Questions on proving and manipulating inequalities.

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Solve the inequality…

Can you please show me how can I solve this inequality. I would like to see how it can be done without the graph of the functions. Thank you! $$2\sqrt{(x-1)(x+2)}\ge|x+1|-2$$
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3answers
54 views

How to prove this inequation?

$$ 1+\frac{2}{3n-2}\leqslant \sqrt[n]{3}\leqslant 1+\frac{2}{n}, n\in \mathbb{Z}^{+} $$ How to prove this inequation?
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11 views

Schur-concave functions, derivative sign help

To establish some inequality I must prove: $$\dfrac{\partial}{\partial ...
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1answer
48 views

Solve the following inequality…

Can you please verify if I've done this exercise correctly, and if you have a better solution, please, show it to me. Thank you! (The exercise is in the left top corner.)
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1answer
149 views
+50

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 ...
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1answer
29 views

How find this value of $x$ such $\log_{\frac{1}{12}}{(x^2+2x-3)}>x^2+2x-16$

if $$\log_{\frac{1}{12}}{(x^2+2x-3)}>x^2+2x-16$$ Find the value of $x$ My idea: since $$x^2+2x-3>0\Longrightarrow x>1 ,or, x<-3$$ ...
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3answers
37 views

Find such anti-symmetric matrix $W$ that $A^T WP \geq 0$

$P$ and $A$ are both n-dimensional vectors with non-negative components. $W$ is an $n\times n$ matrix with $W_{ij}=w_i-w_j$, where all $w_k\geq 0$. So $W$ is an anti-symmetric matrix with some ...
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3answers
43 views

Inequality Exercise in Apostol's Calculus I

Let p and n denote positive integers. Show that: $$n^{p} \lt \frac{(n+1)^{p+1} - n^{p+1}}{p+1} < (n+1)^{p}$$ Attempt at Solution Using the identity $b^{p+1}-a^{p+1} = ...
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0answers
50 views

Inequality, symmetric function

Let $$F(x,y)=\dfrac{f\left(\frac{x}{x+y}\right)+f\left(\frac{y}{x+y}\right)}{x+y},$$ where $f>0$ is a concave function. Using brute force computation (computer based proof) with ...
4
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2answers
70 views

Show that $f(x_0)-f(x)<\vert x_0-x\vert$ for all $x \ne x_0$

Let $f: \mathbb{R}\rightarrow\mathbb{R}$ continuous and bounded below. Show that there exist $x_0 \in \mathbb{R}$ such that $\forall x\ne x_0$, $$f(x_0)-f(x)<\vert x_0-x\vert$$ Since $f$ ...
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1answer
35 views

AM-GM inequality

On the wikipedia page on "Nesbit's inequality", the fifth proof ends as follows: $$ \frac{x+z}{y}+\frac{y+z}{x}+\frac{x+y}{z}\geq 6$$ which is true, by AM-GM inequality. I am wondering if the ...
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0answers
22 views

Muirhead's Inequality (software?)

I just started learning about inequalities: Schur's, Karamata's, Muirhead's, etc... They are beautiful but it seems that in the case of more than two variables, some of the computations become a ...
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16 views

Inequality with nesting floors and ceilings

EDIT: I changed the inequality to a simpler more indicative format. I need to solve the following inequality for $x$ (find the minimum value of $x$) but I have trouble removing the ceiling and floors ...
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2answers
39 views

Prove a inequality about integral and summation

If $f(x)$ is monotonic increasing on the interval $a\leq x < \infty$, could we prove following inequality formally? \begin{equation} f(a+k) \leq \int_{a+k}^{a+k+1} f(t) dt \leq f(a+k+1) ...
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2answers
43 views

help with (simple?) inequality

Let $x\in(0,1)$ be any and let $0<a<b<\frac{1}{2}$, I need to show that $$1-(1-x^a)(1-x^{1-a})>1-(1-x^b)(1-x^{1-b}).$$ Any suggestions? References? In practice I need to solve a more ...
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3answers
56 views

How to apply the Hölder's inequality in a clever way?

Here is the problem: Let $f\in L^p(\mathbb R^n)\cap L^q(\mathbb R^n)$ and $s\in[p,q]$. Show that $f\in L^s(\mathbb R^n)$ I'm almost sure that this is a simple exercise on Hölder's inequality yet ...
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1answer
26 views

How to prove lower and upper bound for exponential sum?

A paper I'm reading implicitly uses the fact $$\sum\limits_{t=1}^n e^{-ta^2} \in \theta(\frac{1}{a^2})$$ (It uses the both $\leq$ and $\geq$ sides in the proofs). I'm able to prove that ...
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2answers
35 views

Solving Problem by different Method ( non-induction)

I have this problem , which I was able to prove it by induction, but I wonder could be solve by direct method ( for example combinatorial method). I want to find number of solution for $$0 \le ...
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2answers
17 views

Lowerbound on expectation of log sum of Bernoulli random variables?

I'm looking for a lower-bound on $E\left[\log \left(B + \sum_i a_i X_i\right)\right]$ where $X_i$ are Bernoulli random variables with $p(X_i = 1) = q_i$ and $a_i > 0, B > 0$. Because $X_i$ is ...
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2answers
41 views

What is the inequality which is used for prove this inequality?

Let $x,y,z,t$ be real numbers such that $x,y,z,t\geq1$ and $xyzt=16$. How to prove $$x-\frac{1}{x}+y-\frac{1}{y}+z-\frac{1}{z}+t-\frac{1}{t}\geq6$$ I want some hint. thank you very much
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3answers
35 views

Simple inequalities

Suppose $l,t\in[0,1]$ and $l+t\leq1$ I want to prove $1+l+t>6lt$. When $t=0$ or $l=0$, it is trivial, so I started with $l,t\neq0$ but I couldn't reach anywhere. I don't have time to write in ...
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0answers
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L1 norm of a trigonometric polynomial

For a real $x$, $f(x) = \sum_{k=-T}^{T}e^{ikx}$ is the well known Dirichlet kernel. It is also known that $\|f\|_{L_1}=\int|f(x)|dx \le C_1\log T + C_2$ for some $C_1,C_2$ independent of $T$. ...
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1answer
22 views

An integral inequality question.

If we have two functions $f,g:[a,b]\to\mathbb{R}$ and we know they are bounded, so: $\sup_{x\in[a,b]}|f(x)|=K$, and $\sup_{x\in[a,b]}|g(x)|=M$. Where $K,M$, are positive finite constants, which of ...
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38 views

Find the value range of $t$ [closed]

Assume $$f(x)=(x+2014)\ln (x+2014)$$ for $\forall x \geq -2013$, we have $$f(x)\geq \left(\frac{1}{3}t^2+\frac{2}{3}t\right)(x+2013)$$, Find the value range of $t$
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2answers
46 views

Find the minimum of $a+2b+3c$

Assume $a,b,c \in \Bbb{R}$ and all non-negative, satisfy $$c(a+b+c) \geq 2-ab$$. how to find the minimum of $$a+2b+3c$$? My solution: My method is to transform $c(a+b+c)\geq 2-ab$ into multiply ...
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2answers
33 views

let a,b,c be the sides of a triangle and $a+b+c=2$ then prove that $ a^2 + b^2 + c^2 +2abc < 2$

let a,b,c be the sides of a triangle and $a+b+c=2$ then prove that $ a^2 + b^2 + c^2 +2abc < 2$ I used the fact that $ a+b-c>0 $ and multiplied the 3 cyclic equations but coudn't reach the ...
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1answer
32 views

Complex numbers

If someone could help me with this question I would really appreciate it.For some reason I am getting a weaker version of these inequalities when applying triangle inequality. Let S be the interior ...
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1answer
29 views

How to prove an inequality

$a$, $b$, $c$, $d$ are rational numbers and all $> 0$. $\max \left\{\dfrac{a}{b} , \dfrac{c}{d}\right\} \geq \dfrac{a+c}{b+d}\geq \min \left\{\dfrac{a}{b} , \dfrac{c}{d}\right\}$ Hope someone ...
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2answers
29 views

Value range of parameter inequality.

Assume $a,b,c>0$ satisfies :$$5c-3a \leq b \leq 4c-a$$ and $$c\ln b \geq a + c\ln c$$. Find the value range of $\frac{b}{a}$. My approach: because $a>0$ and I want to construct the target ...
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29 views

How to find number of integral solutions [duplicate]

I wanted to know if it is possible to find number of positive integral solutions for the inequality: $$xy < N$$ where N is a positive integer, and $x$ and $y$ also can only be positive integers. ...
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60 views

Finding a uniform bound for this ugly expression

I am asked to find a positive bound $M$ independent of $t$ and $n$ for the following expression : ...
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1answer
29 views

Motivation behind steps in proof of Hoeffding Inequality

The lemma that is proved for proving Hoeffding's inequality is: If $a\leq X\leq b$ and $E[X]=0$, $E[e^{tX}] \leq e^{\frac{t^2(b-a)^2}{8}}$ Here's a link to the proof: ...
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2answers
20 views

How to prove this inequality for operator and function

How to prove this? $\sum_{k=1}^{\infty}|(Tf)_k|^2\leq ||T||^2||f||^2$ where $T$ is an operator and a function $f$. $(Tf)_k$ is the $k$-th coordinate of Tf. Should this involve Cauchy-Schwarz or the ...
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2answers
70 views

$\int_0^1 f(x)^2\le 1$ and $\int_0^1 f'(x)^2\le 1$ $\Rightarrow$ $\left|f(x)\right|\le \sqrt3$

Let $f:[0, 1]\rightarrow \mathbb{R}$ be a function that is continous on $[0,1]$ and derivable on $(0, 1)$. If $\int_0^1 f(x)^2\le 1$ and $\int_0^1 f'(x)^2\le 1$, show that $\left|f(x)\right|\le ...
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Finite Series Inequality [duplicate]

For each $n=1,2,{\dots}$ and $x\in(0,{\pi})$, prove that the series $$S_n(x)=\sum_{k=1}^{n} \frac{\sin(kx)}{k}>0$$
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1answer
53 views

Bell's inequality

Let $\xi, \eta, \zeta$ be random variables such that $|\xi|, |\eta|, |\zeta| \le 1$. I need to prove such inequality: $|\mathbb{E}(\zeta \xi)-\mathbb{E}(\zeta \eta)| \le 1 - \mathbb{E}(\xi \eta)$ ...
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4answers
144 views

Proving a logarithm inequality

$$\frac{1}{n+1}< \log(1+ 1/n)$$ Any ideas? I tried estimating the difference between $1/n$ and the logarithm and comparing with $1/n-1/(n+1)$ but I miserably failed.
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1answer
123 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 ...
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1answer
25 views

A not clear chain of inequalities

Look at the following excerpt from the book "J.Talbot, D.Welsh - Complexity and Cryptography": First of all I don't understand the chain of inequalities. Moreover if in the last term ...
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1answer
60 views

Find $a$ satisfy an inequality for all $t>0$

Let we think about 2-d orthogonal coordinate $xOy$. set point $A(a,a)$, where $a \in \Bbb{R}$. and $P$ is a point in the function $$y=\frac{1}{x}$$. if $$|PA|\geq 2\sqrt{2}$$. find all $a\in \Bbb{R}$ ...
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1answer
114 views

Hard Olympiad Inequality

Let x,y,z be positive real numbers such that $xy+xz+yz=1$. Prove that $$\sqrt{x^3+x}+ \sqrt{y^3+y}+ \sqrt{z^3+z} \geq 2 \cdot \sqrt{x+y+z}$$. I tried to square expand homogenize then majorize. But I ...
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3answers
80 views

if $abc=1$, then $a^2+b^2+c^2\ge a+b+c$

This is supposed to be an application of AM-GM inequality. if $abc=1$, then the following holds true: $a^2+b^2+c^2\ge a+b+c$ First of all, $a^2+b^2+c^2\ge 3$ by a direct application of ...
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2answers
34 views

Inequality with trigonometric functions

Find all values for $a$ such that the following inequality holds: $$\sin^6x + \cos^6x + a\sin x \cos x \ge 0$$ To be fair, I didn't manage to get anything helpful wiht my calculations. I tried to ...
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1answer
19 views

why does this inequality hold with expectations of supremums

I'm reading a proof on criterion for a class to be Glivenko-Cantelli and I don't see why this holds? $$E \sup_{g\in G} \left|E\left[ \frac{1}{n}\sum_{i=1}^n(g(X'_i)-g(X_i))\big|X_1^n\right]\right| ...
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1answer
63 views

Number of solutions for 2 equations involving 4 variables

Given that $a, b, c, d$ are positive integers, What are the number of solutions for the given 2 equations, $\mathbf{ad - bc > 0}$ $\mathbf{a + d = n }$ where, $n$ is a given positive integer.
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2answers
49 views

Help on this inequality

If a,b,c are positive numbers, prove the inequality $$ \frac{1}{a(1+b)}+\frac{1}{b(1+c)}+\frac{1}{c(1+a)} ≥ \frac{3}{1+abc} $$
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2answers
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Inequality $e^x-1+e^x(1-\cos \pi y+\sin \pi y)<1$

Find a ranges of $x,y\in \mathbb R^+$ in which the following inequality is verified $$e^x-1+e^x(1-\cos \pi y+\sin \pi y)<1$$ My approach: $$2-\cos \pi y+\sin \pi y<\frac{2}{e^x}\leq 2$$ Then ...
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3answers
29 views

Wrong way to use MVT to prove inequality

I am asked to show that $|\sin x - \sin y| \leq |x-y|$ using the mean value theorem. What I have done seems 'fishy'. I defined $h(x) = |x-y|-|\sin x - \sin y| $. Then $h'(x) = 1 - |\cos x| \geq 0$ ...
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0answers
26 views

Why are inequalities not accepted in GCSE Statistics Questions?

I was helping somebody with GCSE maths work, and they were doing a GCSE past paper, and they did a statistics question where there is a rubbish question (the typical no time scale and overlapping ...
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3answers
55 views

How to understand Cauchy's proof of AM-GM inequality(the last step)

The AM-GM inequality: $$a_1a_2\cdots a_n\leq\left(\frac{a_1+\cdots + a_n}{n}\right)^n$$ the trivial case: $a_1a_2 \leq \left(\frac{a_1+a_2}{2}\right)^2 $ is self-evident. then cauchy use this fact ...