Questions on proving and manipulating inequalities.

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Prove that $ \left( \frac{M+z_2+\dots+z_{2n}}{2n} \right)^2\ge\left( \frac{x_1+\dots+x_n}{n} \right)\left(\frac{y_1+\dots+y_n}{n} \right). $

Let $n$ be a positive integer and let $(x_1,\ldots,x_n)$, $(y_1,\ldots,y_n)$ be two sequences of positive real numbers. Suppose $(z_2,\ldots,z_{2n})$ is a sequence of positive real numbers such that ...
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29 views

does this inequality imply the Cauchy Schwarz inequliaty?

Let $z_i,w_i \in \mathbb{C}$. My teacher in class proved the following inequality: $$ \Big( \sum_{j=1}^n \mathrm{Re}(z_jw_j) \Big)^2 \leq \sum_j |z_j|^2 \cdot \sum_j |w_j|^2$$ Question: Does this ...
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1answer
55 views

Evaluate $\pi$ using $\arctan(\frac{\sqrt{3}}{3})$

I Have to evaluate $\pi$ using $\arctan(\frac{\sqrt{3}}{3})$ with an error with no more than $10^{-10}$ using taylor approximation $ p_{2n-1}(x) \approx\arctan(x)$ . So, After manipulation, I get: ...
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11 views

Upper-bounds to $B_z(a,b)$

Is there any standard technique to produce nice upper-bounds to the incomplete beta function $$B_z(a,b)=\int_0^z t^{a-1} (1-t)^{b-1} dt \,?$$ Disclaimer: this question is intentionally not too ...
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1answer
30 views

A problem of inequality and request for help

Problem If $n$ be a positive integer, prove that $$\begin{align} \frac{1}{2\sqrt{n+1}} < \frac{1.3.5 \dots (2n-1)}{2.4.6 \dots 2n}< \frac{1}{\sqrt{2n+1}} \end{align}$$ Following is my ...
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1answer
31 views

Bounding the increments of $\sin(e^{-x})$

Show that $|\sin(e^{-b}) -\sin(e^{-a})| \leq {b-a \over e^{-a}}$ for all $a \leq b$ This is part of a basic calculus class so i would appreciate answers suitable for my knowledge.
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61 views

Proving $\sqrt{100,001}-\sqrt{100,000} < \frac{1}{2\sqrt{100,000}}$

Proving $\sqrt{100,001}-\sqrt{100,000} < \frac{1}{2\sqrt{100,000}}$ I squared both sides of the equation to get $100,001 + 100,000+-200\sqrt{10}\sqrt{100,001} < \frac{1}{400,000}$. I am just ...
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1answer
26 views

Simple inequality proof in analysis

Just need verification on whether my proof is valid. I couldn't find a straightforward way to prove this inequality directly, so I tried a proof by contradiction instead. The question: Let $a, b \in ...
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3answers
46 views

Prove this inequality.

Let $S=a_1+...+a_n<1$ where $a_i>0$. Prove that $1+S<(1+a_1)\cdot ... \cdot (1+a_n)<{1\over 1-S}$. I started with the right inequality but I am not sure it iss plausible (I did something ...
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5answers
84 views

Proof for $\forall x \in [0, \frac{\pi}{2}]\quad \sin(x) \ge \frac{x}{2}$

What is the proof for $\forall x \in [0, \frac{\pi}{2}]\quad \sin(x) \ge \frac{x}{2}$ ? Assuming it is true.
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2answers
58 views

If $a,b,c\in\mathbb{R^+}$ such that $ abc = 1 $ and $ ab + bc + ca = 5 $. Prove that $ 17/4 \leq (a+b+c)\leq 1+ \sqrt{32}. $

If $a,b,c\in\mathbb{R^+}$ such that $ abc = 1 $ and $ ab + bc + ca = 5 $. Prove that $$ \frac{17}{4} \leq (a+b+c)\leq 1+ \sqrt{32}. $$ My attempt Tried using Vieta but it didn't work. Also I used ...
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4answers
53 views

What are all values of $x$ in $\mathbb{R}$ that satisfy $4 < |x+2| + |x-1| < 5$?

I am having some problems getting started with this problem, as I never had to deal with an inequality that was between two values with absolute values. Any help is appreciated. The problem is find ...
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2answers
54 views

Show that $a^2 + b^2 + c^2 \geq ab + bc + ca$ for all positive integers $a$, $b$, and $c$ [duplicate]

Show that $a^2 + b^2 + c^2 \geq ab + bc + ca$ for all positive integers $a$, $b$, and $c$. I am not sure how to approach this problem. Should I divide this problem into multiple cases based on ...
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1answer
20 views

Prove that for all $m$, there exist some $k$, such that $(m-n)^2 > m^2$ for all $n>k$

I have a problem where I need to prove: $\forall m \in \mathbb{N}:\exists m \in \mathbb{N} ∋(m−n)^2>m^2~∀n>k$ My thought was since it is only "there exists some k.." can I not say: if $k = ...
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1answer
18 views

Writting an inequality to represent a situation (Help needed) [on hold]

In the course of the day, Annalise will drive her car to work and later use the city rail system to get around her different branch offices. At one rail stop, she needs to ride a bus. Will she stay ...
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1answer
21 views

How to prove the inequality? [on hold]

Set $f(x)=1-(1-\lambda)^x$, where $\lambda \in (0,1)$, show that $f(x)/x \ge f(x)-f(x-1)$ holds for any $x\ge 1$.
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A problem of inequality

Let $a_1, a_2, a_3$; $b_1, b_2, b_3$; $c_1, c_2, c_3$; $d_1, d_2, d_3$ be all real numbers. We need to show that $$\begin{align}(a_1b_1c_1d_1 + a_2b_2c_2d_2 &+ a_3b_3c_3d_3)^4\\ &\leq ...
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4answers
481 views

Given that a,b,c are distinct positive real numbers, prove that (a + b +c)( 1/a + 1/b + 1/c)>9

Given that $a,b,c$ are distinct positive real numbers, prove that $(a + b +c)\big( \frac1{a}+ \frac1{b} + \frac1{c}\big)>9$ This is how I tried doing it: Let $p= a + b + c,$ and $q=\frac1{a}+ ...
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1answer
44 views

How to prove/dispute the following log inequality?

I was wondering if the following inequality is true: $$\forall x,N\in \mathbb N^+: \lceil \log_2\left(\lfloor\frac{N}{x}+1\rfloor\right)\rceil\leq \lceil\log_2 (N+1)\rceil - \lfloor\log_2 (x)\rfloor ...
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1answer
54 views

Proof Involving Generalized Mean

Let $x=(x_1,...,x_n) \in \mathbb R^n$ and $$g(p)=\sqrt[p]{\frac{1}{n}\sum_{k=1}^{n} |x_k|^p)}$$ Using Hölder's inequality, show that $g(p)$ is increasing on $(0,\infty)$. For a sequence with ...
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14 views

Existence of solution for linear matrix inequality?

Suppose $x$ is a $n\times1$ column vector. How to know whether the following matrix inequality has solution or not? $$Ax\leq B$$ where $A$ is a $m\times n$ matrix and $B$ is a $n\times 1$ column ...
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1answer
43 views

Integral inequality $\int_0^x{f(t)^3 dt \leq \left( \int_0^x f(t) dt\right)^2} :\forall x>0$

Let $f(0) = 0$ and $0<f'(x)\leq1$ for all $x \geq0$, then prove: $$\int_0^x{f(t)^3 dt \leq \left( \int_0^x f(t) dt\right)^2} :\forall x>0$$ The hint I was given was "differentiate, factor and ...
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2answers
35 views

Show$\:\frac{1}{\left|x^2+x+1\right|}\:\ge \:\frac{1}{x^2-\left|x\right|-1}$

This is the answer I can come up with. I get the complete opposite of what I'm supposed to get. My mistake is probably in the first part, could anyone help me out? $$\left|x^2+x+1\right|\:\ge ...
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1answer
40 views

Extending Minkowsky inequality to double summation?

I know the Minkowski inequality for sequences as follows : $$\left(\sum_{k=1}^n|x_k+y_k|^p\right)^{1/p} \leq \left(\sum_{k=1}^n|x_k|^p\right)^{1/p}+\left(\sum_{k=1}^n|y_k|^p\right)^{1/p}$$ Now say we ...
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32 views

Four real numbers p,q,r,s satisfy $p+q+r+s = 9$ and $p^2 + q^2 + r^s + s^s = 21$. [duplicate]

Four real numbers p,q,r,s satisfy $ p+q+r+s = 9 $ and $ p^2 + q^2 + r^2 + s^2 = 21 $. Prove that there is a permutation $ (a,b,c,d) $ of $ (p,q,r,s) $ such that $ ab-cd \geq 2 $. My attempt I tried ...
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3answers
86 views

Let a,b,c be positive real numbers numbers such that $ a^2 + b^2 + c^2 = 3 $

Let $a,b,c\in\mathbb{R^+}$ such that $ a^2 + b^2 + c^2 = 3 $. Prove that $$ (a+b+c)(a/b + b/c + c/a) \geq 9. $$ My Attempt I tried AM-GM on the symmetric expression so the $a+b+c \geq 3$, but I ...
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2answers
67 views

Prove that if $a$,$b$,$c$ are non-negative real numbers such that $a+b+c =3$, then $abc(a^2 + b^2 + c^2)\leq 3$

Prove that if $ a,b,c $ are non-negative real numbers such that $a+b+c = 3$, then $$ abc(a^2 + b^2 + c^2) \le 3 $$ My attempt : I tried AM-GM inequality, tried to convert it to $a+b+c$, but I think ...
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51 views

A lower bound for $\log\left( \frac{a+x^2}{b+x^2}\right)$

I am looking for a tight lower bound for $$f(x)=\log\left( \frac{a+x^2}{b+x^2} \right)$$ $x>0$ and $1<b<<a$. I didn't check for convexity analytically, but I plotted this function ...
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1answer
33 views

Find $\min x^TAy+b^Tx+c^Ty$ subject to $1^Tx=1^Ty=1,x\ge 0,y\ge 0$

The problem seems to be easy but I can't find a solution :( Problem: Given $A\in\mathbb{R}^{m\times n}, A\ge 0, b\in\mathbb{R}^{m}, c\in\mathbb{R}^{n}$. Minimize $f(x,y) = x^TAy+b^Tx+c^Ty$ subject to ...
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18 views

Leading up to Young's Inequality

I am trying to prove Young's Inequality by considering the function $$h(u) = \frac{u^p}{p} + \frac{C^q}{qu^q}$$ for $C,u>0$ and $p,q >1$. We also require $$\frac{1}{p}+\frac{1}{q}=1$$ so that ...
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13 views

Kneser Inequality in multivariables

Based on the Kneser Inequality ("Polynomials and Polynomial Inequalities", p. 260) one has $\Vert q \Vert_{[-1, 1]} \Vert r \Vert_{[-1, 1]} \leq C(n, m) \Vert q r \Vert_{[-1, 1]}$ where all norms ...
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0answers
26 views

Values satisfying the inequality [on hold]

if $ 1-\cos x=\frac {\sqrt3}{2} |x| +a$ has no solution then find the complete set of values of $'a'$.Here is the question i got struck.
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Basic inequality problem [on hold]

Here is my problem if $ 16-x^2> |x-a|$ is to be satisfied by atleast one negative value of $x$, then i have to find complete set of values of $'a'$ .Please provide me hint to solve this ...
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1answer
31 views

How to solve Inequality with factorials

Im reading a book in Numerial analysis and I have the following which I dont understand involving inequalities and factorials, What i have is the following: $$\frac{1}{(2n+1)!(2n+1)} \leq 5*10^{-9}$$ ...
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3answers
168 views

Inequalities proven by real analysis or induction.

Let $t\in [-1,1]$. Prove that $(1+t)^p+(1-t)^p\ge2$ when $p\ge 1$ and that $(1+t)^p+(1-t)^p \le 2$ where $0 \le p\le 1$. I am not sure how I should solve it. I tried induction at first and it was ...
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0answers
12 views

Inequality involving expectations of vector/matrix norms

I'm reading a paper and trying to understand the proof of a lemma regarding expectations of norms of random vectors. The author's notation does not distinguish between vector and matrix norms, nor ...
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1answer
39 views

Help with fraction inequality

Let $a,b,c$ be three numbers such that: $a,b\in (0.5,1)$ $c \in (0.25,0.5)$ $c < 0.5a$ $c > 0.5b$ $a + b < 1 + c$ Let $$f(a,b,c) = \frac{1+c}{a+\frac{bc}{c+0.5b}}$$ What is the ...
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1answer
31 views

Inequality $(a+b)^2 + (a+b+4c)^2\ge \frac{kabc}{a+b+c}$ for $a,b,c \in\mathbb{R}$

Find biggest constans k such that $(a+b)^2 + (a+b+4c)^2\ge \frac{kabc}{a+b+c}$ is true for any $a,b,c \in\mathbb{R}$ Could you check up my solution? I'm not sure it's ok - $(a+b)^2 + (a+b+4c)^2 \ge ...
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1answer
38 views

How does the Euler Totient Function apply here?

How many positive integers $< 2013$ are divisible by $2$ Can I somehow use Euler's Totient function to find this?
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1answer
48 views

Inequality $x^4+y^4+(x^2+1)(y^2+1)\ge x^3(1+y) +y^3(1+x)+x+y$ for $x,y \in\mathbb{R}$

Prove for $x,y \in\mathbb{R}$ that such inequality exists ; $x^4+y^4+(x^2+1)(y^2+1)\ge x^3(1+y) +y^3(1+x)+x+y$ And here is what I realised ; because $(x^2+1)(y^2+1) >=1$ and $x^4+y^4 \ge 0$ ...
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1answer
20 views

summation inequality with logarithms

show: $$\sum_{i=1}^n \log_{2}\,i = O(n\log n)$$ Proof by induction: $$\sum_{i=1}^n \log\,i \le n\log n$$ $$\text{Test for n=1: }\sum_{i=1}^1 \log_{2}\,i \le 1\log 1$$ $$0 \le 0\text{ true for ...
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0answers
38 views

Inequality please help [on hold]

Adam is running marathon . He has complete 10 mile in 90 minute . What should his average split be in order to complete the race less than 4 hours
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2answers
30 views

proof by induction summation inequality

show by induction that: $$\sum_{i=1}^n i^2 = O(n^3)$$ what I have so far: $$\sum_{i=1}^n i^2 <= n^3$$ base case: for n=1 $$\sum_{i=1}^1 i^2 <= 1^3$$ ...
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4answers
994 views

Slick proof of exponential inequality

Today I saw that using taylor series, one can show that $e^x+e^{-x}\leq 2e^{x^2/2}$. Is there a slick proof using some sort of Jensen-type inequality or integral bound?
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3answers
76 views

Cannot follow proof that $n! \leq en(n/e)^n$

prove that $n! \leq en(n/e)^n$ skip proof for base (n=1)... Assume it holds for $n-1$, verify for $n$. We have $n! = n* (n-1)! \leq n * e(n-1)(\frac{n-1}{e})^{n-1} $ by inductive assumption. we ...
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1answer
23 views

question on proving inequalities [on hold]

If I need to prove $t(x) \ge0 $, for all $ x>0$ and I prove that $t(x) \gt 0 $, for all $ x>0$ does that make for a proof or is it wrong?
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4answers
36 views

How to study positivity of $x\sqrt{4-x^2}-4\arcsin({\frac x2})$

I have to study where the function is positive/negative. What's the method to solve the inequality $x\sqrt{4-x^2}-4\arcsin({\frac x2})>0$ ?
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2answers
31 views

Olympiad minimum question, minimal value

If the numbers $A, B, C$ are such that the expression $\sqrt{A-B} + \sqrt{(B+3)^2} + C^2 - 4C + 4$ is as small as possible, then $A+B+C$ is? I thought start with, $A > B > C$ without loss of ...
3
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2answers
69 views

Prove, inequality ,positive numbers

$$\frac{a}{e+a+b}+\frac{b}{a+b+c}+\frac{c}{b+c+d}+\frac{d}{c+d+e}+\frac{e}{d+e+a}<2$$ Prove that for positive numbers $a,b,c,d,e$ there is such inequality
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3answers
23 views

How can we make this expression small? [on hold]

How can we make the following expression small: $$(bx-ay)^2+(cx-az)^2+(cy-bz)^2+(ay-bx)^2+(az-cx)^2+(bz-cy)^2$$, where $a,b,c,x,y,z$ are nonnegative reals? Note: I'm not looking for an exact answer, ...