Questions on proving and manipulating inequalities.

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0
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1answer
16 views

Cauchy-Schwartz Inequality - Proof using Projections [Lay P379 Thm 6.7.16]

If $u=0$, then the inequality becomes $ 0 \le 0 $, which is true. See Practice Problem 6.7.1 on P382. If $u\neq 0$, let $W$ be the subspace spanned by $u$. $1.$ How would one determine to ...
3
votes
1answer
42 views

How find the $AP+\frac{1}{2}BP$ minmum value

An equilateral triangle $ABC$ such $$AB=BC=AC=2a>0$$ A circle $O$ is inscribed in triangle $ABC$,and the point $P$ on the circle $O$. Find the minimum $$AP+\dfrac{1}{2}BP$$ My idea: let ...
5
votes
2answers
69 views

Is $\int_x^{\infty}e^{-\frac{t^2}{2}} < \frac{1}{x}e^{-\frac{x^2}{2}}$?

While solving a problem in real analysis, I got stuck. I need to prove $$\int_x^{\infty}e^{-\frac{t^2}{2}}dt < \frac{1}{x}e^{-\frac{x^2}{2}} $$ Clearly I have to use some kind of inequality, but ...
0
votes
1answer
23 views

$C^{-1} (1+|x|^{2})^{\frac{s}{2}} \leq (1+|x|)^{\frac{s}{2}} \leq C (1+|x|^{2})^{\frac{s}{2}}$?

Let $s\in \mathbb R,$ and define $f: \mathbb R^{n}\to [0, \infty)$ such that $f(x)= (1+|x|^{2})^{\frac{s}{2}}, (x\in \mathbb R^{n})$ and $g:\mathbb R^{n}\to [0, \infty)$ such that $g(x)= ...
7
votes
0answers
27 views

How prove this inequality $\sum_{cyc}\frac{x^a\ln{x}}{(x^a+y+z)^2}\ge 0$

Question: let $x,y,z$ be postive numbers,and such $xyz\ge 1$,and such $a$ is real numbers.show that $$\dfrac{x^a\ln{x}}{(x^a+y+z)^2}+\dfrac{y^a\ln{y}}{(y^a+x+z)^2}+\dfrac{z^a\ln{z}}{( ...
6
votes
4answers
76 views

A question about inequality ${(n+1)\over e^n}^n<n!$

How to prove the inequality $${(n+1)\over e^n}^n<n!$$ I have tried mathematical induction, but it doesn't work! Are there other methods to solve it?
4
votes
1answer
71 views

A tough inequality problem with condition $a+b+c+abc=4$

If, $a+b+c+abc=4$, with $a,b,c$ being positive reals, then prove or disprove the following inequality: $$\frac{a}{\sqrt{b+c}}+\frac{b}{\sqrt{a+c}}+\frac{c}{\sqrt{a+b}}\geq\frac{a+b+c}{\sqrt2}$$ I ...
0
votes
1answer
31 views

Inequality, $\left(\frac{2}{x}+2\right)^{n}-\left(\frac{2}{x}-2\right)^{n}\leq \left(\frac 4 x \right)^n$

How do I show that $$\left(\frac{2}{x}+2\right)^{n}-\left(\frac{2}{x}-2\right)^{n}\leq \left(\frac 4 x \right)^n$$ for $x\in\left(0,1\right]$ and $n\in\mathbb N$?
3
votes
1answer
89 views

counterexamples to $ \det \Big(A^2+B^2\Big)\ge \det(AB-BA) $

$n\geq3$. A and B are two $n\times n$ reals matrices. For $n\times n$, Could one give counterexamples to show that $$ \det \Big(A^2+B^2\Big)\ge \det(AB-BA) \tag{$*$}$$ is not necessarily true? ...
2
votes
4answers
35 views

If $9^{x+1} + (t^2 - 4t - 2)3^x + 1 > 0$, then what values can $t$ take?

If $9^{x+1} + (t^2 - 4t - 2)3^x + 1 > 0$, then what values can $t$ take? This is what I have done: Let $y = 3^x$ $$9^{x+1} + (t^2 - 4t - 2)3^x + 1 > 0$$ $$\implies9y^2 + (t^2 - 4t - 2)y + ...
0
votes
2answers
55 views

Minimum value of $\sqrt{(1+1/y)(1+1/z)}$

If $y,z > 0$ and $y + z = c$ where $c$ is a constant, then what's the minimum value of $$\sqrt{\left(1+\frac1y\right)\left(1+\frac1z\right)}$$ I am having a hard time solving this.
7
votes
1answer
85 views

How to learn inequalities and become good at proving them?

I am taking a real analysis course next year and I want to start slowly preparing for that class now, so I hope you can help me. The class is quite challenging and the fail rate is relatively high. ...
1
vote
3answers
45 views

Quadratic equations and inequalites

For every positive integer $n$, prove that $$\sqrt{4n+1}<\sqrt{n} + \sqrt{n+1}<\sqrt{4n+2}$$ Hence or otherwise, prove that $[\sqrt{n}+\sqrt{n+1}] = [\sqrt{4n+1}]$, where $[x]$ ...
0
votes
1answer
17 views

Nonexpansive Affine Operators in Hilbert spaces

Let $H$ be a Hilbert space with real inner product $\langle \cdot, \cdot \rangle : H \times H \rightarrow \mathbb{R}$. Let $T$ be an affine operator. Show that $T$ is nonexpansive, i.e., $\left\| ...
1
vote
0answers
43 views

How prove this $\sin(2a)+\sin(2b)+\sin(2c)<\dfrac{\pi}{2}+2\sin a\cos b+2\sin b\cos c$

let $0<a<b<c<\dfrac{\pi}{2}$, use the integral inequality show that $$\sin(2a)+\sin(2b)+\sin(2c)<\dfrac{\pi}{2}+2\sin a\cos b+2\sin b\cos c$$ I know this problem can use The area of ...
1
vote
3answers
64 views

problem with inequality of modulus

how can I prove the following inequality? $$\frac{\mid x+y\mid}{1+\mid x+y\mid}\leq\frac{\mid x \mid}{1+\mid x\mid}+\frac{\mid y \mid}{1+\mid y \mid}$$ I was trying to prove it by ...
4
votes
1answer
68 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
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4answers
122 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
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1answer
19 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 ...
2
votes
0answers
67 views
+50

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

Question: Let $n\in \mathbf N^{+}$,and define set $S=\{1,2,\cdots,4n\}$, for any $ a<b\in \mathbf R^{+}$, define $$S_{1}=\{\,X\mid X\subseteq S,a\le S(X)\le b,S(X)\equiv 1\pmod 2\,\}$$ ...
-4
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0answers
22 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
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0answers
32 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
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3answers
42 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
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0answers
26 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
22 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
134 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
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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
114 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
15 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
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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 ...
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1answer
28 views
0
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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
39 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
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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
28 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
16 views

2 Linear equation problems [closed]

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 $$\left({1+\frac ab}\right)^n+\left(1+\frac ba\right)^n\ge2^{n+1}.$$
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votes
0answers
55 views

Which of the following is correct?

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
87 views
+50

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

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,\qquad\forall ...
2
votes
3answers
52 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 ...