What is Inequality: Definition and 1000 Discussions
In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, but some authors, especially those writing about elementary geometry, will exclude this possibility, thus leaving out the possibility of equality. If x, y, and z are the lengths of the sides of the triangle, with no side being greater than z, then the triangle inequality states that
z
≤
x
+
y
,
{\displaystyle z\leq x+y,}
with equality only in the degenerate case of a triangle with zero area.
In Euclidean geometry and some other geometries, the triangle inequality is a theorem about distances, and it is written using vectors and vector lengths (norms):
‖
x
+
y
‖
≤
‖
x
‖
+
‖
y
‖
,
{\displaystyle \|\mathbf {x} +\mathbf {y} \|\leq \|\mathbf {x} \|+\|\mathbf {y} \|,}
where the length z of the third side has been replaced by the vector sum x + y. When x and y are real numbers, they can be viewed as vectors in R1, and the triangle inequality expresses a relationship between absolute values.
In Euclidean geometry, for right triangles the triangle inequality is a consequence of the Pythagorean theorem, and for general triangles, a consequence of the law of cosines, although it may be proven without these theorems. The inequality can be viewed intuitively in either R2 or R3. The figure at the right shows three examples beginning with clear inequality (top) and approaching equality (bottom). In the Euclidean case, equality occurs only if the triangle has a 180° angle and two 0° angles, making the three vertices collinear, as shown in the bottom example. Thus, in Euclidean geometry, the shortest distance between two points is a straight line.
In spherical geometry, the shortest distance between two points is an arc of a great circle, but the triangle inequality holds provided the restriction is made that the distance between two points on a sphere is the length of a minor spherical line segment (that is, one with central angle in [0, π]) with those endpoints.The triangle inequality is a defining property of norms and measures of distance. This property must be established as a theorem for any function proposed for such purposes for each particular space: for example, spaces such as the real numbers, Euclidean spaces, the Lp spaces (p ≥ 1), and inner product spaces.
Homework Statement
Let f be an analytic function on the disc |z|<1 and satisfies |f(z)|≤M if |z|<1.
Show that |f(z)| \le M \left| \frac{z-a}{1-a'z} \right| when |z|<1
where a' is the complex conjugate of a
Homework Equations
This section uses maximum modulus principle, but I really don't...
Forum, do you have any idea how to solve the trigonometric inequality \cos (x) < \sin (x) strictly algebraically?
The conventional(?) approach is to first solve \cos(x) = \sin(x) and then draw the graphs for each function in order to find the correct interval. However, I would love to know if...
Homework Statement
Prove Bernoulli's Inequality: if ##h>-1##
(1+h)^n \geq 1+hn
Homework Equations
Binomial Theorem
(a+b)^n=\sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^{k}
The Attempt at a Solution
If ##h=0##
(1+0)^n=1
1=1
If ##h>0##
This
(1+h)^n \geq 1+hn
Implies...
In the EPR scenario the correlation results are explained with the conservation laws of classical mechanics as applied to spin. The Bell type inequalities are derived on expected spin values.
But the violations of these inequalities are then explained with QM: That simultaneous knowledge of...
Homework Statement
If ##A+B+C=\pi##, prove that ##\cos A+\cos B+\cos C \leq 3/2##.
Homework Equations
The Attempt at a Solution
I don't really know how to start. ##A+B=\pi-C##. Taking cos on both sides doesn't seem of much help. I need a few hints to start with.
I'm beginning to read Spivak's Calculus 3ed, and everything is smooth until I reach page 12.
My question is marked, between line 2 and 3. Why there's such sign change suddenly? In fact I tried with simple line 4 case and it's not in fact equal. I'm assuming that a and b is valid for all...
I want to know that is it possible to show that
$$
\int_{0}^{T}\Bigr(a(t
)\Bigr)^{\frac{p+1}{2p}}dt\leq C\Bigr(\int_{0}^{T}a(t)dt\Bigr)^{\frac{p+1}{2p}}
$$
for some ##C>0## where ##a(t)>0## and integrable on ##(0,T)## and ##p\in(\frac{1}{2},1)##. It is worth noting that this range for ##p##...
For what class of functions we have:
$$
\int_{\Omega} [f(x)]^m dx \leq
C\Bigr ( \int_{\Omega} f(x)dx\Bigr)^{m},
$$
where ##\Omega## is open bounded and ##f## is measurable on ##\Omega## and ##C,m>0##.
When entangled photons are generated from a cascade of a Calciums' 6s level
this inequality : n[y+z-] + n[x-y+] ≥ n[x-z-] is derived for what is equivalent to spin in photons.
When the detectors at A and B are parallel the perfect anti correlations are due to conservation
laws of angular...
Homework Statement
Let ##x^2+y^2+xy+1 \geq a(x+y)## for all ##x,y \in R##. Find the possible integer(s) in the range of ##a##.
Homework Equations
The Attempt at a Solution
I can rewrite this into ##(x+y)^2-xy+1 \geq a(x+y) \Rightarrow (x+y)(x+y-a)+1-xy \geq 0## but I don't think...
Homework Statement
3\sqrt{x}-\sqrt{x+3}>1
Homework Equations
The Attempt at a Solution
As obvious from the given inequality, x must be greater than zero.
Rearranging and squaring both the sides,
9x>1+x+3+2\sqrt{x+3} \Rightarrow 4x-2>\sqrt{x+3}
Squaring again,
16x^2+4-16x>x+3...
Homework Statement
Find all numbers ##a## for each of which the least value of the quadratic trinomial ##4x^2-4ax+a^2-2a+2## on the interval ##0\leq x \leq 2## is equal to 3.Homework Equations
The Attempt at a Solution
I don't really know what should be the best way to start with this type of...
I guess the best way to start this is by admitting that my conceptual understanding of the Cauchy-Schwarz Inequality and the Lagrange Identity, as the title suggests, is not as deep as it could be.
I'm working through Marsden's 3e "Basic Complex Analysis" and it contains a proof of the Cauchy...
i found this problem interesting in stack exchange unfortunately i will participate in discussion for 4 days(vacation)
inequality - Prove $\sum_{i=1}^{n}\frac{a_{i}}{a_{i+1}}\ge\sum_{i=1}^{n}\frac{1-a_{i+1}}{1-a_{i}}$ if $a_{i}>0$ and $a_{1}+a_{2}+\cdots+a_{n}=1$ - Mathematics Stack Exchange...
Homework Statement
Let ##a,b,c## be positive real numbers with sum 3. Prove that
\sqrt{a}+\sqrt{b}+\sqrt{c} \geq ab+bc+ca
Homework Equations
AM-GM inequalityThe Attempt at a Solution
I don't really know how to start with. We are given ##a+b+c=3##.
Also, ##2(ab+bc+ca)=(a+b+c)^2-(a^2+b^2+c^2)##...
Homework Statement
2x>3sinx-xcosx, 0<x<∏/2
Homework Equations
The Attempt at a Solution
One possible way is to draw the graph of the functions and compare but plotting a graph manually is not easy in this case. I want some other methods.
Homework Statement
Find inverse of each.
1. y<x+1
2. y=2x/(x-2)
Homework Equations
Switch y and x?
The Attempt at a Solution
For 1. I switched y and x, so x<y+1. Do I have to switch the sign also?
For 2. I switched y and x, so x=2y/(y-2). But I have to express the inverse...
Here's yet another assigned problem that I'm having difficulty with. I think I'm close to the end but am "nervous" (for lack of a better word) about whether or not I have used summation notation properly throughout the problem. Here it is:
"Use the Schwarz inequality to establish that
(...
Hello all,
I am currently working through a proof in my Real Analysis book, by Royden/Fitzpatrick and I'm confused on a part.
if f is a measurable function on E, f is integrable over E, and A is a measurable subset of E with measure less than δ, then ∫|f| < ε...
Question:
Find all numbers x for which \frac{1}{x}+\frac{1}{1-x}>0.
Solution:
If \frac{1}{x}+\frac{1}{1-x}>0,
then \frac{1-x}{x(1-x)}+\frac{x}{x(1-x)}>0;
hence \frac{1}{x(1-x)}>0.
Now we note that
\frac{1}{x(1-x)} \rightarrow ∞ as x \rightarrow 0
and \frac{1}{x(1-x)} \rightarrow 0 as x...
Homework Statement
Use the triangle inequality to prove that \left| s_n - s \right| < 1 \implies \left| s_n \right| < \left| s \right| +1
Homework Equations
The triangle inequality states that \left| a-b \right| \leq \left| a-c \right| + \left| c-b \right|
The Attempt at a Solution...
Homework Statement
If f and g are two distinct linear functions defined on R such that they map[-1,1] onto [0,2] and h:R-{-1,0,1}→R defined by h(x)=f(x)/g(x) then show that |h(h(x))+h(h(1/x))|>2
Homework Equations
The Attempt at a Solution
I assume f(x) to be ax+b and g(x) to be lx+m so...
Homework Statement
Prove the following
a>0, X is a non-negative function
Ʃ_{n\in N} P(X>an)≥\frac{1}{a}(E[X]-a)
Ʃ_{n\in N} P(X>an)≤\frac{E[X]}{a}
The Attempt at a Solution
I know that
\sum_{n\in N} P(X>an)=\sum_{k \in N} kP((k+1)a≥X>ka)=\sum_{k \in N} E[k1_{[(k+1)a,ka)}(X)]...
Hi! I have to do this exercise:
Define a finite probabilistic Space (Ω; Pr[ ]) and 2 events A,B⊆ Ω and Pr[A] ≠ Pr[B] so that we can verify that
Pr[A∩B]>=9*Pr[A]*Pr[B] > 0. (1)
___________________________________________
I've been trying it but i have reached this conclusion:
If Pr[A]>0...
in the clausius inequality is the temperature that of the system or of the surroundings? or is it temperature of the body receiving positive heat?
(assuming the irreversibility is due to heat transfer with finite temperature difference)
[borgnakke and sonntag-principle of entropy increase for...
I need a bit of help proving the following statement
(n + 2)^n ≤ (n + 1)^n+1 where n is a positive integer. The (n+2) and (n+1) bases are making it hard for me solve this. I tried several time, I can't get the inductive step. Can someone lend me a little hand here?
The base case is real...
Homework Statement
Prove that |a + b| ≤ |a| + |b|.
Homework Equations
|a| = √a2
The Attempt at a Solution
Since |a| = √a2, then
|a + b| = √(a + b)2 = √(a2 + 2ab + b2) = √a2 + √b2 + √(2ab) = |a| + |b| + √(2ab).
And since the square root of a negative number is not defined...
Homework Statement
I was discussing the proof for the Cauchy-Schwarz inequality used in our lectures, and another student suggested an easier way of doing it. It's really, really simple. But I haven't seen it anywhere online or in textbooks, so I'm wondering if it's either wrong or is only...
Hi, I was hoping that someone might be able to please help me with this proof.
Prove that var(x+y) ≤ 2(var(x) + var(y)).
So far I have:
var(x+y) = var(x) + var(y) + 2cov(x,y)
where the cov(x,y) = E(xy) - E(x)E(y), but I'm not really sure to go from there.
Any insight would be very...
Homework Statement
lx/(x-2)l < 5
Homework Equations
The Attempt at a Solution
x/(x-2) < 5
x< 5x-10
10 < 4x
5/2 < x
x/(x-2) > -5
x > -5x+10
6x > 10
x > 5/3
The answer is x < 5/3 and x > 5/2
so where did I go wrong on the second one?
I have worked my way though the proof of the Cauchy Schwarz inequality in Rudin but I am struggling to understand how one could have arrived at that proof in the first place. The essence of the proof is that this sum:
##\sum |B a_j - C b_j|^2##
is shown to be equivalent to the following...
I am stuck at the inequality proof of this convext set problem.
$\Omega = \{ \textbf{x} \in \mathbb{R}^2 | x_1^2 - x_2 \leq 6 \}$
The set should be a convex set, meaning for $\textbf{x}, \textbf{y} \in \mathbb{R}^2$ and $\theta \in [0,1]$, $\theta \textbf{x} + (1-\theta)\textbf{y}$ also belong...
Hi, so here is my question that I am totally stumped on.
for all real values of x and y, show that |x|+|y|≥ √(x^2+y^2 )
and find the real values of x and y in which equality holds.
I sort of thought I could do the second part, but it confuses me with two pronumerals and how to get rid...