Questions on proving and manipulating inequalities.

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15
votes
4answers
962 views

An Inequality problem relating $\prod\limits^n(1+a_i^2)$ and $\sum\limits^n a_i$

Let $(a_1,\space a_2,\space \cdots, \space a_n) \in \mathbb R^n_+$ such that $\displaystyle \prod^n_{i=1 }a_i = 1$. Prove that $$\displaystyle \prod^n_{i=1} (1+a_i^2) \le \cfrac ...
16
votes
3answers
1k 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 ...
13
votes
2answers
376 views

Prove that: $x_1\cdot x_2\cdots x_n>y_1\cdot y_2\cdots y_m$.

For two positive integer sequences $x_1,x_2,\ldots,x_n$ and $y_1,y_2,\ldots,y_m$ satisfying $x_i\neq x_j\quad \text{and}\quad y_i\neq y_j\quad \forall i,j, i \ne j$ ...
12
votes
2answers
677 views

Inequality. $\sqrt{\frac{11a}{5a+6b}}+\sqrt{\frac{11b}{5b+6c}}+\sqrt{\frac{11c}{5c+6a}} \leq 3$

Let $a,b,c$ be positive numbers . Prove the following inequality: $$\sqrt{\frac{11a}{5a+6b}}+\sqrt{\frac{11b}{5b+6c}}+\sqrt{\frac{11c}{5c+6a}} \leq 3.$$ What I tried: I used ...
11
votes
1answer
619 views

How to prove $\frac{1}{4}(\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{d}+\frac{d^2}{a})\ge \sqrt[4]{\frac{a^4+b^4+c^4+d^4}{4}}$

Let $a,b,c,d>0$, show that $$\dfrac{1}{4}\left(\dfrac{a^2}{b}+\dfrac{b^2}{c}+\dfrac{c^2}{d}+\dfrac{d^2}{a}\right)\ge \sqrt[4]{\dfrac{a^4+b^4+c^4+d^4}{4}}$$ I know this is interesting ...
11
votes
8answers
2k views

$a+b+c =0$; $a^2+b^2+c^2=1$. Prove that: $a^2 b^2 c^2 \le \frac{1}{54}$

If a,b,c are real numbers satisfying $a+b+c =0; a^2+b^2+c^2=1$. Prove that $a^2 b^2 c^2 \le \frac{1}{54}$.
21
votes
2answers
490 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 ...
14
votes
2answers
1k views

Sum inequality: $\sum_{k=1}^n \frac{\sin k}{k} \le \pi-1$

I'm interested in finding an elementary proof for the following sum inequality: $$\sum_{k=1}^n \frac{\sin k}{k} \le \pi-1$$ If this inequality is easy to prove, then one may easily prove that the sum ...
9
votes
1answer
213 views

To show that $P(|X-Y| \leq 2) \leq 3P(|X-Y| \leq 1)$

I found this question while browsing through "The Probabilistic Method", by Noga Elon. Let X and Y be 2 independent and identically distributed real valued random variables. Prove that: $$P(|X-Y| ...
11
votes
6answers
1k views

Proving that $\frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + \cdots + \frac{1}{\sqrt{100}} < 20$

How am I suppose to prove that: $$\frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + \cdots + \frac{1}{\sqrt{100}} < 20$$ Do I use the way like how we count $1+2+ \cdots+100$ to estimate? So $1/5050 \lt ...
7
votes
2answers
324 views

How prove this inequality $\sum_{cyc}\frac{a^2}{b(a^2-ab+b^2)}\ge\frac{9}{a+b+c}$

let $a,b,c>0$, show that $$\dfrac{a^2}{b(a^2-ab+b^2)}+\dfrac{b^2}{c(b^2-bc+c^2)}+\dfrac{c^2}{a(c^2-ca+a^2)}\ge\dfrac{9}{a+b+c}$$ My try: since this inequality is homogeneous ,without loss of ...
12
votes
8answers
779 views

prove $\frac{1}{ n+1}+\frac{1}{ n+2}+\cdots+\frac{1}{2n}<\frac{25}{36}$ by mathematical induction

How to prove $$\frac{1}{ n+1}+\frac{1}{ n+2}+\cdots+\frac{1}{2n}<\frac{25}{36}$$ by Mathematical induction,n$\ge $1
10
votes
2answers
347 views

determinant inequality $ \det(A^2+B^2+(A-B)^2)\ge 3\det(AB-BA) $

A and B are two $2\times2$ reals matrices. then $$ \det \Big(A^2+B^2+(A-B)^2\Big)\ge 3\det(AB-BA) $$ well, it is seems interesting, but it is really hard to get started Thank you very much!
5
votes
1answer
151 views

How prove this number theory inequality $\left(\dfrac{1}{N}\sum_{n=1}^{N}(\omega{(n)})^k\right)^{\frac{1}{k}}\le k+\sum_{q\le N}\frac{1}{q}$

show that: for any positive numbers $k$ and $N$, have $$\left(\dfrac{1}{N}\sum_{n=1}^{N}(\omega{(n)})^k\right)^{\frac{1}{k}}\le k+\sum_{q\le N}\dfrac{1}{q}$$ where $\displaystyle\sum_{q\le N}$ is ...
15
votes
3answers
723 views

Prove $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$

Let $a,b,c$ are non-negative numbers, such that $a+b+c = 3$. Prove that $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$ Here's my idea: $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$ ...
15
votes
3answers
1k views

Inequality. $\frac{1}{16}(a+b+c+d)^3 \geq abc+bcd+cda+dab$

I want to prove the following inequality : $$\frac{1}{16}(a+b+c+d)^3 \geq abc+bcd+cda+dab, $$ $a,b,c,d \in \mathbb{R}_{+} .$ In my book, at the answers chapter the author uses AM $\geq$ GM, ...
11
votes
2answers
1k views

Relationship between diameter and radius of a point set

Consider a set of $n$ points in $\mathbb{R}^k$. The diameter of this set is the maximum distance between two of its points; its radius is the radius of the smallest (closed) k-ball that contains all ...
4
votes
1answer
309 views

Prove that: $\cos(x) -1 < -\frac{x^2}{2} + \frac{x^4}{24}$

Prove that: $$\cos(x) -1 < -\frac{x^2}{2} + \frac{x^4}{24}$$ for $x \ne 0$ I need to prove this using Cauchy's mean value theorem. What I did: $f(x) = \cos(x) -1$ $$g(x) = ...
3
votes
1answer
126 views

Show that $\frac1{\sqrt{(n+\frac12) \pi}} \le\frac{1\cdot 3\cdot 5 … (2n-1)}{2\cdot 4\cdot 6 … (2n)} \le \frac1{\sqrt{n \pi}} $

Show that, if $n$ is a positive integer, $$\frac1{\sqrt{(n+\frac12) \pi}} \le\frac{1\cdot 3\cdot 5 ... (2n-1)}{2\cdot 4\cdot 6 ... (2n)} \le \frac1{\sqrt{n \pi}} . $$ This result is in a current ...
3
votes
3answers
167 views

Inequality: $ab^2+bc^2+ca^2 \le 4$, when $a+b+c=3$.

Let $a,b,c $ are non-negative real numbers, and $a+b+c=3$. How to prove inequality $$ ab^2+bc^2+ca^2\le 4.\tag{*} $$ In other words, if $a,b,c$ are non-negative real numbers, then how to prove ...
14
votes
2answers
355 views

A conjecture concerning primes and algebra

A monoid morphism $\psi:\mathbb Z_+\!\!\rightarrow\mathbb Z_+$ is defined by an arbitrary function $f:\mathbb Z_+\!\!\rightarrow\mathbb Z_+$ and defines a group homomorphism $\varphi:\mathbb ...
12
votes
6answers
435 views

Asymptotic behaviour of a multiple integral on the unit hypercube

A few days ago I found an interesting limit on the "problems blackboard" of my University: $$\lim_{n\to +\infty}\int_{(0,1)^n}\frac{\sum_{j=1}^n x_j^2}{\sum_{j=1}^n x_j}d\mu = 1.$$ The correct claim, ...
12
votes
2answers
320 views

Prove that $\dfrac{a}{b^2+5}+ \dfrac{b}{c^2+5} + \dfrac{c}{a^2+5} \le \dfrac 12$

Let $a,b,c>0$ and $a^3+b^3+c^3=3$. Prove that $$\dfrac{a}{b^2+5}+ \dfrac{b}{c^2+5} + \dfrac{c}{a^2+5} \le \dfrac 12$$ I have an ugly solution for this solution.
10
votes
3answers
365 views

Let $a,b,c>0$ and $a+b+c= 1$, how to prove the inequality $\frac{\sqrt{a}}{1-a}+\frac{\sqrt{b}}{1-b}+\frac{\sqrt{c}}{1-c}\geq \frac{3\sqrt{3}}{2}$?

Let $a,b,c>0$ and $a+b+c= 1$, how to prove the inequality $$\frac{\sqrt{a}}{1-a}+\frac{\sqrt{b}}{1-b}+\frac{\sqrt{c}}{1-c}\geq \frac{3\sqrt{3}}{2}$$?
6
votes
1answer
157 views

How prove $\frac{(k+1)^{k+1}}{k^k}\sum_{t=k+1}^{n}\frac{1}{t^2}<e$

Let $k,n\in \mathbb{N},n\ge k$, prove that $$\dfrac{(k+1)^{k+1}}{k^k}\sum_{t=k+1}^{n}\dfrac{1}{t^2}<e.$$ I got the impression that this inequality is very sharp. My idea: ...
5
votes
2answers
63 views

Prove the inequality: $\frac{a}{c+a-b}+\frac{b}{a+b-c}+\frac{c}{b+c-a}\ge{3}$

Prove the inequality: $\frac{a}{c+a-b}+\frac{b}{a+b-c}+\frac{c}{b+c-a}\ge{3}$ Where $a,b,c$ are sides of a triangle. It is clear that $c+a-b$ is positive but how to use it?
4
votes
2answers
161 views

Prove that $\sqrt{n} \le \sum_{k=1}^n \frac{1}{\sqrt{k}} \le 2 \sqrt{n} - 1$ is true for $n \in \mathbb{N}^{\ge 1}$

I'm trying to solve these induction exercises proposed by the department of mathematics of Oxford University. I don't know how to give a valid proof for the third one which says the following: ...
2
votes
1answer
148 views

Rudin: Problem Chp3.11 and need advice.

I am working on the following problems and I have a couple of questions. Suppose $a_n>0, s_n = \sum_{i = 1}^{n}$ and $\Sigma a_n$ diverges. RTP (a) $\Sigma \frac{a_n}{1+a_n}$ ...
1
vote
1answer
72 views

Find minimum of $P=\frac{\sqrt{3(2x^2+2x+1)}}{3}+\frac{1}{\sqrt{2x^2+(3-\sqrt{3})x +3}}+\frac{1}{\sqrt{2x^2+(3+\sqrt{3})x +3}}$

For $x\in\mathbb{R}$ find minimum of $P$. $P=\dfrac{\sqrt{3(2x^2+2x+1)}}{3}+\dfrac{1}{\sqrt{2x^2+(3-\sqrt{3})x +3}}+\dfrac{1}{\sqrt{2x^2+(3+\sqrt{3})x +3}}$ Source : Viet Nam national test for high ...
18
votes
3answers
542 views

An Inequality Involving Bell Numbers: $B_n^2 \leq B_{n-1}B_{n+1}$

The following inequality came up while trying to resolve a conjecture about a certain class of partitions (the context is not particularly enlightening): $$ B_n^2 \leq B_{n-1}B_{n+1} $$ for $n \geq ...
10
votes
3answers
428 views

proving :$\frac{ab}{a^2+3b^2}+\frac{cb}{b^2+3c^2}+\frac{ac}{c^2+3a^2}\le\frac{3}{4}$.

Let $a,b,c>0$ how to prove that : $$\frac{ab}{a^2+3b^2}+\frac{cb}{b^2+3c^2}+\frac{ac}{c^2+3a^2}\le\frac{3}{4}$$ I find that $$\ ...
9
votes
2answers
628 views

Proof that $t-1-\log t \geq 0$ for $t > 0$

Using basic calculus, I can prove that $f(t)=t-1-\log t \geq 0$ for $t > 0$ by setting the first derivative to zero \begin{align} \frac{df}{dt} = 1 - 1/t = 0 \end{align} And so I have a critical ...
8
votes
1answer
263 views

If $f$ is a positive, monotone decreasing function, prove that $\int_0^1xf(x)^2dx \int_0^1f(x)dx\le \int_0^1f(x)^2dx \int_0^1xf(x)dx$

If $f$ is a positive, monotone decreasing function, prove that $\int_0^1xf(x)^2dx \int_0^1f(x)dx\le \int_0^1f(x)^2dx \int_0^1xf(x)dx$
8
votes
4answers
278 views

How many sequence of integers ($j_1 , j_2 , . . . , j_k$) are there such that $0 ≤ j_1 ≤ j_2 ≤ . . . ≤ j_k ≤ n$?

I need to solve the problem, How many sequence of integers ($j_1 , j_2 , . . . , j_k$) are there such that $0 ≤ j_1 ≤ j_2 ≤ . . . ≤ j_k ≤ n$? I've been given a hint, (Hint: Reduce the ...
7
votes
5answers
722 views

Arithmetic mean is less than geometric mean (Spivak Calculus 3rd Chapter 2 Problem 22)

If $a_1, \ldots, a_n \ge 0$, the arithmetic mean $$A_n={a_1 + \cdots + a_n \over n}$$ and the geometric mean $$G_n = \sqrt[n]{a_1 \cdots a_n}$$ satisfy $G_n \le A_n$. As a first step to prove this ...
7
votes
1answer
171 views

Prove inequality: $\sum \frac{a^4}{a^3+b^3} \ge \frac{a+b+c}{2}$

Prove inequality with $a,b,c >0$ $$\frac{a^4}{a^3+b^3} + \frac{b^4}{b^3+c^3}+\frac{c^4}{c^3+a^3} \ge \frac{a+b+c}{2}$$ I tried the inequality: $\sum \frac {a^4+b^4}{a^3+b^3} \ge \sum ...
6
votes
3answers
94 views

Prove $(x+r_1) \cdots (x+r_n) \geq (x+(r_1 \cdots r_n)^{1/n})^{n}$.

I can show that for $x > 0$ and $r_{i} > 0$ we have $$ \left(\, x + r_{1}\,\right)\ldots\left(\, x + r_{n}\,\right)\ \geq\ \left[\, x + \left(\, r_{1}\ldots r_{n}\,\right)^{1/n}\,\right]^{n}.$$ ...
6
votes
1answer
116 views

Solving an inequality : $n \geq 3$ , $n^{n} \lt (n!)^{2}$.

I proved this inequality in the following way: Lemma: $r \in \Bbb N, r \geq 3$. We have $r^r \gt (r+1)^{r-1}$. Proof: We apply the AM-GM inequality to the $r$ positive integers where there are ...
5
votes
1answer
94 views

Prove that $\|a\|+\|b\| + \|c\| + \|a+b+c\| \geq \|a+b\| + \|b+c\| + \|c +a\|$ in the plane.

Prove that $\|a\| + \|b\| + \|c\| + \|a+b+c\| \geq \|a+b\| + \|b+c\| + \|c +a\|$ in the plane. Gentle hints only, please! I know that attempting to decompose R.H.S. into $$\alpha a + \beta b + ...
5
votes
3answers
803 views

Derivatives of the Riemann zeta function at $s=0$

It's a curious fact that for $n>0$, $\zeta^{(n)}(0)\approx -n!$. Apostol gave a table for $\frac{\zeta^{(n)}(0)}{n!}$, among other results on $\zeta^{(n)}(0)$ . the sequence : $$\delta_{n}=\left | ...
5
votes
2answers
202 views

Inequality involving sides of a triangle

How can one show that for triangles of sides $a,b,c$ that $$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b} < 2$$ My proof is long winded, which is why I am posting the problem here. Step 1: let ...
4
votes
4answers
220 views

Working with proofs help?

I'm trying to study for my midterm and doing some random practise questions to work with proofs. However I'm stuck on, as the only way I know how to prove it is through plugging in numbers, however as ...
3
votes
1answer
104 views

If $f(0) = f(1)=0$ and $|f'' | \leq 1$ on $[0,1]$, then $|f'(1/2)|\le 1/4$

Let $f : [0,1] \rightarrow \mathbb{R}$ be a function whose second order derivative $f''(x)$ is continuous on $[0,1]$. Suppose that $f(0) = f(1)=0$ and that $|f''(x)| \leq 1$ for any $x \in [0,1]$. ...
3
votes
2answers
189 views

Find maximum of $P$

Let $$P = \frac{{{x^2}}}{{{x^2} + yz + x + 1}} + \frac{{y + z}}{{x + y + z + 1}} - \frac{{1 + yz}}{9}.$$ Find maximum of $P$ where $x, y,z$ are nonnegative real numbers such that ${x^2} + {y^2} + ...
3
votes
1answer
147 views

little inequality conjecture

proof or disproof for $n\geq2$ even and $x>0$ $$\sum\limits_{i=0}^{n}x^i\geq \frac{(1+2\sum\limits_{i=1}^{ \frac{n}{2} }x^i)^2}{ \frac{n}{2} (x+1)+1}$$ I came up with this little inequality while ...
3
votes
2answers
161 views

Prove that $\frac{\int_0^1xf^2(x) \mathrm{d}x}{\int_0^1 xf(x) \mathrm{d}x}\le\frac{\int_0^1 f^2(x) \mathrm{d}x}{\int_0^1 f(x) \mathrm{d}x}$ [duplicate]

Let $f:[0,1]\rightarrow\mathbb{R_+}$ be a monotone decreasing function. We want to prove that $$\frac{\int_0^1x(f(x))^2 \,\mathrm{d}x}{\int_0^1 xf(x) \,\mathrm{d}x}\le\frac{\int_0^1 (f(x))^2 ...
2
votes
2answers
71 views

Prove QM-AM inequality

$$\dfrac{x_1^2+ x_2^2 + \cdots + x_n^2}n \geq \left(\dfrac{x_1+x_2+\cdots+x_n}n\right)^2$$ I don't think AM, GM can be used here. And simple expansion doesn't help too. What should I do?
2
votes
2answers
125 views

An inequality, which is supposed to be simple

Let $x,y,z\in\mathbb{R}$.Let $xy+yz+xz=1$. Prove:$\displaystyle \frac{x}{\sqrt{x^2+1}}+\frac{y}{\sqrt{y^2+1}}+\frac{z}{\sqrt{z^2+1}}\leq \frac{3}{2}$
2
votes
1answer
121 views

If $m,n\in \mathbb N$ and $n>m$, prove that $\text{lcm}(m,n)+\text{lcm}(m+1,n+1)>\frac{2mn}{\sqrt{n-m}}$.

Where $\text{lcm}$ is the least common multiple. I've changed it to: $$\frac{mn}{\gcd(m,n)}+\frac{(m+1)(n+1)}{\gcd(m+1,n+1)}>\frac{2mn}{\sqrt{n-m}}$$ Can't see how to continue. Is there a way to ...
2
votes
2answers
320 views

Mean Value Theorem: Real Analysis

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 ...