Questions tagged [inequality]

Questions on proving, manipulating and applying inequalities. Do not use this tag just because an inequality appears somewhere in your question.

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59 votes
14 answers
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How to prove that $\log(x)<x$ when $x>1$? [duplicate]

It's very basic but I'm having trouble to find a way to prove this inequality $\log(x)<x$ when $x>1$ ($\log(x)$ is the natural logarithm) I can think about the two graphs but I can't find ...
Gianolepo's user avatar
  • 2,487
108 votes
15 answers
98k views

How to prove that exponential grows faster than polynomial?

In other words, how to prove: For all real constants $a$ and $b$ such that $a > 1$, $$\lim_{n\to\infty}\frac{n^b}{a^n} = 0$$ I know the definition of limit but I feel that it's not enough to ...
faceclean's user avatar
  • 7,149
125 votes
7 answers
216k views

Prove that $||x|-|y||\le |x-y|$

I've seen the full proof of the Triangle Inequality \begin{equation*} |x+y|\le|x|+|y|. \end{equation*} However, I haven't seen the proof of the reverse triangle inequality: \begin{equation*} ||x|-|...
Anonymous's user avatar
  • 2,378
28 votes
1 answer
10k views

Inequality involving $\limsup$ and $\liminf$: $ \liminf(a_{n+1}/a_n) \le \liminf a_n^{1/n} \le \limsup a_n^{1/n} \le \limsup(a_{n+1}/a_n)$

This may have been asked before, however I was unable to find any duplicate. This comes from pg. 52 of "Mathematical Analysis: An Introduction" by Browder. Problem 14: If $(a_n)$ is a ...
podiki's user avatar
  • 2,265
131 votes
27 answers
51k views

Proofs of AM-GM inequality

The arithmetic - geometric mean inequality states that $$\frac{x_1+ \ldots + x_n}{n} \geq \sqrt[n]{x_1 \cdots x_n}$$ I'm looking for some original proofs of this inequality. I can find the usual ...
Michiel Van Couwenberghe's user avatar
42 votes
7 answers
11k views

Elementary central binomial coefficient estimates

How to prove that $\quad\displaystyle\frac{4^{n}}{\sqrt{4n}}<\binom{2n}{n}<\frac{4^{n}}{\sqrt{3n+1}}\quad$ for all $n$ > 1 ? Does anyone know any better elementary estimates? Attempt. We ...
Robert William Hanks's user avatar
54 votes
3 answers
4k views

How do you prove Gautschi's inequality for the gamma function?

A few answers here on math.SE have used as an intermediate step the following inequality that is due to Walter Gautschi: $$x^{1-s} < \frac{\Gamma(x+1)}{\Gamma(x+s)} < (x+1)^{1-s},\qquad x > ...
J. M. ain't a mathematician's user avatar
119 votes
26 answers
80k 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 ...
Ashley Montanaro's user avatar
17 votes
1 answer
3k views

Could this approximation be made simpler ? Solve $n!=a^n 10^k$

I need to find the smallest value of $n$ such that $$\frac{a^n}{n!}\leq 10^{-k}$$ in which $a$ and $k$ are given (these can be large numbers). I set the problem as : solve for $n$ the equation $$n!...
Claude Leibovici's user avatar
106 votes
15 answers
16k views

Comparing $\pi^e$ and $e^\pi$ without calculating them

How can I compare (without calculator or similar device) the values of $\pi^e$ and $e^\pi$ ?
Mirzodaler's user avatar
  • 1,317
67 votes
9 answers
39k views

How to prove and interpret $\operatorname{rank}(AB) \leq \operatorname{min}(\operatorname{rank}(A), \operatorname{rank}(B))$?

Let $A$ and $B$ be two matrices which can be multiplied. Then $$\operatorname{rank}(AB) \leq \operatorname{min}(\operatorname{rank}(A), \operatorname{rank}(B)).$$ I proved $\operatorname{rank}(AB) \...
user avatar
66 votes
6 answers
23k views

How do you show monotonicity of the $\ell^p$ norms?

I can't seem to work out the inequality $(\sum |x_n|^q)^{1/q} \leq (\sum |x_n|^p)^{1/p}$ for $p \leq q$ (which I'm assuming is the way to go about it).
user1736's user avatar
  • 8,553
43 votes
2 answers
32k views

Proof of upper-tail inequality for standard normal distribution

$X \sim \mathcal{N}(0,1)$, then to show that for $x > 0$, $$ \mathbb{P}(X>x) \leq \frac{\exp(-x^2/2)}{x \sqrt{2 \pi}} \>. $$
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29 votes
4 answers
16k views

Is the Euler phi function bounded below?

I am working on a question for my number theory class that asks: Prove that for every integer $n \geq 1$, $\phi(n) \geq \frac{\sqrt{n}}{\sqrt{2}}$. However, I was searching around Google, and on ...
user avatar
24 votes
3 answers
28k views

Prove the inequality $n! \geq 2^n$ by induction

I'm having difficulty solving an exercise in my course. The question is: Prove that $n!\geq 2^n$. We have to do this by induction. I started like this: The lowest natural number where the ...
Floris Devriendt's user avatar
17 votes
4 answers
3k views

Show that $\left|\frac{\alpha - \beta}{1-\bar{\alpha}\beta}\right| < 1$ when $|\alpha|,|\beta| < 1$

This is the question I'm stumbling with: When $|\alpha| < 1$ and $|\beta| < 1$, show that: $$\left|\cfrac{\alpha - \beta}{1-\bar{\alpha}\beta}\right| < 1$$ The chapter that contains this ...
Andomar's user avatar
  • 536
7 votes
9 answers
9k views

Show that if $n>2$, then $(n!)^2>n^n$.

Show that if $n>2$, then $(n!)^2>n^n$. My work: I tried to apply induction. So, at the induction step, I need to prove, $n^n>(n+1)^{n-1}$ Here, I tried to use induction again without any ...
Hawk's user avatar
  • 6,540
68 votes
4 answers
69k views

Intuition behind logarithm inequality: $1 - \frac1x \leq \log x \leq x-1$

One of fundamental inequalities on logarithm is: $$ 1 - \frac1x \leq \log x \leq x-1 \quad\text{for all $x > 0$},$$ which you may prefer write in the form of $$ \frac{x}{1+x} \leq \log{(1+x)} \leq ...
Federico Magallanez's user avatar
77 votes
11 answers
235k views

Proof of triangle inequality

I understand intuitively that this is true, but I'm embarrassed to say I'm having a hard time constructing a rigorous proof that $|a+b| \leq |a|+|b|$. Any help would be appreciated :)
ivan's user avatar
  • 3,217
31 votes
5 answers
17k views

Showing the inequality $|\alpha + \beta|^p \leq 2^{p-1}(|\alpha|^p + |\beta|^p)$

I have a small question that I think is very basic but I am unsure how to tackle since my background in computing inequalities is embarrassingly weak - I would like to show that, for a real number $...
harlekin's user avatar
  • 8,690
52 votes
4 answers
23k views

Purely "algebraic" proof of Young's Inequality

Young's inequality states that if $a, b \geq 0$, $p, q > 0$, and $\frac{1}{p} + \frac{1}{q} = 1$, then $$ab\leq \frac{a^p}{p} + \frac{b^q}{q}$$ (with equality only when $a^p = b^q$). Back when I ...
asmeurer's user avatar
  • 9,736
17 votes
6 answers
11k views

Proving $\sum\limits_{k=1}^{n}{\frac{1}{\sqrt{k}}\ge\sqrt{n}}$ with induction

I am just starting out learning mathematical induction and I got this homework question to prove with induction but I am not managing. $$\sum\limits_{k=1}^{n}{\frac{1}{\sqrt{k}}\ge\sqrt{n}}$$ ...
Jason's user avatar
  • 2,367
21 votes
4 answers
9k views

Intuition: If $a\leq b+\epsilon$ for all $\epsilon>0$ then $a\leq b$?

I am reading Tom Apostol's Analysis and come across this theorem. Should $a \leq b$ if $a\leq b+\epsilon$ for all $\epsilon >0$? I don't doubt the proof in the book but I don't understand the ...
MathNoobe's user avatar
  • 391
33 votes
6 answers
18k views

How do you prove ${n \choose k}$ is maximum when $k$ is $ \lceil \tfrac n2 \rceil$ or $ \lfloor \tfrac n2\rfloor $?

How do you prove $n \choose k$ is maximum when $k$ is $\lceil n/2 \rceil$ or $\lfloor n/2 \rfloor$? This link provides a proof of sorts but it is not satisfying. From what I understand, it focuses on ...
curryage's user avatar
  • 1,183
6 votes
20 answers
23k views

How to prove this inequality $ x + \frac{1}{x} \geq 2 $

I was asked to prove that: $$x + \frac{1}{x}\geqslant 2$$ for all values of $ x > 0 $ I tried substituting random numbers into $x$ and I did get the answer greater than $2$. But I have a ...
Aspiring Mathlete's user avatar
18 votes
3 answers
3k views

If $\sigma_n=\frac{s_1+s_2+\cdots+s_n}{n}$ then $\operatorname{{lim sup}}\sigma_n \leq \operatorname{lim sup} s_n$

This is a question from the book Methods of Real Analysis by R. R. Goldberg. If $(s_n)$ is a sequence of real numbers and if $$\sigma_n=\frac{s_1+s_2+\cdots+s_n}{n}$$ then prove that: $\...
Sayantan's user avatar
  • 3,388
12 votes
5 answers
8k views

Proving the AM-GM inequality for 2 numbers $\sqrt{xy}\le\frac{x+y}2$

I am having trouble with this problem from my latest homework. Prove the arithmetic-geometric mean inequality. That is, for two positive real numbers $x,y$, we have $$ \sqrt{xy}≤ \frac{x+y}{2} .$$ ...
Steve's user avatar
  • 155
50 votes
5 answers
49k views

Prove $\limsup\limits_{n \to \infty} (a_n+b_n) \le \limsup\limits_{n \to \infty} a_n + \limsup\limits_{n \to \infty} b_n$

I am stuck with the following problem. Prove that $$\limsup_{n \to \infty} (a_n+b_n) \le \limsup_{n \to \infty} a_n + \limsup_{n \to \infty} b_n$$ I was thinking of using the triangle inequality ...
hyg17's user avatar
  • 5,017
27 votes
6 answers
27k views

lim sup inequality $\limsup ( a_n b_n ) \leq \limsup a_n \limsup b_n $

I´m not sure how to start with this proof, how can I do it? $$ \limsup ( a_n b_n ) \leqslant \limsup a_n \limsup b_n $$ I also have to prove, if $ \lim a_n $ exists then: $$ \limsup ( a_n b_n ) = ...
August's user avatar
  • 3,513
38 votes
2 answers
6k views

Prove that $(p+q)^m \leq p^m+q^m$

If $p,q$ are positive quantities and $0 \leq m\leq 1$ then Prove that $$(p+q)^m \leq p^m+q^m$$ Trial: For $m=0$, $(p+q)^0=1 < 2= p^0+q^0$ and for $m=1$, $(p+q)^1=p+q =p^1+q^1$. So, For $m=0,1$ ...
Argha's user avatar
  • 4,671
29 votes
10 answers
28k views

Prove by induction that for all $n \geq 3$: $n^{n+1} > (n+1)^n$

I am currently helping a friend of mine with his preperations for his next exam. A big topic of the exam will be induction, thus I told him he should practice this a lot. As at the beginning he had no ...
Huy's user avatar
  • 6,644
38 votes
5 answers
129k views

How to prove $\text{Rank}(AB)\leq \min(\text{Rank}(A), \text{Rank}(B))$? [duplicate]

How to prove $\text{Rank}(AB)\leq \min(\text{Rank}(A), \text{Rank}(B))$?
Maysam's user avatar
  • 693
18 votes
6 answers
18k views

Why $x<\tan{x}$ while $0<x<\frac{\pi}{2}$?

In proof of $\displaystyle\lim_{x\rightarrow0}\frac{\sin{x}}{x}=1$ is assumed that $\sin{x}\leq{x}\leq\tan{x}$ while $0<x<\frac{\pi}{2}$. First comparison is clear, arc length must be greater ...
nakajuice's user avatar
  • 2,549
16 votes
12 answers
10k views

How to prove $a^2 + b^2 + c^2 \ge ab + bc + ca$?

How can the following inequation be proven? $$a^2 + b^2 + c^2 \ge ab + bc + ca$$
Paul Manta's user avatar
  • 3,495
10 votes
10 answers
10k views

Proving inequality $(a+\frac{1}{a})^2 + (b+\frac{1}{b})^2 \geq \frac{25}{2}$ for $a+b=1$

If $a, b$ are positive real numbers and $a+b = 1$, prove that : $$\left(a+\frac{1}{a}\right)^2 + \left(b+\frac{1}{b}\right)^2 \geq \frac{25}{2}$$ I can see that the value $\frac{25}2$ is attained ...
user93765's user avatar
  • 335
1 vote
8 answers
664 views

Prove that $\left(\sum^n_{k=1}x_k\right)\left(\sum^n_{k=1}y_k\right)\geq n^2$

If $x_1,...,x_n$ are positive real numbers and if $y_k=1/x_k$, prove that $$\left(\sum^n_{k=1}x_k\right)\left(\sum^n_{k=1}y_k\right)\geq n^2.$$ I've been learning induction, and I've come across this ...
Jackson's user avatar
  • 420
46 votes
6 answers
43k views

Sylvester rank inequality: $\operatorname{rank} A + \operatorname{rank}B \leq \operatorname{rank} AB + n$ [duplicate]

If $A$ and $B$ are two matrices of the same order $n$, then $$ \operatorname{rank} A + \operatorname{rank}B \leq \operatorname{rank} AB + n. $$ I don't know how to start proving this inequality. I ...
Ben Ward's user avatar
  • 665
36 votes
6 answers
3k views

Proving : $ \bigl(1+\frac{1}{n+1}\bigr)^{n+1} \gt (1+\frac{1}{n})^{n} $

How could we prove that this inequality holds $$ \left(1+\frac{1}{n+1}\right)^{n+1} \gt \left(1+\frac{1}{n} \right)^{n} $$ where $n \in \mathbb{N}$, I think we could use the AM-GM inequality for ...
Quixotic's user avatar
  • 22.4k
2 votes
5 answers
24k views

Proving $ 1+\frac{1}{4}+\frac{1}{9}+\cdots+\frac{1}{n^2}\leq 2-\frac{1}{n}$ for all $n\geq 2$ by induction

Question: Let $P(n)$ be the statement that $1+\dfrac{1}{4}+\dfrac{1}{9}+\cdots +\dfrac{1}{n^2} <2- \dfrac{1}{n}$. Prove by mathematical induction. Use $P(2)$ for base case. Attempt at solution: ...
Donald Dang's user avatar
2 votes
6 answers
774 views

Prove by induction that $n^2<n!$

How can I show that $n^2<n!$ for all $n\geq 4$ Step 1 For $n=1$, the LHS=$4^2=16$ and RHS=$4!=24$. So LHS$<$ RHS. Step 2 Suppose the result be true for $n=k$ i.e., $k^2<k!$ Step 3 For $n=k+1$...
John1's user avatar
  • 21
22 votes
3 answers
59k views

Proof by induction of Bernoulli's inequality: $(1 + x)^n \geq 1 + nx$

I'm asked to used induction to prove Bernoulli's Inequality: If $1+x>0$, then $(1+x)^n\geq 1+nx$ for all $n\in\mathbb{N}$. This what I have so far: Let $n=1$. Then $1+x\geq 1+x$. This is true. Now ...
Loie Benedicte's user avatar
50 votes
3 answers
23k views

On the equality case of the Hölder and Minkowski inequalities

I'm following the book Measure and Integral of Richard L. Wheeden and Antoni Zygmund. This is problem 4 of chapter 8. Consider $E\subseteq \mathbb{R}^n$ a measurable set. In the following all the ...
leo's user avatar
  • 10.4k
10 votes
5 answers
9k views

Proving that $x - \frac{x^3}{3!} < \sin x < x$ for all $x>0$

Prove that $x - \frac{x^3}{3!} < \sin(x) < x$ for all $x>0$ This should be fairly straightforward but the proof seems to be alluding me. I want to show $x - \frac{x^3}{3!} < \sin(x) &...
user23658's user avatar
  • 453
8 votes
3 answers
1k views

Inductive Proof that $k!<k^k$, for $k\geq 2$.

Call $P(k): k!<k^k$, for $k\geq 2$ Test it out with 2, and it's true ($2<4$). Assume that $P(k)$ is true for some $k\geq2$. Then show that $P(k+1)$ is true. $P(k+1): (k+1)!<(k+1)^{k+1}$ ...
Bob's user avatar
  • 637
14 votes
3 answers
1k views

Prove $(a_1+b_1)^{1/n}\cdots(a_n+b_n)^{1/n}\ge \left(a_1\cdots a_n\right)^{1/n}+\left(b_1\cdots b_n\right)^{1/n}$

consider positive numbers $a_1,a_2,a_3,\ldots,a_n$ and $b_1,b_2,\ldots,b_n$. does the following in-equality holds and if it does then how to prove it $\left[(a_1+b_1)(a_2+b_2)\cdots(a_n+b_n)\right]^{...
Mia's user avatar
  • 691
9 votes
3 answers
6k views

Mean Value Theorem: $\frac{2}{\pi}<\frac{\sin x}{x}<1$

I need to show that $\dfrac{2}{\pi}<\dfrac{\sin(x)}{x}<1$ for $0<x<\dfrac{\pi}{2}$. I know I need to use the mean value theorem, would I just say that since $f$ is continuous in the ...
user avatar
9 votes
4 answers
7k views

Show that, for all $n > 1: \frac{1}{n + 1} < \log(1 + \frac1n) < \frac1n.$ [duplicate]

I'm learning calculus, specifically derivatives and applications of MVT, and need help with the following exercice: Show that, for all $n > 1$ $$\frac{1}{n + 1} < \log(1 + \frac1n) < \...
glpsx's user avatar
  • 2,122
7 votes
5 answers
4k views

Suppose that $(s_n)$ converges to $s$, $(t_n)$ converges to $t$, and $s_n \leq t_n \: \forall \: n$. Prove that $s \leq t$.

I'm stuck with the proof of the following: Suppose that $(s_n)$ converges to $s$, $(t_n)$ converges to $t$, and $s_n \leq t_n \: \forall \: n$. Prove that $s \leq t$. I've tried starting with $s_n \...
manan's user avatar
  • 187
129 votes
14 answers
30k views

Olympiad Inequality $\sum\limits_{cyc} \frac{x^4}{8x^3+5y^3} \geqslant \frac{x+y+z}{13}$

$x,y,z >0$, prove $$\frac{x^4}{8x^3+5y^3}+\frac{y^4}{8y^3+5z^3}+\frac{z^4}{8z^3+5x^3} \geqslant \frac{x+y+z}{13}$$ Note: Often Stack Exchange asked to show some work before answering the ...
HN_NH's user avatar
  • 4,351
85 votes
3 answers
14k views

Fastest way to check if $x^y > y^x$?

What is the fastest way to check if $x^y > y^x$ if I were writing a computer program to do that? The issue is that $x$ and $y$ can be very large.
learner's user avatar
  • 905

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