Questions tagged [inequality]

Questions on proving, manipulating and applying inequalities.

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24
votes
1answer
2k 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 $m=0,1$ ...
24
votes
2answers
2k views

An information theory inequality which relates to Shannon Entropy

For $a_1,...,a_n,b_1,...,b_n>0,\quad$ define $a:=\sum a_i,\ b:=\sum b_i,\ s:=\sum \sqrt{a_ib_i}$. Is the following inequality true?: $${\frac{\Bigl(\prod a_i^{a_i}\Bigr)^\frac1a}a \cdot \frac{\...
24
votes
1answer
975 views

Prove $\left(\frac{a+1}{a+b}\right)^a+\left(\frac{b+1}{b+c}\right)^b+\left(\frac{c+1}{c+a}\right)^c \geqslant 3$

$a,b,c \geqslant 0,$$ a+b+c=3$, and $(a+b)(b+c)(c+a) \neq 0$ , prove $$\left(\frac{a+1}{a+b}\right)^a+\left(\frac{b+1}{b+c}\right)^b+\left(\frac{c+1}{c+a}\right)^c \geqslant 3$$ I try Bernouli's ...
24
votes
0answers
384 views

Prove that $\frac{1}{\phi}<\int_0^\infty \frac{e^{-x}}{\Gamma(x)} dx< \frac{24+\sqrt{2}}{41} $

I'm sure that's a coincidence, but the Laplace transform of $1/\Gamma(x)$ at $s=1$ turns out to be pretty close to the inverse of the Golden ratio: $$F(1)=\int_0^\infty \frac{e^{-x}}{\Gamma(x)} dx=0....
23
votes
7answers
3k views

Why does the sign have to be flipped in this inequality?

We are learning about inequalities. I originally assumed it would be the same as equations, except with a different sign. And so far, it has been - except for this. Take the simple inequality: $-5m&...
23
votes
3answers
931 views

A (probably trivial) induction problem: $\sum_2^nk^{-2}\lt1$

So I'm a bit stuck on the following problem I'm attempting to solve. Essentially, I'm required to prove that $\frac{1}{2^2}+\frac{1}{3^2}+\cdots+\frac{1}{n^2} < 1$ for all $n$. I've been toiling ...
23
votes
2answers
20k views

Proof of upper-tail inequality for standard normal distribution

$X \sim \mathcal{N}(0,1)$, then to show that for $x > 0$, $$ \mathbb{P}(X>x) \leq \frac{\exp(-x^2/2)}{x \sqrt{2 \pi}} \>. $$
23
votes
16answers
929 views

Prove that $2018^{2019}> 2019^{2018}$ without induction, without Newton's binomial formula and without Calculus.

Prove that $2018^{2019}> 2019^{2018}$ without induction, without Newton's binomial formula and without Calculus. This inequality is equivalent to $$ 2018^{1/2018}>2019^{1/2019} $$ One of my '...
23
votes
4answers
2k views

Prove: $\sin (\tan x) \geq {x}$

I bumped into this question: Question: Prove that for $x\in \Bigl[0,\dfrac {\pi}{4}\Bigr]$, $$\sin (\tan x) \geq {x}$$ This seems to be an innocent inequality but I am already exhausted trying to ...
23
votes
2answers
666 views

Trigonometric Inequality. $\sin{1}+\sin{2}+\ldots+\sin{n} <2$ .

How can I prove the following trigonometric inequality : $$\sin1+\sin2 +\ldots+\sin n <2$$ with $n \in \mathbb{N}^{*}$. The problem is that I don't know how to start this problem, I try to ...
23
votes
1answer
1k views

The series $\sum\limits_{n=1}^\infty \frac n{\frac1{a_1}+\frac1{a_2}+\dotsb+\frac1{a_n}}$ is convergent

If a series $\sum\limits_{n=1}^\infty a_n$ is convergent, and $a_n\gt0$... Do not refer to Carleman's inequality or Hardy's inequality, show that the series $$\sum_{n=1}^\infty \frac n{\frac1{a_1}+\...
23
votes
1answer
1k views

How prove this inequality $\frac{a}{b+3}+\frac{b}{c+3}+\frac{c}{d+3}+\frac{d}{a+3}\le 1$

Question: let $a,b,c,d\ge 0$,such $$a^2+b^2+c^2+d^2=4$$ show that $$\dfrac{a}{b+3}+\dfrac{b}{c+3}+\dfrac{c}{d+3}+\dfrac{d}{a+3}\le 1$$ My try: By Cauchy-Schwarz inequality,we have $$\sum_{cyc}\dfrac{...
23
votes
4answers
1k views

Prove $\sqrt{x^2+yz+2}+\sqrt{y^2+zx+2}+\sqrt{z^2+xy+2}\ge 6$, given $x+y+z=3$ and $x,y,z\ge0$

Let $x+y+z=3,x,y,z\ge 0$,show that $$\sqrt{x^2+yz+2}+\sqrt{y^2+zx+2}+\sqrt{z^2+xy+2}\ge 6$$ Additional information I have seen the following problem: $x,y,z>0,x+y+z=3$, prove that $$\sqrt{x^2+...
23
votes
2answers
425 views

How prove that:$\varphi(2)+\varphi(3)+\varphi(4)+\cdots+\varphi(n)\ge\frac{n(n-1)}{4}+1$

Prove that for $n\ge 3$, $$\varphi(2)+\varphi(3)+\varphi(4)+\cdots+\varphi(n)\ge\dfrac{n(n-1)}{4}+1$$ where $\varphi$ is the Euler's totient function I think we must use this $$\sum_{k=1}^{n}\varphi(...
23
votes
4answers
741 views

How to prove that $ \sin \angle{GAB}+\sin \angle{GBC}+\sin \angle{GCA} \le \frac{3}{2} $ for a triangle $ABC$ with centroid $G$?

Let $ G $ be the centroid of $ \triangle ABC $ , such that $ \measuredangle{GAB}=x,\measuredangle{GBC}=y,\measuredangle{GCA}=z $. How do I prove that : $$ \sin x +\sin y +\sin z\le \frac{3}{2} $$
22
votes
14answers
4k views

Functions that are always less than their derivatives

I was wondering if there are functions for which $$f'(x) > f(x)$$ for all $x$. Only examples I could think of were $e^x - c$ and simply $- c$ in which $c > 0$. Also, is there any significance in ...
22
votes
8answers
3k views

Showing n! is greater than n to the tenth power

I'd like to show $n!>n^{10} $ for large enough n ( namely $ n \geq 15 $). By induction, I do not know how to proceed at this step: $$ (n+1)\cdot n!>(n+1)^{10} $$ As I can't see how to ...
22
votes
8answers
38k views

Is a prime factor of a number always less than its square root?

I was going through the fundamental theorem in Number Theory where any non zero integer n can be represented as a product of distinct primes. A related problem with this theorem is to prove that for ...
22
votes
7answers
737 views

How prove this $f(n)\le f(n+1)$ where $f(n)=\sum_{k=1}^{n}\frac{n}{n^2+k^2}$

let $$f(n)=\sum_{k=1}^{n}\dfrac{n}{n^2+k^2}$$ prove or disprove $$f(n)\le f(n+1)$$ this inequality is found when I deal this follow limit: $$\lim_{n\to\infty}f(n)=\lim_{n\to\infty}\dfrac{1}{n}\...
22
votes
6answers
2k views

How to prove the inequality between mathematical expectations?

Let $X$ and $Y$ be independent random variables having the same distribution and the finite mathematical expectation. How to prove the inequality $$ E(|X-Y|) \le E(|X+Y|)?$$
22
votes
1answer
748 views

Interpretation of $\frac{22}{7}-\pi$

Integral and series proofs that $\frac{22}{7}>\pi$ We can prove that $\frac{22}{7}$ exceeds $\pi$ by using Dalzell integral $$\int_0^1 \frac{x^4(1-x)^4}{1+x^2}dx=\frac{22}{7}-\pi$$ or its ...
22
votes
1answer
342 views

Let $P(x)\in \mathbb{R}[x]$ be of degree $n$ and for any $x \in \left(0,1\right]$, we have $x\cdot P^2(x) \le 1$. Calculate $\max P(0)$.

Let $P(x)$ be a polynomial with real coefficient and with degree $n$ such that for any $x \in \left(0,1\right]$, we have $$x\cdot P^2(x) \le 1$$ Find the maximum of $P(0)$. Note: $P^2(x) = (P(x))^2$. ...
22
votes
2answers
592 views

An “AGM-GAM” inequality

For positive real numbers $x_1,x_2,\ldots,x_n$ and any $1\leq r\leq n$ let $A_r$ and $G_r$ be , respectively, the arithmetic mean and geometric mean of $x_1,x_2,\ldots,x_r$. Is it true that the ...
22
votes
3answers
723 views

Prove $a^{ab}b+b^{bc}c+c^{ca}a \geqslant \sqrt[6]{5}$

$a,b,c >0$, and $a+b+c=3$, prove $$ a^{ab}b+b^{bc}c+c^{ca}a \geqslant \sqrt[6]{5}$$ I try to substitute $c=3-a-b$ to reduce the number of variables, but cannot further proceed to solve the problem....
21
votes
8answers
988 views

Comparing $2013!$ and $1007^{2013}$

I have to compare the following two numbers: $$2013! \text{ and } 1007^{2013}$$ where $n! = 1 \times 2 \times \cdots \times (n-1) \times n$. I tried in different ways to group the $1 \times 2 \...
21
votes
3answers
41k views

Triangle inequality for subtraction?

Is the following inequality(that looks like the triangle inequality) valid: $|a - b| \leq |a| - |b|$ Why?
21
votes
7answers
18k views

Inequality: $(x + y + z)^3 \geq 27 xyz$

Edit: $a,b,c$ and $x,y,z$ are positive, real numbers. Since $(a-b)^2 \geq 0~$, $a^2 + b^2 - 2ab\geq0~$ and $a^2 + b^2 \geq 2ab~$. Similarly, $a^2 + c^2 \geq 2ac~$ and $b^2 + c^2 \geq 2bc~$. ...
21
votes
4answers
1k views

Inequality with complex numbers

Consider the following problem. Fix $n \in \mathbb N$. Prove that for every set of complex numbers $\{z_i\}_{1\le i \le n}$, there is a subset $J\subset \{1,\dots , n\}$ such that $$\left|\sum_{j\...
21
votes
2answers
601 views

How to show that $\frac{\pi}{5}\leq\int_0^1 x^x\,dx\leq\frac{\pi}{4}$

Show that: $$\frac{\pi}{5}\leq\int_0^1 x^x\,dx\leq\frac{\pi}{4}$$ All I've got so far is that the minimum of $x^x$ is $e^{-1/e}$. At this point I could compare $\pi/5$ to $e^{-1/e}$ but I'm ...
21
votes
2answers
1k views

How to prove this inequality $(a^2+bc^4)(b^2+ca^4)(c^2+ab^4) \leq 64$?

Question: If $a,b,c$ are nonnegative real numbers such that $a+b+c=3,$ then $$(a^2+bc^4)(b^2+ca^4)(c^2+ab^4) \leq 64$$ My try: I found the equality holds only if $(a,b,c)=(2,0,1)$ or all of its ...
21
votes
5answers
666 views

Show that $\sum^{6}_{i=1} a_{i}=\frac{15}{2}$ and $ \sum^{6}_{i=1} a^{2}_{i}=\frac{45}{4} \implies \prod_{i=1}^{6} a_{i} \leq \frac{5}{2}$

Let $a_{i}$, $1 \leq i \leq 6,$ be real numbers such that $\displaystyle\hspace{1.2 in}\sum^{6}_{i=1} a_{i}=\frac{15}{2}\;\;$ and $\;\;\displaystyle\sum^{6}_{i=1} a^{2}_{i}=\frac{45}{4}$. Prove ...
21
votes
2answers
822 views

A tricky integral inequality

A friend has submitted this problem to me: Let $0<a<b<1$ and $f:[0,1]\to \mathbb R$ be a differentiable function such that $$\displaystyle \frac{\int_0^a f(x) dx}{a(1-a)}+\frac{\int_b^1 f(...
21
votes
2answers
1k views

Prove $\frac{xy}{5y^3+4}+\frac{yz}{5z^3+4}+\frac{zx}{5x^3+4} \leqslant \frac13$

$x,y,z >0$ and $x+y+z=3$, prove $$\tag{1}\frac{xy}{5y^3+4}+\frac{yz}{5z^3+4}+\frac{zx}{5x^3+4} \leqslant \frac13$$ My first attempt is to use Jensen's inequality. Hence I consider the function $$...
21
votes
2answers
856 views

Prove that $\frac{x^x}{x+y}+\frac{y^y}{y+z}+\frac{z^z}{z+x} \geqslant \frac32$

$x,y,z >0$, prove $$\frac{x^x}{x+y}+\frac{y^y}{y+z}+\frac{z^z}{z+x} \geqslant \frac32$$ I have a solution for this beautiful and elegant inequality. I am posting this inequality to share and see ...
21
votes
1answer
393 views

Integers $n$ for which the digit sum of $n$ exceeds the digit sum of $n^5$

This question is strongly inspired by The smallest integer whose digit sum is larger than that of its cube? by Bernardo Recamán Santos. The number $n=124499$ has digit sum $1+2+4+4+9+9=29$ while its ...
20
votes
3answers
2k views

Stuck trying to prove an inequality

I have been trying to prove (the left half of) the following inequality: $$ \underbrace{\sum_i \sum_j |x_i| \le \sum_i \sum_j |x_i + x_j|}_\textrm{?} \le 2 \sum_i \sum_j |x_i|$$ (All $x_i$s are ...
20
votes
1answer
89k views

Proof for triangle inequality for vectors

Generally,the length of the sum of two vectors is not equal to the sum of their lengths. To see this consider the vectors $u$ and $v$ as shown below. By considering $u$ and $v$ as two sides of a ...
20
votes
4answers
394 views

How to prove this inequality $\frac{a_{1}a_{2}+a_{2}a_{3}+\cdots+a_{n-1}a_{n}}{a^2_{1}+a^2_{2}+\cdots+a^2_{n}}\le\cos{\frac{\pi}{n+1}}$

Let $a_{1},a_{2},\cdots,a_{n},n\ge 2$ be real numbers,show that $$\dfrac{a_{1}a_{2}+a_{2}a_{3}+\cdots+a_{n-1}a_{n}}{a^2_{1}+a^2_{2}+\cdots+a^2_{n}}\le\cos{\dfrac{\pi}{n+1}}$$ I think this result is ...
20
votes
2answers
1k views

Let $a_{i} \in\mathbb{R}$ ($i=1,2,\dots,n$), and $f(x)=\sum_{i=0}^{n}a_{i}x^i$ such that if $|x|\leqslant 1$, then $|f(x)|\leqslant 1$. Prove that:

Let $a_{i} \in\mathbb{R}$ ($i=1,2,\dots,n$), and $f(x)=\sum_{i=0}^{n}a_{i}x^i$ such that if $|x|\leqslant 1$, then $|f(x)|\leqslant 1$. Prove that: $|a_{n}|+|a_{n-1} | \leqslant 2^{n-1}$. $|a_{n}|+|...
20
votes
1answer
21k views

Integral Inequality Absolute Value: $\left| \int_{a}^{b} f(x) g(x) \ dx \right| \leq \int_{a}^{b} |f(x)|\cdot |g(x)| \ dx$

Suppose we are given the following: $$\left| \int_{a}^{b} f(x) g(x) \ dx \right| \leq \int_{a}^{b} |f(x)|\cdot |g(x)| \ dx$$ How would we prove this? Does this follow from Cauchy Schwarz? Intuitively ...
20
votes
6answers
20k views

Proofs of the Cauchy-Schwarz Inequality?

How many proofs of the Cauchy-Schwarz inequality are there? Is there some kind of reference that lists all of these proofs?
20
votes
2answers
1k views

Geometric proof for inequality

While on AOPS, I saw this interesting problem. I was wondering how many different approaches could be used to tackle the problem. In other words I am looking for interesting and unique ways to solve ...
20
votes
3answers
613 views

An inequality involving three consecutive primes

Can you provide a proof or a counterexample to the following claim : Let $p,q,r$ be three consecutive prime numbers such that $p\ge 11 $ and $p<q<r$ , then $\frac{1}{p^2}< \frac{1}{q^2} + \...
20
votes
1answer
361 views

Slick proofs that if $\sum\limits_{k=1}^\infty \frac{a_k}{k}$ converges then $\lim\limits_{n\to\infty} \frac{1}{n}\sum\limits_{k=1}^n a_k=0$

I'm looking for slick proofs that if $a_n$ is a sequence of complex numbers such that $\sum\limits_{k=1}^\infty \frac{a_k}{k}$ converges then $\lim\limits_{n\to\infty} \frac{1}{n}\sum\limits_{k=1}^n ...
20
votes
1answer
938 views

Prove that if $x_1,x_2,\ldots,x_n>0$, then $(1+x_1)(1+x_1+x_2)\ldots(1+x_1+x_2+\ldots+x_n) \ge \sqrt{(n+1)^{n+1}}\sqrt{x_1x_2\ldots x_n}$.

Prove that if $x_1,x_2,\ldots,x_n>0$, then $$(1+x_1)(1+x_1+x_2)\ldots(1+x_1+x_2+\ldots+x_n) \ge \sqrt{(n+1)^{n+1}}\sqrt{x_1x_2\ldots x_n}$$ I've tried using the AM-GM inequality: $$(1+x_1)(1+x_1+...
20
votes
3answers
287 views

The inequality $\frac{MA}{BC}+\frac{MB}{CA}+\frac{MC}{AB}\geq \sqrt{3}$

Given a triangle $ABC$, and $M$ is an interior point. Prove that: $\dfrac{MA}{BC}+\dfrac{MB}{CA}+\dfrac{MC}{AB}\geq \sqrt{3}$. When does equality hold?
20
votes
4answers
909 views

How prove this inequality $\frac{2}{(a+b)(4-ab)}+\frac{2}{(b+c)(4-bc)}+\frac{2}{(a+c)(4-ac)}\ge 1$

let $a,b,c>0$,and such $a+b+c=3$, show that $$\dfrac{2}{(a+b)(4-ab)}+\dfrac{2}{(b+c)(4-bc)}+\dfrac{2}{(a+c)(4-ac)}\ge 1$$ I think this inequality use this $$ab\le\dfrac{(a+b)^2}{4}$$
20
votes
1answer
634 views

How might one prove the following inequality?

Let $r$ be a natural number. I wish to prove that $x + \dfrac{x^2}{2} + \dfrac{x^3}{3} + \cdots + \dfrac{x^r}{r} \leq x^{r+1} + \log(r+1)$ for all $x>0$. Some friends and I have tried using ...
20
votes
2answers
348 views

How prove this inequality $2a^ab^bc^cd^d\ge ac+bd$

Let $a,b,c,d$ be positive numbers such that $a+b+c+d=2$. Show that $$2a^ab^bc^cd^d\ge ac+bd$$ My try: I think maybe I can use this inequality $$(1+x)^n\ge 1+nx \hspace{12pt} (n>1)$$ then I can't ...
20
votes
1answer
890 views

Prove that:$f(f(x)) = x^2 \implies \int_{0}^{1}{(f(x))^2dx} \geq \frac{3}{13}$

Let $f: [0,\infty) \to [0,\infty)$ be a continuous function such that $f(f(x)) = x^2, \forall x \in [0,\infty)$. Prove that $\displaystyle{\int_{0}^{1}{(f(x))^2dx} \geq \frac{3}{13}}$. All I know ...