Questions on proving and manipulating inequalities.

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5
votes
4answers
101 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 ...
2
votes
4answers
30 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
1answer
20 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
22 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
73 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?
6
votes
3answers
122 views

How to prove $(\frac{n+1}{e})^n<n!<e(\frac{n+1}{e})^{n+1}$ without integrating method?

How to prove $$\left(\frac{n+1}{e}\right)^n<n!<e\left(\frac{n+1}{e}\right)^{n+1}$$ without integrating method? In fact we could prove this by noticing that $$i<x<i+1\Rightarrow \ln ...
13
votes
3answers
301 views

when does $\det(AB^T+BA^T)\le \det(AA^T+BB^T)$ hold?

When does the following matrix inequality hold? $$\det(AB+B^TA^T)\le \det(AA^T+BB^T)$$ $A$ and $B$ are any real matrices. My reply gives a counter example. The question is under what condition ...
4
votes
1answer
70 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 ...
3
votes
1answer
84 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? ...
0
votes
1answer
29 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 $n\in\mathbb N$?
14
votes
0answers
309 views
+50

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

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 ...
5
votes
2answers
146 views

Isoperimetric inequality, isodiametric inequality, hyperplane conjecture… what are the inequalities of this kind known or conjectured?

Question: Which inequalities similar to the famous isoperimetric inequality is known? conjectured? I recently learned about some inequalities which are all similar to the famous isoperimetric ...
0
votes
2answers
51 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.
5
votes
1answer
131 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!
7
votes
1answer
82 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
15 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\| ...
7
votes
3answers
92 views

If $a,b,c,d>0$ and $a+b+c+d=4$, then $\frac{1}{11+a^2}+\frac{1}{11+b^2}+\frac{1}{11+c^2}+\frac{1}{11+d^2} \leq \frac {1}{3}$

Prove if $a,b,c,d>0$ and $a+b+c+d=4$, then $$\dfrac{1}{11+a^2}+\dfrac{1}{11+b^2}+\dfrac{1}{11+c^2}+\dfrac{1}{11+d^2} \leq \dfrac {1}{3}$$ This was an Inequality Olympiad Problem1. I proved by ...
-3
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) ...
7
votes
1answer
350 views

Putnam PigeonHole

This is from page 12 of Putnam and Beyond. Problem: Prove that for every set $X ={x_1,x_2, \ldots ,x_n}$ of $n$ real numbers, there exists a nonempty subset $S$ of $X$ and an integer $m$ such that ...
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
0answers
52 views

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

Question: 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 ...
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 ...
0
votes
1answer
223 views

Combining inequalies into one inequality

Let's say we are given $a$, $b$, $d$ with $1 \leq a, b, d \leq 1000$ and inequalities $x \geq a$, $y \geq b$, and $a+b < x + y \leq a+b+d$. I need to combine all this and the following into one ...
4
votes
1answer
67 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 = ...
0
votes
1answer
18 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 ...
10
votes
5answers
334 views

How prove this $2(x^4+y^4+z^4)+2xyz+7\ge 5(x^2+y^2+z^2)$

let $x,y,z\ge 0$,show that $$2(x^4+y^4+z^4)+2xyz+7\ge 5(x^2+y^2+z^2)$$ my idea: let $$x+y+z=p,xy+yz+xz=q,xyz=r$$ since $$x^2+y^2+z^2=(x+y+z)^2-2(xy+yz+xz)=p^2-2q$$ and ...
-4
votes
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
2
votes
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 ...
6
votes
3answers
161 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 $a,b,c > 0$ ?
3
votes
1answer
126 views

Extra help on inequality

Someone very helpfully provided an answer to an inequality. See Hard Olympiad Inequality However I don't get part of their answer. How did they get the last factorization??? Thanks so much for any ...
1
vote
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 ...
1
vote
0answers
37 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 ...
1
vote
2answers
166 views

Average limit superior [duplicate]

Let $\mathcal{l}_\mathbb{R}^\infty$ be the space of bounded sequences in $\mathbb{R}$. We define a map $p: \mathcal{l}_\mathbb{R}^\infty\to\mathbb{R}$ by $$p(\underline x)=\limsup_{n\to\infty} ...
3
votes
0answers
64 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 ...
4
votes
2answers
227 views

How prove this $|1+x|^a\ge 1+ax+\dfrac{1}{1000}|x|^a$

let $2\le a\le 13,a\in R$,and $x\in R$,show that: $$|1+x|^a\ge 1+ax+\dfrac{1}{1000}|x|^a\tag{1}$$ My try: let $$f(x)=|1+x|^a-1-ax-\dfrac{1}{1000}|x|^a$$ and since if $x>-1$,then ...
0
votes
1answer
372 views

generalized inequalities defined by proper cones

The generalized inequality defined by a proper cone $K$ is that $x \ge_{K} y$ if $x-y \in K$ for $x,y \in K$. Does this means that for any $x \in K$, we have $x \ge_{K} 0$ since $x - 0 = x \in K$ ? ...
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) ...
2
votes
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 ...
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
0
votes
2answers
21 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
1answer
110 views

How can I prove that $1+x \geq \exp({x/(1+x)})$?

I am dilemma how can I prove that $1+x \geq \exp({x/(1+x)})$. Any hints and suggestions will be much appreciated.
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
7answers
113 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
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 ...
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 ...
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 ...
0
votes
1answer
14 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, ...
14
votes
3answers
368 views

How to prove this inequality $\frac{x^y}{y^x}+\frac{y^z}{z^y}+\frac{z^x}{x^z}\ge 3$

let $x,y,z$ be positive numbers, and such $x+y+z=1$ show that $$\dfrac{x^y}{y^x}+\dfrac{y^z}{z^y}+\dfrac{z^x}{x^z}\ge 3$$ My try: let ...