Questions on proving, manipulating and applying inequalities.

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3
votes
2answers
79 views

Inequality Proof, need help simplifying

I'm new here and unsure if this is the right way to format a problem, but here goes nothing. I'm currently trying to solve an inequality proof to show that $n^3 > 2n+1$ for all $n \geq 2$. I ...
6
votes
3answers
535 views

How prove this inequality: $\sum_{i,j=1}^{n}|x_{i}+x_{j}|\ge n\sum_{i=1}^{n}|x_{i}|$? [duplicate]

Let $x_{1},x_{2},\cdots,x_{n}$ be real numbers. Show that $$\sum_{i,j=1}^{n}|x_{i}+x_{j}|\ge n\sum_{i=1}^{n}|x_{i}|.$$ I think this problem may be solved using nice methods, but I can't find ...
7
votes
2answers
285 views

Limit superior inequalities proof: $\limsup_{n\to \infty} \left(\frac{a_1+a_{n+1}}{a_n}\right)^n\ge e$

Let $a_n$ be a positive sequence. Prove that $$\limsup_{n\to \infty} \left(\frac{a_1+a_{n+1}}{a_n}\right)^n\geqslant e.$$
1
vote
2answers
36 views

threshold of n to satisfy $a^n <n^a$

How to find the minimum of $n$ when we know $a$, to satisfy: $a^n<n^a$ $a^m>m^a$ for each $m>n$ $n$ and $m$ are natural numbers.
0
votes
1answer
88 views

Prove that $\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}\geq 27$

How can I prove $\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}\geq 27$, given that $(x+y+z)(\frac{1}{x}+\frac{1}{y}+\frac{1}{z})\geq 9$ and $x+y+z=1$. I've already tried using that: $\frac{1}{x} ...
1
vote
1answer
70 views

If the product of two numbers is positive and less than one, what can I conclude about the quotient?

If I'm given that $0 < ab < 1$, how do I figure out what values $\frac ab$ can take on? I have $0 < ab < 1$ and am asked if $\frac ab < 1$. I can see that if I set $a = \frac 12$ and ...
2
votes
3answers
84 views

How to prove $(1+x)^n\geq 1+nx+\frac{n(n-1)}{2}x^2$ for all $x\geq 0$ and $n\geq 1$?

I've got most of the inductive work done but I'm stuck near the very end. I'm not so great with using induction when inequalities are involved, so I have no idea how to get what I need... ...
3
votes
2answers
117 views

Help checking proof of reverse triangle inequality $|x| - |y| \le |x + y|$?

Let $x, y \in \mathbb{R}$. Prove $|x| - |y| \le |x + y|$. By the the triangle inequality $|x| + |y| \ge |x + y|$, hence $$ \begin{align} &|y| \ge |x+y| - |x| \\ &|x+y| \ge |x+y| - |y| \\ ...
1
vote
3answers
217 views

Prove, that $\sin x- a^3\cos x\leq \frac 1 3 \sqrt{1+a^6}$

Let a and $x$ be natural numbers with the property that $\sin x\leq a\cos x$. Prove that $\sin x- a^3 \cos x\leq \frac 1 3 \sqrt{1+a^6}$. Again, I'm looking for a second solution. I don't know how to ...
2
votes
1answer
44 views

Direct proof that $\sum_{k=0}^m \binom{n}{k} \leq n^m$

Is there a short direct proof that $\sum_{k=0}^m \binom{n}{k} \leq n^m$ ? I can prove it by showing it is true for $m=2$ and then proving by induction. Is there a direct non-inductive proof?
2
votes
2answers
167 views

Proof of $X+\csc{\left(\frac{\pi}{X}\right)}>2\csc{\left(\frac{\pi}{2X}\right)}$

When $X$ is greater than 1, I want to prove that $X+\csc{\left(\frac{\pi}{X}\right)}>2\csc{\left(\frac{\pi}{2X}\right)}$ where $\csc{(\cdot)}=\frac{1}{\sin{(\cdot)}}$. Plotting the above ...
1
vote
3answers
350 views

Proof for limit superior's property: $\limsup (a_n b_n ) \leq \limsup a_n \cdot \limsup b_n$ [duplicate]

Let $a_n,b_n>0$ for all $n\in\mathbb N$. Prove that $\limsup (a_n b_n ) \leq \limsup a_n \cdot \limsup b_n$ I know that $\limsup (a_n+b_n ) \leq \limsup a_n + \limsup b_n$. But I don't know how ...
1
vote
1answer
45 views

I need help with this inequality

$$ \frac{|x|}{(x + 2)|x - 5|} \le 0 $$ I thought to multiply the denominator to $0$, but I didn't think it is correct.
0
votes
4answers
77 views

Help solving $ | x + 2| + 3| x - 1| > 2x - 1$

$$ | x + 2| + 3| x - 1| > 2x - 1$$ My idea was to solve $| x - 1|$ and $| x + 2|$ it in two cases (when is positive and when is negative). It didn't work!
1
vote
2answers
86 views

Help solving the following inequality

$$ \frac{x|x + 1|(x + 2)}{|x - 1|} \ge 0 $$ My idea was to multiply the denominator to $0$ and $| x + 1 |$ to solve it in two cases (when is positive and when is negative). No luck
2
votes
0answers
217 views

Weak/Variational Gronwall type inequality

I came across the following weak differential inequality while looking through F.Otto's paper on $L^{1}$ contraction and uniqueness of quasilinear elliptic-parabolic equation: \begin{align*} - ...
1
vote
2answers
626 views

Solving a non-linear inequality involving square root

What approach should be taken to solve inequalities similar to the following one? $x < \sqrt{2 - x}$ p.s. The answer as I have been told is $x < 1$. Any help is much appreciated. Thank you. ...
7
votes
3answers
186 views

How prove this $ \sqrt{\frac{a}{a+3b+5bc}}+\sqrt{\frac{b}{b+3c+5ca}}+\sqrt{\frac{c}{c+3a+5ab}}\geq 1.$

Let $a,b,c$ be nonnegative real numbers such that $a+b+c=3$, Prove that $$ \sqrt{\frac{a}{a+3b+5bc}}+\sqrt{\frac{b}{b+3c+5ca}}+\sqrt{\frac{c}{c+3a+5ab}}\geq 1.$$ This problem is from ...
1
vote
1answer
50 views

How prove this equality $(\sum_{k=1}^{n}(-1)^{k+1}a_{k})^r\le \sum_{k=1}^{n}(-1)^{k+1}a^r_{k+1}$

let $\{a_{n}\}_{\ge 0},n\in \Bbb N^{+},a_{n}\ge 0$ is decreasing. show that $$\left(\sum_{k=1}^{n}(-1)^{k+1}a_{k}\right)^r\le \sum_{k=1}^{n}(-1)^{k+1}a^r_{k},(r>0)$$ My try: ...
2
votes
1answer
105 views

Why are strict inequalities stronger than non-strict inequalities?

I'm working with induction proofs involving inequalities and I am encountering example proofs that wish to show things of the sort, $n!\le\ n^n$ for every positive integer. The proof given in the ...
4
votes
1answer
196 views

How prove this inequality $\frac{2^x-1}{3^x-2^x}\le3\left(\frac{1}{x-1}-\frac{1}{x}\right)$

show that $$\dfrac{2^x-1}{3^x-2^x}\le3\left(\dfrac{1}{x-1}-\dfrac{1}{x}\right)\cdots(1)$$ This problem (1) is from when I solve following ...
6
votes
3answers
222 views

Short and intuitive proof that $\left(\frac{n}{k}\right)^k \leq \binom{n}{k}$

The simple inequality that $\left(\frac{n}{k}\right)^k \leq \binom{n}{k}$ has a number of different proofs. But is there a particularly intuitive, short and elegant proof that uses the natural ...
3
votes
1answer
106 views

Simplest proof that $\binom{n}{k} \leq \left(\frac{en}{k}\right)^k$

The inequality $\binom{n}{k} \leq \left(\frac{en}{k}\right)^k$ is very useful in the analysis of algorithms. There are a number of proofs online but is there a particularly elegant and/or simple proof ...
38
votes
20answers
3k views

Simplest or nicest proof that $1+x \le e^x$

The elementary but very useful inequality that $1+x \le e^x$ for all real $x$ has a number of different proofs, some of which can be found online. But is there a particularly slick, intuitive or ...
1
vote
3answers
66 views

If $A,B,C \leq M$, how can we show $\alpha A + \alpha B + (1-2\alpha )C \leq M$?

If $A,B,C \leq M$, how can we show $$\alpha A + \alpha B + (1-2\alpha )C \leq M,$$ where $0 \leq \alpha \leq 1/2$? What is a counter-example if $\alpha > 1/2$? My first thought was to try to ...
0
votes
1answer
81 views

Inequality with a sum and factorial

For a homework assignment we have the following question that I'm stuck on. Let $ 0 \leq y \leq 1 $ be given. $\forall m \in \mathbb{N}$, define $ \displaystyle S_m(y)=\sum_{k=0}^m \binom{m}{k}y^k$. ...
3
votes
2answers
133 views

Prove through induction that $3^n > n^3$ for $n \geq 4$

I'm new to induction and have not done induction with inequalities before, so I get stuck at proving after the 3rd step. The question is: Use induction to show that $3^n > n^3$ for $n \geq ...
1
vote
1answer
206 views

How prove this inequlaity $xyz+4\ge xy+yz+zx$

let $x,y,z\in [-1,2]$ show that $$xyz+4\ge xy+yz+zx$$ my try let $$f(x,y,z)=xyz-4-xy-yz-xz=z(xy-y-x)-4$$ and I can't any works. and Have nice methods? Thank you
2
votes
2answers
109 views

Strange deduction about relation of median and mean

On his blog T. Tao's proves the following concentration inequality, due to Talagrand. Let $K>0$, and let $X_{1},..., X_{n}$ be iid complex random variables all bounded by $K$. Let ...
1
vote
1answer
363 views

Distance between mean and median

I want to solve the following problem in T.Tao's random matrix theory book. Let $X$ be a random variable with finite second momment. A median $M(X)$ of $X$ saisfies ...
1
vote
1answer
65 views

How do I bound ${{n \log n} \choose {\log n}}\frac{1}{n ^ {\log n}}$ tightly?

I'm trying to get a tight upper bound for the following expression: ${{n \log n} \choose {\log n}}\frac{1}{n ^ {\log n}}$ The bound that I've obtained is too loose, which is as follows: $ m = nM, ...
5
votes
1answer
238 views

A tight lower bound for the entropy of the XOR of two random variables

Let $U$ be the uniform random variable over $n$-bit binary strings, and let $X$ be another random variable that is dependent on $U$ and ranges over $n$-bit binary strings. Assuming $I(X;U) \le ...
1
vote
5answers
69 views

how to prove $|e^{i \langle u,x \rangle}-e^{i \langle u,y \rangle }|<|u|\cdot|x-y|$?

I'm reading Bernt Oksendal's "Stochasticc Differential Equations" and this is one of the result that I don't see the proof. This is from Appendix A, page 309 (sixth edition): $$\large \lvert e^{i ...
2
votes
1answer
85 views

Trigonometric inequality for the sum of sin and cos

I need a proof for the following trigonometric inequality $$\frac{|\sin x| + |\cos x|}{\sqrt2} \leq 1- \frac{\cos^2(2x)}{8}$$ Can someone please help me with this?
10
votes
1answer
609 views

How find the range of $m$

Let $m$ be a positive integer and let $$I_{m}(x,y)=\sum_{k=0}^{m-1}(e^{\cos{(x-y+2k\pi/m)}}-e^{\cos{(-x-y+2k\pi/m)}})$$ for all $x,y\in [0,\dfrac{\pi}{m}]$. For which $m$ is $I_m(x,y) \ge 0$? ...
0
votes
1answer
63 views

Relation of $e$ to other numbers…

I found the following result, When i was working on my calculator . $$x^y < y^x \quad ,x < y \quad \text{ for } x,y<e$$ $$x^y > y^x \quad ,x < y \quad \text{ for } x,y>e$$ I can't ...
3
votes
2answers
280 views

How prove this mathematical analysis by zorich from page 233

Let $f$ be twice differentiable on an interval $I$,Let $$M_{0}=\sup_{x\in I}{|f(x)|},M_{1}=\sup_{x\in I}{|f'(x)|},M_{2}=\sup_{x\in I}{|f''(x)|}$$ show that (a):$$M_{1}\le ...
7
votes
2answers
151 views

Prove $ \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{4} + \frac{\sqrt{4}}{6} + \cdots + \frac{\sqrt{n+1}}{2n} > \frac{\sqrt{n}}{2} $ by induction

Prove by induction that for all $n > 0$, $$ \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{4} + \frac{\sqrt{4}}{6} + \cdots + \frac{\sqrt{n+1}}{2n} > \frac{\sqrt{n}}{2} $$ I have done the basis ...
-1
votes
3answers
156 views

If b and d are positive real numbers, then prove that $\frac{b+d}{2}>(bd)^{\frac 12}$.

Prove the following statement: If $b$ and $d$ are positive real numbers, then $\frac{b+d}{2}\geq(bd)^{\frac 12}$. Thank you.
7
votes
6answers
1k views

How to prove: $a+b+c\le a^2+b^2+c^2$, if $abc=1$?

Let $a,b,c \in \mathbb{R}$, and $abc=1$. What is the simple(st) way to prove inequality $$ a+b+c \le a^2+b^2+c^2. $$ (Of course, it can be generalized to $n$ variables).
4
votes
2answers
92 views

Simple looking inequality

I would to find the smallest possible constant $c$ that satisfies $$\frac{3^{3k}e\sqrt{3}}{\pi\sqrt{k}2^{3/2+2k}} \leq 2^{ck}$$ assuming $k\geq 1$ is an integer. I tried taking logs base $2$ of ...
1
vote
1answer
72 views

How to prove this inequality? $ | z-1 | \le | | z | -1 | + | z| \cdot | \arg z | $

If $z$ is any non-zero complex number, how to prove the following inequality? $$ | z-1 | \le | | z | -1 | + | z| \cdot | \arg z | $$ Hints please!
4
votes
1answer
111 views

How prove this inequality $ \frac{b^3+c^3}{a}+\frac{c^3+a^3}{b}+\frac{a^3+b^3}{c} \ge 2(a^2+b^2+c^2)+3\left((b-c)^2+(c-a)^2+(a-b)^2\right)$

let $a,b,c$ are positive numbers, show that $$ \frac{b^3+c^3}{a}+\frac{c^3+a^3}{b}+\frac{a^3+b^3}{c} \ge 2(a^2+b^2+c^2)+3\left((b-c)^2+(c-a)^2+(a-b)^2\right)\cdots (1)$$ my try: ...
1
vote
1answer
81 views

Prove $1 + \sum_{i=0}^n(\frac1{x_i}\prod_{j\neq i}(1+\frac1{x_j-x_i}))=\prod_{i=0}^n(1+\frac1{x_i})$

Prove the identity $$1 + \sum_{i=0}^n \left(\frac1{x_i}\prod_{j\neq i} \left(1+\frac1{x_j-x_i} \right) \right)=\prod_{i=0}^n \left(1+\frac1{x_i} \right)$$ and hence deduce the inequality in Problem ...
0
votes
1answer
61 views

Writing equation of curve using sin(s) in Lax's proof

I am going through Peter Lax's proof of the isoperimetric inequality. It seems elegant, however, it doesn't seem to be written in a manner that undergraduate students can understand quickly. There are ...
4
votes
2answers
104 views

How prove this inequality

in $\Delta ABC$, if $A,B,C\in (0,\pi/2]$,show that $$\sin{A}+\sin{B}+\sin{C}>2$$ This problem have many nice methods? Thank you
2
votes
1answer
40 views

Exponent of 2 in $m!$

I'm having trouble proving this inequality. Let $p(m!)$ be the exponent of 2 in the prime factorization of $m!$ Prove that $$ p(m!) \leq m-1 $$ I guessed that $$ p(2^k!) = 2^k-1 $$ but that doesn't ...
5
votes
1answer
110 views

How prove this $ b^2c^2+abc(b+c)+a(b^3+c^3)+a^3(b+c)\ge 2a^2(b^2+c^2+bc)$

let $a,b,c\ge 0$, show that $$ b^2c^2+abc(b+c)+a(b^3+c^3)+a^3(b+c)\ge 2a^2(b^2+c^2+bc)$$ my idea use the SOS methods, But I don't work at last. Thank you
9
votes
5answers
540 views

Improving bound on $\sqrt{2 \sqrt{3 \sqrt{4 \ldots}}}$

An old challenge problem I saw asked to prove that $\sqrt{2 \sqrt{3 \sqrt{4 \ldots}}} < 3$. A simple calculation shows the actual value seems to be around $2.8$, which is pretty close to $3$ but ...
1
vote
2answers
46 views

A simple Inequality problem

Can't I eliminate $p^3$ from both side and say that quantity A is greater?