Questions tagged [inequality]

Questions on proving, manipulating and applying inequalities.

-1
votes
1answer
37 views

proof of inequality with absolute-value and function

Prove that if $|f'(x)| \leq M$, for all $x \in [a,b]$, then $$f(a)-M(b-a)\leq f(b) \leq f(a)+M(b-a)$$ What I tried so far: Proof. Let $x \in [a,b]$ Assume $|f'(x)|\leq M$ Show $f(a)-(Mb-Ma)\leq f(...
2
votes
1answer
52 views

Given $ a_{k+1} \ge \frac{k a_{k}}{(a_{k}^{2} + k-1)}, \:\: k > 0$, prove $ S_{n} = a_{1} + .. + a_{n} \ge n, \:\: n \ge 2 $

Suppose a sequence of positive real numbers with $$ a_{k+1} \ge \frac{k a_{k}}{(a_{k}^{2} + k-1)}, \:\: k > 0$$ prove that $$ S_{n} = a_{1} + .. + a_{n} \ge n, \:\: n \ge 2 $$ Solution: I will ...
4
votes
1answer
106 views

Prob. 5, Sec. 6.2, in Bartle & Sherbert's INTRO TO REAL ANALYSIS, 4th ed: How to show this function is strictly decreasing using derivative

Here is Prob. 5, Sec. 6.2, in the book Introduction To Real Analysis by Robert G. Bartle & Donald R. Sherbert, 4th edition: Let $a > b > 0$ and let $n \in \mathbb{N}$ satisfy $n \geq 2$. ...
2
votes
1answer
103 views

Nice olympiad inequality

Let's go for an olympiad inequality : let $a,b,c>0$ then we have : $$\sum_{cyc}\frac{ab}{a+b}\geq \frac{3\sqrt{3}}{2}\sqrt{\frac{abc}{a+b+c}}$$ My proof : $$\sum_{cyc}\frac{ab}{a+b}=\frac{(a^...
2
votes
2answers
122 views

Why AM-GM inequality showing different results?

I was given to find the minimum of $$1+a_1+a_2+a_3+...a_n$$. It was given that $$a_1\times{a_2}\times{a_3}...a_n =c$$ My approach Using AM-GM inequality $1+a_1+a_2+...a_n\ge (n+1)c^{\frac{1}{n+1}}$...
1
vote
2answers
43 views

Put the values of $a , b$ and $c$ in the equation $b^2 - 4ac \geq 0$

In the equation $a = (y - 1) , b = 0 , c = y$. Put the values in the following equation $$b^2 - 4ac \geq 0$$ Just tell me what answer you guys got. The answer should be $0 \leq y \leq 1$ but my ...
0
votes
0answers
46 views

Prove that the function $f(x)$ is increasing for $x<0$ and decreasing for $x>0$

I'm interested by the following problem : Let $a,b,c>0$ and $x>0$ then the following function is decreasing : $$f(x)=\sqrt{\frac{(abc)^x}{a^x+b^x+c^x}}\Big(\frac{1}{7a^x+b^x}+\frac{1}{7b^x+...
0
votes
1answer
29 views

Series with exponential upper bound

Let $K > 0$. Is it then true that there is some constant $C$ independent of $K$ such that $$\sum_{n=0}^\infty e^{-2^n K} \leq C e^{-K/C}$$ Thanks for the help!
1
vote
2answers
47 views

Show that $\frac{j(j-1)}{2n}> \frac{j^2}{4nr}$

In Lemmas 8.5 and 8.6 in book Irrational Numbers by Ivan M. Niven it uses the following : $$\frac{j(j-1)}{2n}> \dfrac{j^2}{4nr}$$ $n \ge 2$, $2 \le j \le n$ and $r \ge 2$, that's it! How the ...
3
votes
3answers
113 views

Prove that $(x + y + z)^3 + 9xyz \ge 4(x + y + z)(xy + yz + zx)$ where $x, y, z \ge 0$.

Prove that $(x + y + z)^3 + 9xyz \ge 4(x + y + z)(xy + yz + zx)$ where $x, y, z \ge 0$. This has become the norm now... This problem is adapted from a recent competition. We have that $6(x^2y + xy^2 ...
2
votes
1answer
61 views

Can we use QM-AM inequality to solve this?

There are two sequences (${a_1,a_2,a_3,....,a_n })$ and $( {b_1,b_2,b_3,....,b_n})$ such that $$\sum_{i=1}^n a_i = \sum_{i=1}^n b_i$$ Prove that: $$\sum_{i=1}^n \frac{a_i^2}{a_i+b_i} \ge \...
2
votes
2answers
79 views

Calculate the maximum value of $\frac{x^2}{x^4 + yz} + \frac{y^2}{y^4 + zx} + \frac{z^2}{z^4 + xy}$ where $x, y, z > 0$ and $x^2 + y^2 + z^2 = 3xyz$.

$x$, $y$ and $z$ are positives such that $x^2 + y^2 + z^2 = 3xyz$. Calculate the maximum value of $$\large \frac{x^2}{x^4 + yz} + \frac{y^2}{y^4 + zx} + \frac{z^2}{z^4 + xy}$$ This is (obviously) ...
13
votes
5answers
362 views

Interesting inequality $\frac{x^{m+1}+1}{x^m+1} \ge \sqrt[2m+1]{\frac{1+x^{2m+1}}{2}}$

Prove that $$\frac{x^{m+1}+1}{x^m+1} \ge \sqrt[k]{\frac{1+x^k}{2}}$$ for all real $x \ge 1$ and for all positive integers $m$ and $k \le 2m+1$. My work. If $k \le 2m+1$ then $$\sqrt[k]{\frac{1+x^k}{...
0
votes
1answer
34 views

A Euler summation like inequality.

Let $f: \mathbb{R} \to \mathbb{C} $ be a continuously differentiable function. Let $n$ be an integer. Why do we have: $$\int_n^{n+1} f(t) dt = f(n)+O\left(\int_n^{n+1} |f’(t)| dt\right)?$$
1
vote
3answers
58 views

$\frac{xy}{z^2(x + y)} + \frac{yz}{x^2(y + z)} + \frac{zx}{y^2(z + x)} \ge xy + yz + zx$ given that where $x, y, z > 0$ and $xyz = \frac{1}{2}$.

$x$, $y$ and $z$ are positives such that $xyz = \dfrac{1}{2}$. Prove that $$ \frac{xy}{z^2(x + y)} + \frac{yz}{x^2(y + z)} + \frac{zx}{y^2(z + x)} \ge xy + yz + zx$$ Before you complain, this problem ...
1
vote
1answer
29 views

Bound $\|(A+B)-(A^{1/4}(1+A^{-1/2}BA^{-1/2})^{1/2}A^{1/4})^2\|$ in terms of commutator $\|AB-BA\|$

For positive definite matrices $A$ and $B$, can $$ \|(A+B)-(A^{1/4}(1+A^{-1/2}BA^{-1/2})^{1/2}A^{1/4})^2\| $$ be bounded in terms of $\|AB-BA\|$? Note that if the matrices commute, then both norms ...
-1
votes
1answer
62 views

Nice refinement of an inequality by Michael Rozenberg

It's related to this If $a+b+c=abc$ then $\sum\limits_{cyc}\frac{1}{7a+b}\leq\frac{\sqrt3}{8}$: In fact we have this refinement (wich I think much easier) : Let $a,b,c>0$ then we have : $$\...
0
votes
2answers
29 views

Doubt in basics of inequalities

While solving a problem I encountered with this $ 4m(4m-1) \leq 0 $ Solving this one gets the solution $ 0 \leq m \leq 1/4 $ Now, if the inequation is multiplied with -1 on both sides we get $ 4m(1-4m)...
0
votes
2answers
40 views

Induction with nth root of n

I am trying to prove by induction that $\sqrt[n]{n}<2-\frac{1}{n}$ where $n\ge2$. It seemed simple at first, but I am stuck with $log(2n-1)$ in the RHS. I am in an elementary undergraduate Maths ...
0
votes
1answer
41 views

Should $x=2$ be included in the solution?

Consider an inequation $\frac{(x-2)^6 (x-3)^3 (x-1)}{(x-2)^5 (x-4)^3}>0$. $(x-2)^5$ can be cancelled from both the numerator and denominator with the condition that $x\neq2$ leaving $(x-2)$ in the ...
7
votes
1answer
235 views

New bound for Am-Gm of 2 variables

Today I'm interested by the following problem : Let $x,y>0$ then we have : $$x+y-\sqrt{xy}\leq\exp\Big(\frac{x\ln(x)+y\ln(y)}{x+y}\Big)$$ The equality case comes when $x=y$ My proof uses ...
2
votes
2answers
79 views

Calculate the maximum value of $\frac{ab}{ab + a + b} + \frac{2ca}{ca + c + a} + \frac{3bc}{bc + b + c}$ where $3a + 4b + 5c = 12$

$a$, $b$ and $c$ are positives such that $3a + 4b + 5c = 12$. Calculate the maximum value of $$\frac{ab}{ab + a + b} + \frac{2ca}{ca + c + a} + \frac{3bc}{bc + b + c}$$ I want to know if there are ...
0
votes
3answers
45 views

How to solve $\left(1+\frac{1}{n+2}\right)^{n+2}\geq \left(1+\frac{1}{n+1}\right)^{n+1}$ for all $n\in \mathbb{N}$ [duplicate]

any impulses, suggestions? I have been trying for a while but it doesn't get me anywhere... Kind regards
0
votes
2answers
34 views

Solve a system of linear inequalities without graphing, i.e. using pure algebra?

I have a system of linear inequalities such as: $y\geq 2x+1$ $y>(x/2)-1$ I know how to solve this system of inequalities (in order to find possible values that will satisfy it) by graphing the ...
6
votes
1answer
113 views

For $a,b,c,d > 0$ and $abcd = 1$, show that $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} + \frac{12}{a + b + c + d} \geq 7$

Question: For $a,b,c,d > 0$ and $abcd = 1$, show that $$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} + \frac{12}{a + b + c + d} \geq 7$$ My Attempts: Trivial to see that equality ...
0
votes
1answer
50 views

Find the maximum of an expression [duplicate]

If $a,b,c$ and $d$ are positive integers such that $$a+b+c+d = 63$$ Find the maximum value of $$ab +bc +cd$$ My turn : Using AM-GM $$4ab \leq (a+b)^2$$ Then $$4(ab + bc + cd) \leq (a+b)^2 + (b+c)...
2
votes
1answer
57 views

Prove that $\sum_{cyc}\frac{1}{x^2 + 1} \ge \frac{2}{3}\bigl(\sum_{cyc}\frac{x}{\sqrt{x^2 + 1}}\bigr)^3$ where $x, y, z > 0$ and $xy + yz + zx = 1$.

$x$, $y$ and $z$ are positives such that $xy + yz + zx = 1$. Prove that $$\large \frac{1}{x^2 + 1} + \frac{1}{y^2 + 1} + \frac{1}{z^2 + 1} \ge \frac{2}{3}\left(\frac{x}{\sqrt{x^2 + 1}} + \frac{y}{\...
3
votes
1answer
80 views

Prove inequality without using Möbius transformations

Is there a simple way to prove the following inequality without using Möbius transformations: $\left| \frac{{z}_{1}-{z}_{3}}{1-{z}_{3}\bar{{z}_{1}}} \right| \leq \left| \frac{{z}_{1}-{z}_{2}}{1-{z}_{...
0
votes
1answer
21 views

relations betweem Schaten norms

Let for a matrix A $\sigma(A)=(\sigma_1(A) \ldots \sigma_n(A))$ be a sequence of it singular values. The p-th Schatten norm is defined as $$ \|A\|_{S_p}=\|\sigma(A)\|_p, \quad 1\leq p \leq \infty. $$ ...
5
votes
3answers
127 views

Let $A, B$ and $C$ be the angles of an acute triangle. Show that: $\sin A+\sin B +\sin C > 2$. [duplicate]

Let $A, B$ and $C$ be the angles of an acute triangle. Show that: $\sin A+\sin B +\sin C > 2$. I started from considering $$\begin{align}\sin A+\sin B+\sin (180^o-A-B) &= \sin A+\sin B+\sin(A+B)...
0
votes
0answers
23 views

Extend an interval

A few days ago I asked this question, that basically, it said: If it is stated that, $-2.2\leq p \leq 2.3$. Can you say that, $|p| \leq 2.4$? This is true, because, $[2.2,2.3] \subset [-2.4, 2.4]$ ...
-1
votes
0answers
59 views

Show that $(\sqrt2-1)\cdots(\sqrt[n+1]{(n+1)!} - \sqrt[n]{n!}) < \frac{n!}{(n+1)^n}$ [duplicate]

If we let $\xi(n) = \prod_{j=1}^n (\sqrt[J+1]{(j+1)!} - \sqrt[j]{j!})$ then I want to show that $\xi(n)< \frac{n!}{(n+1)^n}$. Here is my previous failed attempt! First prove this holds for $n=...
3
votes
2answers
106 views

Show that $n! < \frac{(n+1)^{n^2+n+1}}{(n+2)^{n^2+1}}$ for $n\ge 2$

$n! < \frac{(n+1)^{n^2+n+1}}{(n+2)^{n^2+1}}$ For very small values of $n$ (i.e. $2\le n\le 6$) the function on the right nicely approximates $n!$ before significantly overtaking it. I don't have ...
2
votes
9answers
170 views

Minimize this real function on $\mathbb{R}^{2}$ without calculus?

When it comes to minimizing a differentiable real function, calculus comes into play immediately. If $f: (x,y) \mapsto (x+y-1)^{2} + (x+2y-3)^{2} + (x+3y-6)^{2}$ on $\mathbb{R}^{2}$, and if one is ...
0
votes
1answer
21 views

Reverse triangle inequality for square of euclidean norm?

The reverse triangle inequality as listed on https://en.wikipedia.org/wiki/Triangle_inequality#Reverse_triangle_inequality shows that $| ||x||_2 - ||y||_2 | \leq ||x -y||_2$ for $x,y \in \mathbb R^n$,...
-1
votes
0answers
68 views

Prove that $n!^2 > n^n$ [duplicate]

I am unable to solve this following inequality: Prove that $n!^2 > n^n$, where $n>1$
3
votes
0answers
53 views

Inequality on a “slice” of the Gaussian (standard normal) integral.

Let $\phi$ denote the standard Gaussian pdf, i.e., $\phi(x) = \frac{1}{\sqrt{2\pi}}e^{-x^2/2}$ for $x\in \mathbb{R}$; and $\alpha,\delta\in(0,1]$. Define $t\in\mathbb{R}$ as $$ \alpha = \int_t^\infty ...
0
votes
1answer
10 views

Maximum Singular Value of Sum of Positive Semidefinite Matrices

We have two real matrices $A$ and $B$. Let $\sigma_{\max}(A)$ and $\sigma_{\max}(B)$ denote the maximum singular value of matrices $A$ and $B$, respectively. Intuitively, the maximum singular value of ...
1
vote
1answer
27 views

How to get $\mid y\Big[\frac{x}{y}\Big]-a\mid <\mid x-a\mid + \mid y \mid \quad x,y\in R$?

The solution of a problem I had to work on states that : $$\mid y\Big[\frac{x}{y}\Big]-a\mid <\mid x-a\mid + \mid y \mid \quad x,y\in R$$ Where $[x]$ is the integer part of $x$. I don't know ...
0
votes
3answers
55 views

Question about inequality?

if: $-2 \leq x \leq 3$ , then what interval does $x^{2}$ belongs to ? my solution was $: 0 \leq x^{2} \leq 9$ according to high school knowledge. if it true why did the negative side turn to be 0$?$ ...
0
votes
2answers
40 views

Why are these inequalities contradictory? How would I go about proving that algebraically?

You don't need to understand the specifics of this explanation or the context, just the inequalities: I don't understand how we can prove that there's a contradiction. If $w_{0} > -w_{1}$ and $w_{...
1
vote
5answers
95 views

Prove for all positive a,b,c that $\frac{a+b}{c}+\frac{b+c}{a}+\frac{c+a}{b} \geq6$

Prove for all positive a,b,c $$\frac{a+b}{c}+\frac{b+c}{a}+\frac{c+a}{b} \geq6$$ My Try I tried taking common denominator of the expression, $\frac{a^2b+ab^2+b^2c+c^2b+ac^2+a^2c}{abc}$ How to ...
0
votes
1answer
27 views

Jensen gap for a real-valued random variable $X \geq 0$

For a real-valued random variable $X \geq 0$, We have $1 - \frac{1}{1+E \left[x\right]} \geq E \left[ 1 - \frac{1}{1+x} \right]$ (Jensen's inequality). We want to get a tight constant gap between $1 ...
0
votes
0answers
5 views

inequality in the proof for expected number of isolated vertices in $G(n,p)$ graph for $p>\log(n)/n$

In the following paper (https://people.maths.bris.ac.uk/~maajg/teaching/complexnets/connected-giantcompt.pdf) they give a proof for the connectivity of the ER random graph $G(n,p)$ for $p=\frac{log(n)+...
-1
votes
1answer
62 views

Prove $0< \frac{x(\,1- x\,)y(\,1- y\,)}{1- xy}\leqq \left \{ \frac{\sqrt{5}- 1}{2} \right \}^{\,5}$ at $0< x,\,y< 1$ [duplicate]

At $0< x,\,y< 1$ $$\displaystyle 0< \frac{x(\,1- x\,)y(\,1- y\,)}{1- xy}\leqq \left \{ \frac{\sqrt{5}- 1}{2} \right \}^{\,5}$$ $\lceil$ IT HOLDS ! $\rfloor$ I found many well-known series ...
0
votes
1answer
38 views

Find constant for upper bound

I want to show that there is a constant $C_{\epsilon}$ such that $$C\left(\frac{\sqrt{t}+x}{\sqrt{t}}\right)^{\alpha}\leq C_{\epsilon}\exp\left(\epsilon\frac{x^{2}}{t}\right)$$ for every $\epsilon>...
0
votes
2answers
71 views

Find the minimum value of $\sum_{cyc}\frac{1}{\sqrt{a^2 - ab + 3b^2 + 1}}$ where $a, b, c > 0$ and $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} \le 3$.

$a$, $b$ and $c$ are positives such that $\dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c} \le 3$. Calculate the maximum value of $$\large \frac{1}{\sqrt{a^2 - ab + 3b^2 + 1}} + \frac{1}{\sqrt{b^2 - bc + 3c^...
0
votes
2answers
24 views

Relatively compact set theorem from Billingsley's Convergence of Probability Measure

x is a continuous function on [0,1] with uniform topology. The part I don't understand is second from last equation, where we have inequality of x(t) and x(0) with sum. How can I make sense of that ...
-1
votes
3answers
135 views

Find $\max\,2\,x- y,\,\min\,2\,x- y$

$x^{\,2}+ y^{\,2}= e^{\,2(\,x- 2\,y\,)}$. Find $$\max\,2\,x- y \tag{and min}$$ I used this to prep for another senior student's university entrance exam in $\lceil$ diendantoanhoc.net $\rfloor$ But ...
2
votes
0answers
34 views

For positive $a$ and $b$, and for $u\in[0,1]$, can we solve for $v$ in $a>b\sqrt{1-v^2}+\sqrt{1-(u+v)^2}$?

Let $a$ and $b$ be positive numbers and $u \in [0, 1]$. Can we solve this inequality for $v$? $$ a > b \sqrt{1-v^2}+ \sqrt{1-(u+v)^2} $$ I need the points for $v$ in the interval $[-1, 1-u]$ for ...