Questions on proving, manipulating and applying inequalities.

learn more… | top users | synonyms (1)

1
vote
6answers
83 views

Prove by induction the particular inequality $\left(1.3\right)^n \ge 1 + \left(0.3\right)n$ for every $n \in \mathbb N$

$\left(1.3\right)^n \ge 1 + \left(0.3\right)n$ for every $n \in \mathbb N$ Not sure where I'm going wrong in my Algebra, but I assume it's because I'm adding an extra term. Is the extra term ...
2
votes
1answer
58 views

Solution of $1+\frac{\cos 2x}{\sin x}+\tan x \geq 30$

Solve the given inequality: $$1+\frac{\cos 2x}{\sin x}+\tan x \geq 30$$ I am trying to convert L.H.S. in terms of one trigonometric ratio but it is not happening here. Could someone suggest some ...
1
vote
2answers
79 views

Proof of an inequality involving three numbers $(a,b,c)\gt 0$

If $(a,b,c)\gt 0$ the following inequality holds: $$\dfrac{a^2}{2a^2+(b+c)^2}+\dfrac{b^2}{2b^2+(c+a)^2}+\dfrac{c^2}{2c^2+(a+b)^2}\lt\dfrac{2}{3}$$ I am stuck to find a proof of it. Can someone help me?...
1
vote
0answers
33 views

Upper bounding a sum of products

Let $a_k$ be an integer valued sequence, $a_k \in \mathbb{N}^+$ and let $b_k = \#\{i: a_i=1,\; i \leq k\}$ and assume that $b_k=o(k)$ (little o notation). How to prove that there exists a constant $...
6
votes
1answer
113 views

Prove inequality $\frac a{1+bc}+\frac b{1+ac}+\frac c{1+ab}+abc\le \frac52$

For real numbers $a,b,c \in [0,1]$ prove inequality $$\frac a{1+bc}+\frac b{1+ac}+\frac c{1+ab}+abc\le \frac52$$ I tried AM-GM, Buffalo way. I do not know how to solve this problem
0
votes
0answers
29 views

On inequalities related with $f(s):=-(1-\frac{2}{2^s})^{-1}$

My Question. a) How can you prove easily that the multivariable function in LHS is positive on $x^2+y^2<1$ $$2^{1-x}\cos(y\log 2)-1>0?$$ b) Let $s=\sigma+it$ the complex variable, ...
1
vote
1answer
57 views

Question based on $\triangle$ and Arithmetic progression.

If $a,b,c$ are the sides of a $\triangle ABC$ which are in Arithmetic progression. Then Range of $\displaystyle \frac{b}{c}$ $\bf{My\; Try::}$ In a Triangle Given that sides $a,b,c$ are in ...
0
votes
0answers
29 views

$|xy(x^2-y^2)+yz(y^2-z^2)+xz(z^2-x^2)|\le J (x^2 + y^2 + z^2)^2$, What is the smallest value of J that allows inequality

$\forall (x,y,z)\;\in\;\mathbb R$ $|xy(x^2-y^2)+yz(y^2-z^2)+xz(z^2-x^2)|\le J (x^2 + y^2 + z^2)^2$ How we can find for the smallest value of "$J$" when providing to inequality Effort: I could only ...
2
votes
1answer
37 views

Asymptotic lower bound of this function

Suppose that $n$ is an even number. Let $$f(n)=\frac{\sum_{j=1}^{n/2}\binom{n}{2j}\log(2j)}{2^{n-1}}.$$ Can we find some function $g(n)$ (e.g. $\log(n)$ or $n^\alpha$) such that $f(n)=\Omega(g(n))$? ...
1
vote
2answers
58 views

Finding solution for irrational inequality

I have given an inequality: $\frac{x-2\sqrt{x} - 3}{x + \sqrt{x} - 2} < 0$. Now trying to solve it, I'm doing substitution for $a = \sqrt{x}$ and then solving irrational equations as quadratics $a^...
3
votes
1answer
37 views

Proof of inequality with local martingale and stopping time..

Let $M$ be continuous local martingale starting from zero. For $a>0$, let $\tau_a=\inf \left\{t \ge 0: |M_t|>a \right\}$. Show that for every $t \ge 0$ we have: $a^2\mathbb{P}(\tau_a\le t)\le\...
4
votes
0answers
90 views

A Matrix Norm Inequality $\|A^{1/2}B^{1/2}(A+B)^{-1/2}\|_F \geq \|A^{1/2}(A+B)^{-1/2}B^{1/2}\|_F$

Let $\|X\|_F:= \sqrt{\text{Tr} \left(XX^\dagger\right) }$ denote the Frobenius norm. Does anyone know how to show the norm inequality: $\left\|A^{\frac12}B^{\frac12}(A+B)^{-\frac12}\right\|_F \geq ...
1
vote
1answer
41 views

How to prove $(xy+yz+zx)^3 \geq xyz(x+y+z)^3$ for $x,y,z \geq 0$?

This inequality seems to be true for all $x,y,z \geq 0$, but I'm not sure how to prove it: $$(xy+yz+zx)^3 \geq xyz(x+y+z)^3$$ After expanding and simplifying we obtain: $$x^3y^3+y^3z^3+z^3x^3 \...
2
votes
1answer
50 views

Species of Machin’s formula?

I read the following problem: $$A=6\arctan(\frac 18)+2\arctan(\frac{1}{57})+\arctan (\frac{1}{239})$$ $$B=24\arctan(\frac{1}{12943})-12\arctan(\frac{1}{682})+44\arctan((\frac{1}{57})+7\arctan(\frac{1}{...
-3
votes
1answer
64 views

Check if it's true that $e^{\sin(3.14)} e ^{3.14} \leqslant e^{\sin(3.15)} e ^{3.15} $ [closed]

Is that true that $e^{\sin(3.14)} e ^{3.14} \leqslant e^{\sin(3.15)} e ^{3.15}$? If yes, does the equality takes place? I transformated it into: $\sin(3.14) - \sin(3.15) \leqslant 0.01$. How to ...
0
votes
3answers
94 views

How to prove $\left | \sin(x)-\sin(c) \right |\leq \left | x-c \right |$ for $c$ constant [closed]

I have no idea about it.Your help will be appreciated!
0
votes
1answer
72 views

The inequality related with normal distribution

When $Z \sim N(0,1)$, then $P(\vert Z\vert \gt t)\le\sqrt\frac{2}{\pi}\times \frac{exp(-t^2/2)}t$. My question is: I can get the aboving inequality, but how can I get the inequality such that $P(\...
1
vote
2answers
38 views

Prove that: $\left( \sum_{k=1}^{6}a_kb_k\right)^2 \leq \sum_{k=1}^{6}a_kb_k^2$

Q: Let $a_1, a_2, a_3,a_4, a_5, a_6$ be positive real number whose sum is 1 and $b_1,b_2,b_3,b_4,b_5,b_6$ real number. Prove that: $$\left( \sum_{k=1}^{6}a_kb_k\right)^2 \leq \sum_{k=1}^{6}a_kb_k^2$$...
1
vote
1answer
333 views

Ramsey number $R(n,n) > (n-1)^2$

I got an home work assignment, prove that: $R(n,n) > (n-1)^2$ Note that I saw on Wikipedia that for subgraph of $K_n$ with k vertices, $R(k,k) > 2^{k/2}$. I tried to work with that, but ...
3
votes
1answer
70 views

Prove that $ A=\sqrt{\frac{(b-c)^2}{a^2}+\frac{(c-a)^2}{b^2}+\frac{(a-b)^2}{c^2}} $ is also rational number.

Let $a,b,c\in\mathbb{Q}$ distinct and none of them equal to $0$ satisfying $\frac{a^2}{(b-c)^2}+\frac{b^2}{(c-a)^2}+\frac{c^2}{(a-b)^2}\leq 2. $ Prove that $ A=\sqrt{\frac{(b-c)^2}{a^2}+\frac{(c-a)^2}{...
0
votes
1answer
45 views

Solving irrational inequality

Given inequality $(x - 2)\sqrt{x^2 + 1} > x^2 + 2$, find it's solution as intervals. And I have problem solving it. So at first, both $\sqrt{x^2 + 1} > 0$ and $x^2 + 2 > 0$. That means, that ...
0
votes
1answer
53 views

$a_ia_j+b_ib_j\leq \sqrt{a_i^2+b_i^2} \sqrt{a_j^2+b_j^2} $

In Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ with the standard inner product, the Cauchy–Schwarz inequality is $$\left(\sum_{i=1}^n u_i v_i\right)^2\leq \left(\sum_{i=1}^n u_i^2\right) \left(...
1
vote
1answer
124 views

Find the maximum of the value $\frac{\sin^2{A}+\sin^2{B}+\sin^2{(A+B)}}{\sin{(A+B)}}$ [closed]

For all triangles $\Delta ABC$ that satisfy $$\sin{A}\sin{B}=\sin{C},$$ define $$f(A,B)=\dfrac{\sin^2{A}+\sin^2{B}+\sin^2{(A+B)}}{\sin{(A+B)}}=\dfrac{\sin^2{A}+\sin^2{B}}{\sin{C}}+\sin{C}=\dfrac{\...
1
vote
2answers
57 views

Prove $\frac{3\sqrt{xyz}}{\sqrt{x}+\sqrt{y}+\sqrt{z}} \leq \frac{2}{3} \left(\frac{xy}{x+y}+\frac{yz}{y+z}+\frac{zx}{z+x} \right)$ for $x,y,z \geq 0$

This inequality is wrong - see the accepted answer (it appears there is no general inequality for these two expressions). On the left we have harmonic mean of pairwise geometric means, which obeys: $...
-2
votes
2answers
40 views

Let $S_n$ be the sum of all integers $k$ such that $2^n<k<2^{n+1}$ [closed]

Then $9$ divides $S_n$ if a) $n$ is odd b) $n$ is even c) $n$ is of the form $3k +1$ d) $n$ is of the form $3k +2$
0
votes
1answer
62 views

How could we obtain $\lim_{n \to \infty} \frac{\lambda_n}{n}=\frac{4 \pi}{ab}$?

Related to the example on the rectangle in the book Strauss W.A. Partial differential equations - an introduction (Wiley, $2008$, $2$nd Ed.) page $326$, is there anyone could explain to me how is it ...
1
vote
1answer
30 views

Liminf, Limsup inequalities in Cesàro's Lemma proof

Probability with Martingales: I tried writing out the details of the proof avoiding $\ge$ if I felt it was unnecessary. Please tell me if I got any steps wrong. $$\liminf \frac{1}{b_n} \...
1
vote
1answer
35 views

$|1-e^{2\pi i n z}|<1$ for $z$ in the upper half plane

If $z$ is in the upper half plane, then $|e^{2 \pi ni z}|<1$ for every $n\in\mathbb{N}$. But why is also $|1-e^{2\pi i n z}|<1$? I just get $$|1-e^{2\pi i n z}|\leq1+|e^{2 \pi ni z}|<1+1=2....
2
votes
1answer
40 views

Prove $\frac{\sqrt{x^2+y^2}+\sqrt{y^2+z^2}+\sqrt{z^2+x^2}}{3 \sqrt{2}} \geq \frac{\sqrt{(x+y)^2+(y+z)^2+(z+x)^2}}{2 \sqrt{3}}$ for $x,y,z \geq 0$

On the left we have arithmetic mean of pairwise quadratic means, which obeys: $$\sqrt{\frac{x^2+y^2+z^2}{3}} \geq \color{blue}{ \frac{\sqrt{x^2+y^2}+\sqrt{y^2+z^2}+\sqrt{z^2+x^2}}{3 \sqrt{2}} } \geq ...
1
vote
1answer
31 views

For what value of $x$: $ n^ {(x+1)} + n^ {2x} < n^2$ ? Where, $0\leq x <1$ and $n$ is constant integer value & $n>1$.

How to find the optimal value of $x$ and what is the relation between $x$ and $n$ i.e. How to get dependency between $x$ and $n$? As per my understanding, solution should be in term of $n$ like like ...
0
votes
0answers
29 views

Proof check: Isoperimetric inequality

My Proof: Let the arc-length parametrization of the curve be $\gamma(s) = \langle x(s),y(s)\rangle$. By Green's Theorem, the area $\mathcal{A}$ is $$ \mathcal{A} = \int_{0}^{2\pi} -y(s)x'(s) dx$$ ...
0
votes
1answer
42 views

Is that inequality true that for every a,b,c?

Is that inequality true that for every real a,b,c? I don't have any idea how to even start this problem.
1
vote
0answers
29 views

Inequalities in weak $L^1$ norm

I have the following lemma: Suppose that for $j=1,2,\dots,$ $g_j(x)$ is a nonnegative function on a measure space for which $\left|\left\{x: g_j (x) >s\right\}\right|<1/s$. Let $\{c_j\}$ be a ...
1
vote
0answers
23 views

prove Fisher's inequality not using matrix method .

Fisher's inequality, is a necessary condition for the existence of a balanced incomplete block design which satisfies certain prescribed conditions in combinatorial mathematics. Let: v : be the ...
2
votes
2answers
100 views

Prove $\frac{\sqrt[3]{(x+y)(y+z)(z+x)}}{2} \geq \sqrt{\frac{xy+yz+zx}{3}}$ for $x,y,z \geq 0$

We have geometric mean of pairwise arithmetic means on the left, which obeys the following inequality: $$\frac{x+y+z}{3} \geq \color{blue}{ \frac{\sqrt[3]{(x+y)(y+z)(z+x)}}{2} } \geq \sqrt[3]{xyz}$$ ...
2
votes
1answer
36 views

How to prove an inequality or read Mathematica's proof

Given $0 < \alpha_1, \alpha_2 <1$ and $1/2 < p <1$, how to prove that the following expression is always non-negative? \begin{align*} (1 - \alpha_2)^2 p^2 (2 p -1) + \alpha_1^2 \, p\, \...
2
votes
1answer
135 views

Prove $(a+b)^3(b+c)^3(c+d)^3(d+a)^3\ge 16a^2b^2c^2d^2$

Let $a,b,c,d>0$ and such $a+b+c+d=1$, show that $$(a+b)^3(b+c)^3(c+d)^3(d+a)^3\ge 16a^2b^2c^2d^2$$ since $$a+b\ge 2\sqrt{ab}$$ I think this will not hold.because we have $256abcd\le 1$
5
votes
0answers
32 views

$\small| U-\frac{m}{n}\small| \leq \frac{1}{n^3}$

Let $U$ be uniform distributed in $[0,1]$ . Show that with probability $1$ there's maximum a finite amount of $n \in \mathbb N$, so that the inequality $\small| U-\frac{m}{n}\small| \leq \frac{1}{n^3}...
-1
votes
1answer
44 views

$ab(a^2-b^2)+bc(b^2-c^2)+ac(c^2-a^2)\geq 9.\left[\dfrac{(a-b)(a-c)(b-c)}{ab+ac+bc}\right]a.b.c$

I created this problem but I want to learn ,if exists, other solution ways. $ab(a^2-b^2)+bc(b^2-c^2)+ac(c^2-a^2)\geq 9.\left[\dfrac{(a-b)(a-c)(b-c)}{ab+ac+bc}\right]a.b.c$ My answer; $ab(a^2-b^2)+...
2
votes
2answers
54 views

How $2 \sqrt[3]{\frac{x^2y^2z^2}{(x+y)(y+z)(z+x)}}$ and $\frac{3xyz}{x\sqrt{yz}+y\sqrt{zx}+z\sqrt{xy}}$ are related for $x,y,z \geq 0$?

The first expression is a heometric mean of pairwise harmonic means and obeys the following inequality: $$\sqrt[3]{xyz} \geq \color{blue}{ 2 \sqrt[3]{\frac{x^2y^2z^2}{(x+y)(y+z)(z+x)}}} \geq \frac{...
-3
votes
4answers
43 views

Algebra Problem regarding powers [closed]

if $a \gt b$ prove that $a^3+2ab^2\gt b^3+2a^2b$
0
votes
3answers
68 views

Prove $n^2 \geq n$ for every integer

I am having some trouble with this proof. Part of it is that I have to prove it for every integer. Does this mean I have an inductive step that goes for $P(k+1)$ and $P(k-1)$? Assuming my base case ...
0
votes
0answers
21 views

Arithmetic mean and geometric mean inequality - proving $[(1+a)(1+b)(1+c)]^7 > 7^7a^4b^4c^4$ [duplicate]

If $a,b,c > 0$, then prove that $[(1+a)(1+b)(1+c)]^7 > 7^7a^4b^4c^4$ I reached till : $8^7 > 7^7 \sqrt{abc}$ Can't move forward from there. Please help!
3
votes
2answers
185 views

Extending the ordered sequence of 'three-number means' beyond AM, GM and HM

I want to create an ordered sequence of various 'three-number means' with as many different elements in it as possible. So far I've got $12$ ($8$ unusual ones are highlighted): $$\sqrt{\frac{x^2+y^2+...
2
votes
2answers
69 views

How does $\sqrt{x^2+y^2}+\sqrt{y^2+z^2}+\sqrt{z^2+x^2}$ relate to $\sqrt{x^2+y^2+z^2}$?

Another possible 'mean' for three positive real numbers $x,y,z$ is made of pairwise quadratic means: $$\frac{\sqrt{x^2+y^2}+\sqrt{y^2+z^2}+\sqrt{z^2+x^2}}{3 \sqrt{2}}$$ By QM-AM inequality it is ...
3
votes
0answers
36 views

On an inequality concerning vectors with norm less than one

Consider two sets of vectors $\{v_i\}_{i=1}^{n_1}$ and $\{w_j\}_{j=1}^{n_2}$ with $v_i,w_j\in\mathbb{R}^n$ such that $\|v_i\|_2< 1$ and $\|w_j\|_2< 1$ for all $i=1,\dots,n_1$ and $j=1,\dots,n_2$....
0
votes
3answers
37 views

Why is solution to this inequality equal to $\mathbb{R}$?

Given inequality $$-5(1- x)^2 < 3x + 11$$ which after algebraic manipulation looks like $$5x^2 - 7x + 16 > 0,$$ it is obvious that it's discriminant is equal to $\Delta = -271$. In my book it is ...
1
vote
3answers
52 views

Arithmetic Mean And Geometric Mean [duplicate]

If $A.M$ and $G.M$ are Arithmetic Mean And Geometric Mean respectively then prove that $A.M >=G.M$. My Attempt : Let $a$ and $b$ are any two real positive numbers. Then: $$A.M=\frac{a+b}{2}$$ $...
3
votes
3answers
26 views

How to show that $\frac{R_1R_2}{R_1+r_2}<(R_1,R_2)$ strictly using AM-GM inequality?

I was reading about parallel circuits in Physics.Equivalent resistance of $n$ resistors in parallel is given by $\displaystyle\frac1{R_{eq}}=\frac{1}{R_1}+\frac{1}{R_2}+...+\frac{1}{R_n}$. I tried to ...
0
votes
1answer
45 views

Solving $\log(x-2) + \log(9-x) \lt 1$.

The solution. Now, in comment section, a person has mentioned (and it's given in the answer behind the book) that another solution could be $2 \lt x \lt 4$. I've tried numerous times, but have not ...