^n > 1 + nx$ for $x>-1$, $n$ a positive integer [duplicate]

For any positive integer $n$ and real number $x > -1$, show that $(1 + x)^n > 1 + nx$. This is Bernoulli’s inequality but I can't figure out how to start with this. Can someone help? Thanks
0
votes
1answer
19 views

Proving elementary inequalities with equations

Assume $b > 0,\ d > 0$. Assume: $$ \frac{a}{b} < \frac{c}{d} $$. Prove that: $$ \frac{a}{b} < \frac{a + c}{b + d} < \frac{c}{d} $$. I would like to find an intuitive way to solve ...
5
votes
2answers
56 views

How to show $a+b+ad\geq c+d+bc$ given $a\geq c$ and $a+b\geq c+d$?

Let $0\leq a,b,c,d\leq 1$ and $a\geq c$ and $a+b\geq c+d$. Show that $a+b+ad\geq c+d+bc.$ Of course we have $a+b\geq c+d$, but how to relate $ad$ and $bc$?
0
votes
2answers
47 views

Inequality with moments

Let $m$ a probability measure, $f$ a positive measurable function (one can assume it is bounded, the existence of the moments is not a problem here). Is $m(f^3) \le m(f^2) m(f)$?
11
votes
0answers
203 views
+50

How prove this inequatity $\frac{x^2}{y}+\frac{y^2}{z}+\frac{z^2}{x}\ge 4+(x-y)^2$

let $x,y,z>0$,and such $$4\le x+y+z\le 5$$ show that $$\dfrac{x^2}{y}+\dfrac{y^2}{z}+\dfrac{z^2}{x}\ge 4+(x-y)^2$$ It seem $\dfrac{x^2}{y}+\dfrac{y^2}{z}+\dfrac{z^2}{x}\ge 4+(x-y)^2$ maybe is ...
6
votes
2answers
84 views

Prove that positive $x,y$ satisfy $\left(\frac{1}{1+x}\right)^2+\left(\frac{1}{1+y}\right)^2\ge\frac{1}{1+xy}$.

Prove that positive $x,y$ satisfy $$\left(\frac{1}{1+x}\right)^2+\left(\frac{1}{1+y}\right)^2\ge\frac{1}{1+xy}$$ My teacher claims this lemma is often useful. I'm wondering, though: how to prove ...
5
votes
3answers
93 views

Find min of $\frac{\left( \sum kx_k \right) \left( \sum x^2_k \right)}{\left( \sum x_k \right)^3}$

With $n \ge 2$ and $x_1,\ x_2,\ \dots,\ x_n > 0$. Find the minimum of: $$ M = \frac{(x_1 + 2 x_2 + ...+ nx_n)( x^2_1 + x^2_2 +...+x^2_n)} {\left( x_1 + x_2 +...+ x_n \right)^3}$$ For specific ...