Questions on proving, manipulating and applying inequalities.

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0
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0answers
29 views

If $f(x)\le 1$ implies $f(x)\le 1/2$ then $f(x+\delta)\le 1$?

Let $f(x)$ be a nonnegative continuous function of $x\in [0,K)$ with $f(0)\le 1/2$, and satisfies "$f(x)\le 1$ implies $f(x)\le 1/2$". Let $x_0\in[0,K)$ (so that $f(x_0)\le 1$ implies $f(x_0)\le ...
2
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0answers
34 views

Inequality involving fourth powers .

I have been into inequalities lately and I am stuck with this. I used a famous inequality at first $\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a} \ge 3 (\frac{a^4+b^4+c^4}{3})^{\frac{1}{4}}$. From this ...
4
votes
1answer
773 views

Equality in Young's inequality for convolution

I am trying to understand how sharp Young's inequality for convolution is. The inequality says $||f \ast g||_r \leq ||f||_p ||g||_q$ where as $1/p+1/q = 1+1/r$. Actually, there are a couple of papers ...
1
vote
1answer
34 views

Inequality with sum of numbers

A have found a very interesting inequality in a Romanian magazine which I use to prepare for the Lithuanian Mathematical Olympiad. Let $a_1,a_2,...,a_n$ be positive real numbers such that $$\frac {1} ...
1
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5answers
51 views

Prove algebraically that, if $x^2 \leq x$ then $0 \leq x \leq 1$

It's easy to just look at the graphs and see that $0 \leq x \leq 1$ satisfies $x^2 \leq x$, but how do I prove it using only the axioms from inequalities? (I mean: trichotomy and given two positive ...
1
vote
1answer
197 views

Three related inequalities (the first being $2(|a|^p + |b|^p) \leq |a + b|^p + |a - b|^p \leq 2^{p-1}(|a|^p + |b|^p)$)

A friend told me this interesting problem. It should be easy enough, but I cannot figure it out completely. If $a, b \in \mathbb{R}, p \geq 2, \frac{1}{p} + \frac{1}{q} = 1$, then $2(|a|^p + |b|^p) ...
0
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0answers
32 views

Computng the floor of a sum [on hold]

Compute: $$\left\lfloor \frac {2^{1/{\sqrt 2}}+ 2^{1/{\sqrt 3}}+... +2^{1/{\sqrt {100}}}} {10} \right\rfloor.$$
1
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1answer
45 views

Prove $(1+x)^p+(1-x)^p \ge 2(1+x^p)$ for $0\le x\le1$ and real number $p\ge2$.

I don't know how to prove the following questions: If $p\ge2$ is real, then $$ (1+x)^p+(1-x)^p \ge 2(1+x^p) \quad \text{for } 0\le x\le1; $$ if $1\le p<2$, then opposite direction of the inequality ...
2
votes
1answer
41 views

Prove that : $\frac{a+b+c+d}{a+b+c+d+f+g}+\frac{c+d+e+f}{c+d+e+f+b+g}>\frac{e+f+a+b}{e+f+a+b+d+g}$

Prove inequality for positive numbers: $$\frac{a+b+c+d}{a+b+c+d+f+g}+\frac{c+d+e+f}{c+d+e+f+b+g}>\frac{e+f+a+b}{e+f+a+b+d+g}$$ My work so far: Lemma: If $x>y>0, t>z>0$, then ...
3
votes
3answers
68 views

Prove that inequality is true for $x>0$: $(e^x-1)\ln(1+x) > x^2$

I was given a task to prove that inequality is true for x>0: $(e^x-1)\ln(1+x) > x^2$. I've tried to use derivatives to show that the $f(x) = (e^x-1)\ln(1+x)-x^2$ is greater than zero, but has never ...
2
votes
3answers
91 views

Prove inequality with $e^x$ and $\ln$ on the same side [duplicate]

The problem is to prove the following inequality: $$ (e^x - 1) \ln(1+x) > x^2 , \quad\text{ for } x >0 $$ Let me introduce notation $f(x) > g(x)$. At $x=0$ both sides are equal to $0$. So, ...
1
vote
2answers
35 views

If $a,b,c>0$ and $abc=1\;,$ Then minimum value of Expression.

If $a,b,c>0$ and $abc=1\;,$ Then minimum value of $$\frac{a^2}{a^2+2}+\frac{b^2}{b^2+2}+\frac{c^2}{c^2+2}$$ $\bf{My\; Try::}$ Using $\bf{Cauchy\; Schwarz}$ Inequality ...
-4
votes
1answer
43 views

Inequality that just won't go up!

I've gotten this result in an exam question on economics, and I can't seem to get this to make sense. Here, $Y$ is unknown. So, how do we know that this is true? $$(1-Y)(1-C) < (1+Y)(1-P), \quad P ...
-1
votes
1answer
39 views

I want to show that $x^2 - x + C\epsilon\ge 0$ under some assumption.

Let $x\ge 0$. For sufficiently small $\epsilon>0$, assume that the property $x\le \sqrt\epsilon$ implies $x\le \frac{1}{2}\sqrt{\epsilon}$. Then I want to show that $$x^2 - x + C\epsilon\ge 0 $$ ...
1
vote
1answer
38 views

A polynomial inequality

Let $f(x)\in\mathbb{R}[x]$ be a polynomial of degree $n$, which has only real zeros. I would like to show that $$(n − 1)(f'(x))^2 \geq nf(x)f''(x),$$ where $f'$ and $f''$ denote the first and second ...
0
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0answers
30 views

Geometric inequality $180^{\circ}\left(1-\frac1n\right)\le AMB$

Point $M$ is located inside a regular $n$-gon. Prove that there exist different vertices $A$ and $B$ that $$180^{\circ}\left(1-\frac1n\right)\le AMB\le 180^{\circ}$$ My work so far: Let ...
9
votes
0answers
272 views

Proving a geometric inequality without Lagrange multipliers

Let $e=(1,1,\ldots,1)$ be the $n$-dimensional vector consisting only of ones. Let $r=\sqrt{\dfrac{n}{n-1}}$ and $\alpha \in (0,1)$ fixed. Given a vector $x=(x_1,x_2,\ldots,x_n) \in \mathbb R^n$, ...
0
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0answers
47 views
8
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5answers
1k views

Prove that $\frac{x^n}{n!} < 1$ for $n$ sufficiently large

Is this statement always true? For any real number $x$, there exists a natural number $n$, such that $\frac{x^n}{n!} < 1$. I reach this conclusion observing the series form of '$e^x$'. I ...
2
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1answer
54 views

Inequality about sums and products

Let $x_1,x_2,...,x_n$ be positive real numbers. Show that $$\frac {1} {2^n \times \sqrt {x_1 \times x_2 \times ... \times x_n} } + \sum_ {k=1}^n \frac {x_k} {(x_1+1)(x_2+1)...(x_k+1)} \ge 1.$$ I tried ...
0
votes
0answers
17 views

Solving an inequality with ceiling [on hold]

I'm stuck on the following equation: $\lceil q/14 \rceil < m\lceil q/40\rceil$ From other answers here, I think you can do: Let $x = \lceil q/14 \rceil$ and $y = \lceil q/40 \rceil$ then ...
1
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0answers
42 views

$1+x^4\leq 2(y-z)^2$ and switching of $x,y,z$

Find all triples of real numbers $x,y,z$ such that $1+x^4\leq 2(y-z)^2$, $1+y^4\leq 2(z-x)^2$, and $1+z^4\leq 2(x-y)^2$. Beside $(1,0,-1)$ and permutations, I can't find any others. We cannot have ...
0
votes
2answers
48 views

Inequality involving square root exponents

Show that $$ 2^ {\frac {1} {\sqrt 2}} + 2^ {\frac {1} {\sqrt 3}} \gt 3.$$ I tried to use AM-GM inequality and Jensen's inequality, but I didn't get to any results.
1
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5answers
54 views

Prove that $\frac{\pi^3}{48} \le \int_0^{\pi/2}\frac{x^2}{2-\sin(x)}\,dx \le \frac{\pi^3}{24}$

Is it possible to prove that $$\frac{\pi^3}{48} \le \int_0^{\pi/2}\frac{x^2}{2-\sin(x)}\,dx \le \frac{\pi^3}{24}$$ without evaluating the integral?
2
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1answer
46 views

inequality involving heights and bisectors

Let $a,b,c,a \le b \le c$ be the sides of the triangle $ABC$, $l_a,l_b,l_c$ the lengths of its bisectors and $h_a,h_b,h_c$ the lengths of its heights. Prove that: $$\frac {h_a+h_c} {h_b} \ge \frac ...
1
vote
1answer
19 views

How to combine inequalities

If $u,v$ are real numbers and $|u-3|<1/3$ and $|v-3|<2/3$ then show that $|v-u|<1$. I'm unsure about how to combine this inequalities and simplify. Thanks in advance.
9
votes
5answers
152 views

How to show $\frac{19}{7}<e$

How can I show $\dfrac{19}{7}<e$ without using a calculator and without knowing any digits of $e$? Using a calculator, it is easy to see that $\frac{19}{7}=2.7142857...$ and $e=2.71828...$ ...
0
votes
0answers
12 views

Bounding the $q$-th moment of a Gaussian random variable

I have come across an inequality which confuses me: Suppose $X$ has a Normal$(0,\sigma^2)$ distribution. Then $$ (\mathbb{E}|X|^q)^{1/q} \leq \text{const.} \sqrt{q} \sigma $$ for $q\geq 1$. I ...
1
vote
0answers
60 views

System of Equations which can be solved by inequalities: $(x^3+y^3)(y^3+z^3)(z^3+x^3)=8$, $\frac{x^2}{x+y}+\frac{y^2}{y+z}+\frac{z^2}{z+x}=\frac32$.

S367. Solve in positive real numbers the system of equations: \begin{gather*} (x^3+y^3)(y^3+z^3)(z^3+x^3)=8,\\ \frac{x^2}{x+y}+\frac{y^2}{y+z}+\frac{z^2}{z+x}=\frac32. \end{gather*} Proposed by ...
1
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0answers
39 views

Prove that if $f(\lambda x) = \left(\frac{1}{\lambda}\right)^N f(x)$ than $|f(x)| \leq c \left|\frac{1}{x}\right|^N$

The task is: Knowing that $\forall \lambda >0, x \neq 0$ $f(\lambda x) = \left(\frac{1}{\lambda}\right)^N f(x)$ prove that: $|f(x)| \leq c \left|\frac{1}{x}\right|^N$ I would really appreciate ...
1
vote
4answers
35 views

Prove $\sup S \leq \inf T$, if $s \leq t$, $\forall s \in S$ and $\forall t \in T$

I have the following exercise: Prove $\sup S \leq \inf T$, if $s \leq t$, forall $s \in S$ and $t \in T$. Note that $S$ is bounded above and $T$ is bounded below. This might seem too obvious, ...
2
votes
2answers
34 views

Prove by induction that $\sum_{k=1}^nk^p < (n+1)^{p+1}/(p+1), \quad n,p \in \mathbb{N}$

For $n=1$, we have at the left side $1^p$, and at the right side: $$ \frac{2^{p+1}}{p+1}\mathrm{~which~is } >1$$ so it holds for $n=1$, but how can we prove that $$ ...
1
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2answers
23 views

Matrix norm inequality proof - does this use Cauchy-Schwarz?

The matrix norm for $A : \mathbb{R}^n \rightarrow \mathbb{R}^m$ (so $A$ is an $m \times n$ matrix) is given by $$\|A\| = \sup_{X \in \mathbb{R}^n \setminus \{0\}} \frac{|AX|}{|X|}$$ where $| \cdot |$ ...
2
votes
2answers
69 views

How to properly find supremum of a function $f(x,y,z)$ on a cube $[0,1]^3$?

Solving an applied problem I was faced with the need to find supremum of the following function $$f(x,y,z)=\frac{(x-xyz)(y-xyz)(z-xyz)}{(1-xyz)^3}$$ where $f\colon\ [0,1]^3\backslash\{(1,1,1)\} ...
6
votes
3answers
142 views
+100

(Elegant) proof of an inequality: $h(x) \geq 1- (1-\frac{x}{1-x})^2$, where $h$ is the binary entropy function

I am looking for the most concise and elegant proof of the following inequality: $$ h(x) \geq 1- \left(1-\frac{x}{1-x}\right)^2, \qquad \forall x\in(0,1) $$ where $h(x) = x \log_2\frac{1}{x}+(1-x) ...
2
votes
2answers
36 views

Help solving the inequality $2^n \leq (n+1)!$, n is integer

I need help solving the following inequality I encountered in the middle of a proof in my calculus I textbook: $2^n \leq (n+1)!$ Where $\mathbf{n}$ in an integer. I've tried applying lg to both ...
1
vote
0answers
27 views

Use Chebyshev’s inequality to choose $n$ such that $P(\bar{X} > 4) > 0.9$

Use Chebyshev’s inequality to choose n such that $$ P(\bar{X_n} > 4) > 0.9 $$ where $$ E[\bar{X_n}] = 5 \ \ \ \ \ Var[\bar{X_n}] = \frac{4}{n} $$ The problem I am having when using Chebyshev's ...
6
votes
6answers
200 views

Extreme of $\cos(A)\cos(B)\cos(C)$ in a triangle without calculus.

If $A,B,C$ angles of a triangle, show extreme value of $$\cos(A)\cos(B)\cos(C)$$ I have tried using $A+B+C=\pi$, and applying all and any trig formulas, also AM-GM, but nothing helps. On this topic ...
3
votes
0answers
79 views
+50

When might some a variable leave the basis?

In the simplex algorithm in linear programming, what are conditions for a variable to leave a basis (not necessarily basis for the/an optimal solution)? I'm supposed to list as many sufficient and ...
0
votes
1answer
21 views

Does a sum of squares become smaller as the number of terms increases?

I am interested in the following question: Let $,kn$ be a positive integeres. Assume $\sum_{i=1}^{k} L_i=\sum_{i=1}^{k+1} \tilde L_i=n$, where $L_i,\tilde L_i$ are positive integers. Is it true ...
11
votes
5answers
1k views

Proof for $\sin(x) > x - \frac{x^3}{3!}$

They are asking me to prove $$\sin(x) > x - \frac{x^3}{3!},\; \text{for} \, x \, \in \, \mathbb{R}_{+}^{*}.$$ I didn't understand how to approach this kind of problem so here is how I tried: ...
2
votes
0answers
37 views

Estimating $n!$ as $e \left(\frac ne \right)^n \le n! \le ne \left(\frac ne \right)^n$

I'm told that for $n \geq 2,$ $$\sum_{k=1}^{n-1} f(k) \leq \int_1^n f(x) \, dx \leq \sum_{k=2}^n f(k)$$ I am then asked to consider $\ln n! = \sum_{k=1}^n \ln k$ and show that for $n \geq 2$ $$n! ...
7
votes
1answer
352 views

Inequality involving factorial $\binom nk<(en/k)^k$

I am trying to prove following inequality: $$\binom{n}{k}<(en/k)^k$$ I tried Stirling approximation but I could not get anything further. Then I get $$\binom{n}{k}\approx \frac{\sqrt{2\pi ...
3
votes
0answers
88 views

prove that $(\frac{n}{3})^n<n!<e\cdot(\frac{n}{2})^n$ [duplicate]

prove that $$(\frac{n}{3})^n<n!<e\cdot(\frac{n}{2})^n$$ I tried to prove by the induction that $(\frac{n}{3})^n<n!$ and $n!<e\cdot(\frac{n}{2})^n$, but I failed my assumption ...
0
votes
1answer
92 views

Proving an inequality involving factorials: $(\frac{n}{2})^n \ge n! \geq (\frac{n}{3})^n$ [duplicate]

For $n \geq 6$, where $n$ is a natural number, prove that $(\frac{n}{2})^n \ge n! \geq (\frac{n}{3})^n$. I tried using induction but could not do it.
14
votes
1answer
2k views

Factorial Inequality problem $\left(\frac n2\right)^n > n! > \left(\frac n3\right)^n$

I met an inequality, I ask, do not mathematical induction to prove that: Prove \[ \left(\frac n2\right)^n > n! > \left(\frac n3\right)^n \] without using induction
0
votes
1answer
26 views

Algebraic Inequality

If a,b,c are positive real numbers and $z = \frac{b^2 + c^2}{b+c} + \frac{c^2 + a^2}{a+c} + \frac{a^2+b^2}{a+b}$ then only one of the following statements is always true , which on is it ? a) ...
-2
votes
0answers
45 views

Inequalities: $e^{-x}> 2-x$ [on hold]

Tried to solve it through the ln method, but didn't know how to proceed from there. Here's what I have done: $$e^{-x} > 2-x$$ $$-x > \ln(2-x)$$ $$x < -\ln(2-x)$$ Really need help to solve ...
2
votes
1answer
66 views

Prove the inequality $\frac{a+c}{a+b}+\frac{b+d}{b+c}+\frac{c+a}{c+d}+\frac{d+b}{d+a}\geq 4$

$a, b, c, d$ are positive reals. How would I prove the inequality $$\frac{a+c}{a+b}+\frac{b+d}{b+c}+\frac{c+a}{c+d}+\frac{d+b}{d+a} \geq 4$$ I have tried using the rearrangement inequality with ...
0
votes
3answers
40 views

Inequality with a square root

If the inequality $ (x+2)^{\frac{1}{2}} > x $ is satisfied. what is the range of x ? My approach - I squared both the sides and proceeded on to solve the quadratic obtained in order to solve the ...