Questions on proving, manipulating and applying inequalities.

learn more… | top users | synonyms (1)

0
votes
0answers
12 views

What is (if there is) the generic term for equalities and inequalities

I'm writing a text about a particular linear programming optimization problem, that is described using a mixture of inequalities (, ...
3
votes
0answers
32 views

Difficulty to prove this inequality in Binomial Coefficient.

This inequality is found in a book titled as Randomized Algorithms, by Rajeev Motwani and Prabhakar Raghavan, in Chapter 3, during explaining Occupancy Problems, to see the book click here PP. 43-44 ...
0
votes
0answers
12 views

How to verify that the following function is monotone increasing?

$\displaystyle f(x)=x\cdot\left(1-\frac{C_Bx^{B}}{\sum\limits_{k=0}^{B}C_kx^k}\right)$, where $0<x<1$, $\displaystyle C_k=\binom{n+k}{k}$, $n,B$ are integers, then, how to verify that $f(x)$ ...
1
vote
2answers
28 views

Prove that from the equalities, $\frac{x(y+z-x)}{\log x}=\frac{y(x+z-y)}{\log y}=\frac{z(y+x-z)}{\log z}$ follows $x^yy^x=y^zz^y=z^xx^z$.

Problem : Prove that from the equalities, $$\frac{x(y+z-x)}{\log x}=\frac{y(x+z-y)}{\log y}=\frac{z(y+x-z)}{\log z}$$ follows $$x^yy^x=y^zz^y=z^xx^z$$. My approach : $$\frac{x(y+z-x)}{\log ...
2
votes
0answers
14 views

Matrix product bound

Consider the following inequality \begin{align*} -AB^{-1}A^\top \preceq cI \end{align*} where $A\in\mathbb{R}^{n\times m}$, $B\in\mathbb{R}^{m\times m}$, $c\in\mathbb{R}$ (given), and $I$ is the ...
5
votes
4answers
87 views

Showing that $\left (\frac{\sin x}{x} \right )^3\geq \cos^{2}x$

Show that $$\left (\frac{\sin x}{x} \right )^3\geq \cos^{2}x,\forall x\in \left ( 0;\frac\pi2 \right )$$ Firstly, I had use the differentiation of $f(x)=\left (\frac{\sin x}{x} \right )^3- ...
0
votes
1answer
28 views

Why is the integral of a square always larger than the square of an integral?

I learned in physics that $\langle x^2 \rangle - \langle x \rangle ^2 = \sigma_x^2 \ge 0$ and thus $\langle x^2 \rangle \ge \langle x \rangle ^2$. In the case of continuous distribution, it becomes ...
0
votes
2answers
36 views

Is it true that $|f(x)|\leq |f^2(x)|$?

Is the following true for all $x\in\mathbb{R}$ and for all real functions f? $$\left| f(x)\right| \leq \left| f^2(x)\right|$$ Also, is it true that $|f(x)|\leq |f^3(x)|$?
3
votes
3answers
72 views

Proving an inequality between $\frac 1{n+1}$ and $\frac 1n$ and a definite integral

For all natural numbers $n$, prove that $$\frac 1{n+1} < \int_n^{n+1} \frac 1t \, dt < \frac 1n$$ I have tried working with $\frac 1{t+1} < \frac 1t < \frac 1{t-1}$ but this doesn't ...
-3
votes
0answers
39 views

$\frac 1{n+1} < \int_n^{n+1} \frac 1t \, dt < \frac 1n$ [on hold]

For all natural numbers $n$, prove that $$\frac 1{n+1} < \int_n^{n+1} \frac 1t \, dt < \frac 1n$$ (Do not use induction.) Please help me on the first step. :)
4
votes
0answers
58 views

Prove $\cos(\sin x)>\sin(\cos x)$ [duplicate]

Prove that $\cos( \sin x)>\sin(\cos x), \forall x\in\mathbb{R}$. I have thought that we should consider their difference and show it is positive for all x, so: Let $$A=\cos\sin x-\sin\cos ...
4
votes
2answers
121 views

A singular Gronwall inequality

Let $f : [0,T] \to R^+$ be a continuous function such that $f(0)=0 $ and : $$ f(t)\le C\int_0^t s^{-1}f(s) ds,\; \forall t\in [0,T] $$ for some constant $C>0.$ Is it true that $f(t)=0,\; \forall ...
0
votes
2answers
39 views

Find the limit of $\frac{x+y+\sin xy}{x^2+y^2+\sin^2 (xy)}$

Find the limit of: $$\lim_{(x,y)\rightarrow(+\infty, +\infty)}\frac{x+y+\sin xy}{x^2+y^2+\sin^2 (xy)}$$ I think the solution could be: $$\frac{x+y+\sin xy}{x^2+y^2+\sin^2 (xy)} \le \frac{x+y+\sin ...
0
votes
0answers
31 views

Finding a function satisfying a certain inequality

This is a continuation of this post where I tried to find a function $f(n)$ that would satisfy the induction step of an inductive argument and it was shown that such function does not exist. Trying ...
3
votes
4answers
77 views

Show $\frac{\sin(x)}{x}>\cos(x)$ for $0<x<\pi$ using the Mean Value Theorem

I'm trying to show the inequality $$\frac{\sin(x)}{x}>\cos(x)$$ by for $0<x<\pi$ using the Mean Value Theorem, but I don't know how to start. I can show that $\sin(x)<x$, but I can't see ...
0
votes
0answers
20 views

helping inequality for cyclic three variable inequality

Let $a\ge b \ge c\ge 0$ be reals and $a+b+c=3$ .Then prove $$c(24a^2b+25)(b^2+ac)+50b(a^2+c^2)+5bc^2\le 200+3b^2c^4$$ this one has a proof replacing $b=3-a-c$ and then using calculus but uggly ...
1
vote
0answers
62 views

cyclic three variable inequality

Let $a,b,c$ be nonnegative real numbers and $a+b+c=3$. Prove the inequality $$ \sqrt{24a^2b+25}+\sqrt{24b^2c+25}+\sqrt{24c^2a+25}\le 21 $$ I have tried to find the solution using classical ...
2
votes
1answer
20 views

Log-determinant ordering for sum of positive definite symmetric matrices

If, for real positive definite symmetric $A, B, C$, $$\log\det (A+B) \geq \log\det(A+C)$$ then can it be said that $$\log\det(B) \geq \log\det(C)?$$ NOTE: A crude form of the reverse is certainly ...
1
vote
1answer
13 views

Are binomial coefficients with fixed “denominator” log-concave?

I'm working on a problem and began suspecting that the following inequality holds. Let $k\in\mathbb{N}$ be fixed, and define $f(n)={n\choose k}$. Then $f(n)$ is log-concave in $n$, in particular if ...
1
vote
2answers
52 views

Can you verify this inequality $\binom {m^2} {m-1} \geq m^{m-1} \geq 2^{n/2}/n$

$N \geq \binom {m^2} {m-1} \geq m^{m-1} \geq 2^{n/2}/n$, given $n = 2 m\log m$. Can you prove it? Where N is the number of subfunction. This question is part of proof on finding lower bound on the ...
1
vote
2answers
26 views

Create some new numbers using $n$ arbitrary positive real numbers

Known facts: Let $a_i$, $b_i$, $i=1, \ldots, n$ be positive real numbers such that $a_1+ \cdots + a_n = a_1b_1 + \cdots +a_nb_n = 1$. Then $$b_1^{a_1}b_2^{a_2} \cdots b_n^{a_n} \leq 1.$$ ...
2
votes
2answers
35 views

Integral values satisfying a inequality

Consider the following inequality : $$\frac{x^2+a^2}{a(4+x)} \ge 1$$ I am trying to find the range of integral values of $a$ for which this inequality holds for all $x$ belongs to $(-1,1)$ I ...
-1
votes
2answers
44 views

Rank of the product of two full rank matrices

I have searched for the above topic and found some results, but the answer I am looking for is not found anywhere. Here is my question: Given $A_{m \times n}$ matrix with rank $m$, and $B_{n ...
1
vote
1answer
30 views

Inequality for the gradient of a power of absolute value

Let $U \subset \mathbb{R}^2$ be open, and let $f : U \to \mathbb{C}$ be a smooth complex-valued function which does not vanish anywhere on $U$. Let $r > 0$ be a real constant. Does the ...
2
votes
3answers
39 views

Trouble understanding inequality proved using AM-GM inequality

I am studying this proof from Secrets in Inequalities Vol 1 using the AM-GM inequality to prove this question from the 1998 IMO Shortlist. However, I'm lost on the very first line of the solution. ...
0
votes
1answer
40 views

An inequality $a_1\leq a_2\leq a_4 , a_1\leq a_3\leq a_4$

$a_1 , a_2, a_3 , a_4 , b_1 , b_2 , b_3 , b_4\in\Bbb R , p\in(0, 1)$. $a_1\leq a_2\leq a_4 , a_1\leq a_3\leq a_4 , b_1\leq b_2\leq b_4 , b_1\leq b_3\leq b_4 $. Show that $$ ...
4
votes
1answer
46 views

Inequality problem: Application of Cauchy-Schwarz inequality

Let $a,b,c \in (1, \infty)$ such that $ \frac{1}{a} + \frac{1}{b} + \frac{1}{c}=2$. Prove that: $$ \sqrt {a-1} + \sqrt {b-1} + \sqrt {c-1} \leq \sqrt {a+b+c}. $$ This is supposed to be solved using ...
0
votes
2answers
54 views

Inequality with Four Numbers

I am trying to prove the following inequality for real numbers $a,b,c,d$ all of which are greater than $1$ $8(abcd+1) > (a+1)(b+1)(c+1)(d+1)$ I tried the following approaches : Used the AM-GM ...
-2
votes
2answers
21 views

Inequality involving summation

Can someone help me with this inequality: $\sum_{i=1}^{n}{\dfrac{1}{\sqrt i}}\leq \dfrac{2n}{\sqrt{n}}$ Thank you.
2
votes
1answer
41 views

Problem understanding this specific AM-GM inequality proof

This is taken from Secrets in Inequalities by Pham Kim Hung So, this part of the proof involves proving that $f(n)$ implies $f(n-1)$. So we define a term as $a_n = \frac{s}{n-1}$. We define $s = a_1 ...
3
votes
1answer
49 views

Find the maximum and minimum of $\sum \limits_{i=1}^n x_i ^3$

Let $x_1,x_2, \dots ,x_n$ be a sequence of integers such that $i) -1\le x_i\le 2$ for $i=1,2,\dots,n$ $ii)x_1+x_2+\dots+x_n=19$ $iii){x_1}^2+{x_2}^2+\dots +{x_n}^2=99$ Determine the minimum and ...
0
votes
0answers
22 views

From inequality on derivatives to inequality on functions

What is the set of differentiable functions $f$ that satisfy the following inequalities for all $x\geq 0$: $0\leq f'(x)\leq e^{-x}$ Initially, I thought I should just integrate the inequality and ...
0
votes
0answers
40 views

What is the maximum value of $M$ when $T$ is set of $\{2,4,8,16,… 2^n\}$ and $S$ is subset of $T$ by given conditions

Qns $T$ is set of $\{2,4,8,16,... 2^n\}$ and $S$ is a subset of $T$ if the sum of no two elements of $S$ is greater than $2^n-2$. let $m$ be $M$ number of elements in $S$. what is ...
2
votes
2answers
57 views

Find min of $M=\frac{1}{2+\cos2A}+\frac{1}{2+\cos2B}+\frac{1}{2-\cos2C}$

Find min of $$M=\frac{1}{2+\cos2A}+\frac{1}{2+\cos2B}+\frac{1}{2-\cos2C}$$, where $A, B, C$ are three angle of triangle $ABC$ Using Cauchy-Schwarz, we obtain: \begin{align*} M &= ...
3
votes
0answers
32 views

Inequality - Why do I not check for these other solutions?

Given, for example: $$ \Big(\frac{2x}{x-2}\Big)^{3x^2-x} \leq \Big(\frac{2x}{x-2}\Big)^{x^2+3x+6} $$ After checking when $x-2 \ne 0$ , the teacher taught us to check 3 cases: 1. $ ...
0
votes
0answers
17 views

Absolute Value Inequality of Differences

I'm hoping someone could give insight as to how I can improve my organization, and/or thought process. Show that $|a-b| \lt c$ if and only if $b -c \lt a \lt b + c$. By the statement $b - c \lt a ...
1
vote
6answers
123 views

Prove $((a+b)/2)^n\leq (a^n+b^n)/2$

Struggling with this proof. Prove that $$\left(\frac{a+b}{2}\right)^n≤\frac{a^n+b^n}{2},$$ where $a$ and $b$ are real numbers such that $a+b≥0$ and $n$ is a positive integer. What technique would ...
2
votes
3answers
74 views

Prove the equation has unique class of solutions

Find the solutions of equation: $$ x^y + y^x = 1 + xy \quad x,y \in \mathbb{R} \quad x,y >0 $$ My quest First, $x=1$ or $y=1$ gives us obvious solutions, so let's suppose $x \not =1$ and $y \not= ...
0
votes
0answers
8 views

How can a nonconvex polytope be defined (not by an LMI)?

A convex polytope can be defined by an LMI (linear matrix inequality) or a list of points. How can a nonconvex polytope be defined?
0
votes
0answers
32 views

Absolute Value Inequality Proof

I realize this is almost identical to another question I posted, but I wanted to ask what the distinction between the two is -- comprehension-wise (other than the $\lt$ vs. $\le$). Show that $|b| ...
16
votes
0answers
356 views
+50

Proving that $e^{\pi}-{\pi}^e\lt 1$ without using a calculator

Prove that $e^{\pi}-{\pi}^e\lt 1$ without using a calculator. I did in the following way. Are there other ways? Proof : Let $f(x)=e\pi\frac{\ln x}{x}$. Then, ...
1
vote
4answers
37 views

Prove that triangle inequality $|a + b| \le |a| + |b|$ holds when $(a + b) \ge 0, a \ge 0, b < 0$

This is what I did: $a + b \ge 0 \rightarrow |a + b| = a + b$ $a \ge 0 \rightarrow |a| = a$ $a + b \le a + |b|$ $b \le |b|$ Which is true $\forall b$. Is this a formal enough way of proving ...
0
votes
1answer
23 views

Upper bound of the function

here you can read my first question on this topic, namely: $$\text{if } f\left(\frac{x}{3}\right)-f\left(\frac{x}{4}\right)\le Ax+B\ln x-C, $$ where $f(x)$ is my function and $A$,$B$,$C$ are ...
1
vote
4answers
36 views

Epsilon-Delta Limit Proof: Arccos(x) Inequalitiy

I'm studying a Calculus proof using notes (proving that $\lim_{x \to 1} \cos(x) = \cos(1)$ from the definition of limit). The text says that we get from: $\cos(1) −\epsilon < \cos(x) < ...
0
votes
2answers
25 views

Can we relax the triangle inequality for $\| v \|$ = $\|v - v_0 + v_0\|$?

Given some vector $v$ on vector space $X$ with a norm $\| \cdot \|$ Then $\| v \|$ = $\|v - v_0 + v_0\|$ where $v_0$ is some other vector is it legal to then write $\| v - v_0 + v_0 \| = \|v -v_0\| ...
0
votes
1answer
42 views

Jensen's inequality problem [on hold]

I want to know an example of a infinite measure space $(\Omega, \mathcal{F},\mu)$, real valued function $g$ and convex function $\phi$ defined on the real line s.t. $$\phi\left(\int g d\mu\right) > ...
3
votes
1answer
52 views

Russian MO 2004 Question involving the AM-GM inequality

I'm reading Secrets in Inequalities by Pham Kim Hung, and I'm having trouble understanding this proof from a problem from the 2004 Russian MO. Let a,b,c be positive real numbers and $a + b +c = 3$. ...
2
votes
2answers
36 views

Rearrangement and Cauchy

Let $a_1, \ldots, a_n$ be distinct positive integers. I want to prove that $$\frac{a_1}{1^2} + \frac{a_2}{2^2} + \cdots + \frac{a_n}{n^2} \geq \frac{1}{1} + \frac{1}{2} + \cdots + \frac{1}{n}.$$ ...
0
votes
1answer
68 views

Find the minimum value of $P=\frac{1}{2-x}+\frac{1}{2-y}+\frac{1}{2-z}$

Let $x,y,z$ be positive real numbers such that $x^3+y^3+z^3=3$. Find the minimum value of $$P=\frac{1}{2-x}+\frac{1}{2-y}+\frac{1}{2-z}.$$ I think that we need to show that $\dfrac{1}{2-x} \ge ...
0
votes
1answer
40 views

How to prove triangle inequality in How to Prove It Sec. 3.5 Question 12c?

(a) Prove that for all real numbers $a$ and $b$, $$|a| \le b \text{ iff } -b \le a \le b.$$ (b) Prove that for any real number $x$, $$-|x| \le x \le |x|.$$ (Hint: Use part (a).) (c) Prove that ...