Questions on proving and manipulating inequalities.

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3answers
37 views

Demonstrate that $\frac{1}{e^e} - 1 + e - \frac{e^2}{2} + \frac{e^3}{6}\ge 0$

How do I prove the inequality? $$\frac{1}{e^e} - 1 + e - \frac{e^2}{2} + \frac{e^3}{6} \geq 0$$ I can see that $e^e = \sum_{k=0}^{\infty} \frac{e^k}{k!} = 1 + e + \frac{e^2}{2} + \frac{e^3}{6}+\dots ...
4
votes
1answer
187 views

Lower bound for multivariate recurrence

I have a recurrence that looks like $$p(i,j,k) = \frac{j}{n}p(i-1,j-1,k-1) + \frac{i-j}{n}p(i-1,j,k-1)$$ $$p(i,0,k) = 1$$ $$p(i,j,0) = 0$$ $$p(0,j,k) = 0$$ The base cases are to be considered in ...
3
votes
1answer
163 views

Norm inequality involving matrices

Let $A$ and $B$ be two definite positive symmetric $n \times n$ matrices. Prove or disprove that $$ \Vert AB - B^{-1} A^{-1} \Vert \geq \Vert AB - I \Vert $$ where $\Vert . \Vert$ is the Frobenius ...
2
votes
1answer
165 views

how to find bounds on (complex) coefficients from bounds on a polynomial?

I'm trying to prove the following two statements about a polynomial $p$ of degree $n$ with complex coefficients: If $|p(x)|\le1$ for all real $x$ with $|x|\le1$, then every coefficient of $p$ has ...
1
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3answers
63 views

Prove that $\frac{1}{\sqrt{2n-1}}-\frac{1}{2n}\geq \frac{1}{2n}$ for $n = 1, 2, 3,…$

Prove that $\frac{1}{\sqrt{2n-1}}-\frac{1}{2n}\geq \frac{1}{2n}$ for $n = 1, 2, 3,...$ This is required to prove that the series $1-\frac{1}{2}+\frac{1}{\sqrt{3}}-\frac{1}{4}$ is divergent, but I ...
5
votes
2answers
344 views

Determinant inequality about positive definite matrices.

Assume $A \in M_n(\Bbb{R})$ (not necessarily symmetric), and for $\forall \alpha\not=0$, $\alpha^TA\alpha>0$. Show that $$\det\left(\frac{A+A^T}{2}\right)\leq \det A.$$
1
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1answer
84 views

Lower bound for polynomial with complex coefficient

Let $p(z)=z^{n}+a_{n-1}z^{n-1}+...+a_{1}z+a_{0}$ be a polynomial with complex coefficients. Define $R:=1+\sum_{k=0}^{n-1}|a_k|$. Show that $|p(z)| > R$ for all $z \in \mathbb C$ with $|z|>R$. ...
1
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1answer
38 views

$P(A \mid B) \leq P(A)$ if and only if $P(A \mid \overline{B}) \geq P(A)$

I am reading a paper that claims (without proof) $$P(A \mid B) \leq P(A)$$ if and only if $$P(A \mid \overline{B}) \geq P(A)$$ for any two events $A$ and $B$. This seems reasonable, but I can't ...
3
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1answer
759 views

Trace inequality

Could you please give me a hint on how to prove the following inequality $$\|u\|_{L^2(\Gamma)}\le C\|u\|^{\frac12}_{L^2(\Omega)}\|u\|^{\frac12}_{H^1}, \quad \forall u\in H^1.$$,
2
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1answer
86 views

Find all functions $f \colon \Bbb R \rightarrow \Bbb R $ that satisfy…

I find the following problem interesting : Find all functions $f \colon \Bbb R \rightarrow \Bbb R $ that satisfy the inequality $f(x+y)+f(y+z)+f(z+x) \geq 3f(x+2y+3z)?$ How Can I tackle the ...
2
votes
0answers
73 views

Probability Theory: How to prove inequality $\mathbb{E}|X - m|^3 \leq \mathbb{E}|X|^3 (1 + \frac{m}{\sigma})^3$

Let's define $X$ - random variable with $F(x)$ distribution function. Also, denote $m = \mathbb{E}X$ and $\sigma^2 = \mathbb{D}X$. Suppose, that $m>0$. How to prove this inequality in these ...
5
votes
2answers
187 views

Does $y(y+1) \leq (x+1)^2$ imply $y(y-1) \leq x^2$?

Can anyone see how to prove the following? If $x$ and $y$ are real numbers with $y\geq 0$ and $y(y+1) \leq (x+1)^2$ then $y(y-1) \leq x^2$. It seems it is true at least according to Mathematica.
1
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1answer
46 views

Complex number inequality calculations

Let $z_1, z_2 \in \mathbb C$. I hope to show that the quantity $$ 3 | z_1 |^4 + 3 |z_2|^4 + 4 |z_1 |^2 |z_2 |^2 + z_2^2 \overline{z_1} ^2 + z_1^2 \overline{z_2}^2 $$ is nonnegative. Does this hold? I ...
1
vote
1answer
42 views

Is this quantity nonnegative?

Let $x_1,x_2, y_1, y_2 \in \mathbb R$. Is $$x_1^2 x_2^2 - x_1^2 y_2^2 + 4 x_1 x_2 y_1 y_2 - x_2^2 y_1^2 + y_1^2 y_2^2 \geq 0 ? $$
1
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2answers
63 views

showing that $\log(N) \leq \prod_{n \leq N} {(1-p^{-1})^{-1}}$

i can't see that $H_n \leq \prod_{n \leq N}{(1-p^{-1})^{-1}}$ and i can't see that $\log(N) \leq \prod_{n \leq N} {(1-p^{-1})^{-1}}$ p is prime and $H_n$ is harmonic series
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0answers
23 views

Some questions about a “relaxed” invariant probability problem $|\mu(P-I)|\leq \epsilon$

Let's consider the set $\mathcal{M}=\{\mu:|\mu(P-I)|_1\leq \epsilon\}$ where $\mu$ is a probability vector, $P$ is the transition matrix of a discrete homogeneous Markov chain, $I$ is the identity ...
1
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1answer
291 views

proving inequality for combinatorial sum

If somone can prove the following for every $d\leq r$ (for $d=0,1$ its easy, see below, the case d=r may be also simple, I didn't find something helpful) $$\frac{(d!)^2}{2^{n-2d}}\sum_{k=0}^{n}{n ...
1
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1answer
76 views

An inequality from PDEs: $|v|^\alpha v -|w|^\alpha w \leq C(|v|^\alpha + |w|^\alpha)|v-w|$

In using Stritcharz estimates to prove well-posedness of the nonlinear Schrödinger equation $$i\partial_tu = \Delta u - \lambda|u|^\alpha u$$ where $\alpha>0$, one requires the following inequality ...
0
votes
1answer
27 views

Compute the point of contraction of a bounded region in $\mathbb{R}^n$

Say we have a list of linear inequalities that define a bounded region in $\mathbb{R}^n$. These inequalities are: $a_1 \cdot x \ge c_1, \dots, a_k \cdot x \ge c_k$. Assume general position (i.e. it ...
2
votes
4answers
469 views

Prove the inequality $4S \sqrt{3}\le a^2+b^2+c^2$

Let a,b,c be the lengths of a triangle, S - the area of the triangle. Prove that $$4S \sqrt{3}\le a^2+b^2+c^2$$
2
votes
1answer
79 views

Inequality in Sobolev Space

Given $\Omega \subset \mathbb{R}^3$, prove $\forall u, v, w \in H^{1,2} (\Omega)$ it holds that $ | \int_{\Omega} u \frac{\partial v}{ \partial x} w dx | \leq \| u \|_{1,2,\Omega}\|v \|_{1,2,\Omega}\| ...
10
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3answers
1k views

How prove this $(abc)^4+abc(a^3c^2+b^3a^2+c^3b^2)\le 4$

let $a,b,c>0$,and such that $a+b+c=3$,prove that $$(abc)^4+abc(a^3c^2+b^3a^2+c^3b^2)\le 4$$ I first consider $$abc\le\left(\dfrac{a+b+c}{3}\right)^3=1$$ so it suffices to show that ...
1
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2answers
133 views

Strategies to prove inequalities with interval notation

How to prove a inequalities with interval notation, for example: Find minimum of $a^3+b^3+c^3$ with $a,b,c \in [-1;\infty), a^2+b^2+c^2=9$
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6answers
568 views

Prove that $4x-x^4 \leq 3, x \in \Bbb R$

How can I tackle the following inequality : Prove that $4x-x^4 \leq 3$, where $x$ is any real number. Can someone point me in the right direction?
0
votes
1answer
186 views

Proving that $\sum_{k=n}^{2n-2} \frac{|\sin k\ |}{k} < 0.7\ln 2$ for $n\ge2$

Prove that $$\sum_{k=n}^{2n-2} \frac{|\sin k|}{k} < 0.7 \ln 2, \qquad (n\ge2)$$ and $$\cot\left(\frac{\pi}{2n}\right) \le \sum_{k=1}^n \left|\sin\left(x+\frac{k\pi}{n}\right)\right| \le ...
0
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1answer
94 views

How prove this inequlity $\sum_{k=n}^{2n-3}\frac{|\sin{k}|}{k}<\frac{1}{\sqrt{2}},n\ge 3$

pove that: $$\sum_{k=n}^{2n-3}\dfrac{|\sin{k}|}{k}<\dfrac{1}{\sqrt{2}},n\ge 3$$ This problem is my frend creat it today,Thank you someone can prove it.Thank you my idea,long ago I have prove it ...
0
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2answers
52 views

Help with real number proof strategy

Let $x, y, z \in \mathbb R$. Prove that $\displaystyle \frac{|x-y|}{1+|x-y|} \leq \frac{|x-z|}{1+|x-z|} + \frac{|z-y|}{1+|z-y|} \; $ Help on how to start with this one? Real lost.
0
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2answers
238 views

Inequality involving $x^2+y^2+z^2 =3$

If $ x^2+y^2+z^2 =3 $ prove that : $3(x+y+z)\ge 3+xy+yz+xz+x^2y+y^2z+z^2x $
6
votes
1answer
101 views

For $p \in [1,2]$, how does one show $\sup\limits_{x\in\mathbb{R}}\frac{|1+x|^p-1-px}{|x|^p}\leq 2^{2-p}?$

The following question arose from Korolëiìuk's Theory of U-statistics: For $p\in[1,2]$, how can we show that $$\sup_{x\in\mathbb{R}}\frac{|1+x|^p-1-px}{|x|^p}\leq 2^{2-p}?$$ My attempts: I ...
8
votes
1answer
244 views

Is it possible to generalize Ramanujan's lower bound for factorials that he used in his proof of Bertrand's Postulate?

I am starting to feel more confident in my understanding of Ramanujan's proof of Bertrand's postulate. I hope that I am not getting overconfident. In particular, Ramanujan's does the following ...
3
votes
1answer
131 views

Understanding the Gamma Function

Is this valid? If $x_1 > x_2 > x_3 > 0$ and $\Delta{t_1} = \Delta{t_2} + \Delta{t_3}$, Does it follow that: $$\frac{\Gamma(x_1 + \Delta{t_1})}{\Gamma(x_1)} \ge \frac{\Gamma(x_2 + ...
2
votes
2answers
31 views

Inequality replacement of numbers

I'm looking for a contradiction or a proof. Suppose we have the following inequality: $a\cdot j \leq b \cdot h$ Where $a,b,j,h \in \mathbb{N}$. We now replace $a$ with $a_{1}$ and $b$ with ...
2
votes
2answers
354 views

Cauchy-Schwarz and Bessel's Inequalities

Deduce the Cauchy-Schwarz Inequality from the case m = 1 of Bessel’s Inequality: the sum of $$\sum_{i=1}^{m}|(v,u_i)|^2 \leq ||v||^2. $$
2
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2answers
148 views

Minimum Value of expression

Given that $x$, $y$ and $z$ are positive real numbers satisfying $xyz=32$, find the minimum value of: $$x^2+4xy+4y^2+2z^2$$ Perhaps AM-GM and manipulation but I'm not quite sure how? Source BMO.
1
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0answers
45 views

Inequality of Partial Taylor Series

For a given $\theta < 1$, and $N$ a positive integer, I am trying to find an $x > 0$ (preferably the smallest such $x$) such that the following inequality holds: $$\sum_{k=0}^{N} \frac{x^k}{k!} ...
0
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2answers
1k views

generalized inequalities defined by proper cones [duplicate]

The generalized inequality defined by a proper cone $K$ is that $x \ge_{K} y$ if $x-y \in K$ for $x,y \in K$. Does this means that for any $x \in K$, we have $x \ge_{K} 0$ since $x - 0 = x \in K$ ? ...
0
votes
3answers
220 views

How to write $b$ between $a$ and $c$ formally?

How to write $b$ between $a$ and $c$ formally ? I mean it could be 1) $a<b<c$ or 2) $a>b>c$ but I want to leave it in the middle which one it is. If I use the sandwich theorem for ...
2
votes
0answers
305 views

Bounds for the exponential integral

In Abramowitz and Stegun: Handbook of Mathematical Functions (on page 229, property 5.1.20) it is found that $$ \frac{1}{2} \log \left(1 + \frac{2}{x} \right) < \exp(x) E_1(x) < \log \left(1 + ...
4
votes
2answers
212 views

Finding the maximum of $\frac{a}{c}+\frac{b}{d}+\frac{c}{a}+\frac{d}{b}$

If $a,b,c,d$ are distinct real numbers such that $\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{d}+\dfrac{d}{a}=4$ and $ac=bd$. Then how would we calculate the maximum value of ...
3
votes
2answers
356 views

Inequalities involving arithmetic, geometric and harmonic means

Let $A$, $G$ and $H$ denote the arithmetic, geometric and harmonic means of a set of $n$ values. It is well-known that $A$, $G$, and $H$ satisfy $$ A \ge G \ge H$$ regardless of the value $n$. ...
2
votes
2answers
239 views

If $ a+b+c = \frac{9}{2}$ and $a,b,c>0$, then what is the minimum value of $\frac{a}{b^3+54}+\frac{b}{c^3+54}+\frac{c}{a^3+54}$

If $a+b+c = \dfrac{9}{2}$ and $a,b,c>0$, then what is the minimum value of $$\dfrac{a}{b^3+54}+\dfrac{b}{c^3+54}+\dfrac{c}{a^3+54} \qquad ?$$ My try: $$\begin{align*} ...
3
votes
3answers
60 views

Intuition behind $(\{a, b, p, q\} \subset \mathbb{R}^{+} \;\wedge\;\; 1/p +1/q = 1) \Rightarrow a^p/p + b^q/q \geq ab$

If $p$ and $q$ are positive real numbers with1 $$ \frac{1}{p} + \frac{1}{q} = 1,$$ then, for any non-negative real numbers $a, b$, $$ \frac{a^p}{p} + \frac{b^q}{q} \geq ab$$ My textbook offers a ...
2
votes
2answers
494 views

Inequality concerning limsup and liminf of Cesaro mean of a sequence [duplicate]

Let $\{x_n \}$ be a sequence of real numbers and let $y_n = \frac{(x_1 + x_2 + ... + x_n)}{n}$. (a) Prove that $\liminf x_n \le \liminf y_n \le \limsup y_n \le \limsup x_n$ (b) Give an example of a ...
1
vote
3answers
103 views

Infinite Series

How can you show that $$\left(1-\frac{2}{n^2}\right)^{n^2/2} \le \frac{1}{e}\:\: \qquad\forall n \ge 2$$ Any ideas? Infinite series have never really been my thing. Thanks
2
votes
1answer
46 views

Inequality in inner product space

Given $V$ an inner product space with norm $(‖v‖_V)^2$=$∫_Ω(v^2 (x)+|∇v|^2 )dx$. Prove that $$(∫_Ω(|v||w|+|∇v||∇w|)dx)^2 ≤ ∫_Ω(|v|^2+|∇v|^2 )dx ∫_Ω(|w|^2+|∇w|^2 )dx=(‖v‖_V)^2(‖w‖_V)^2.$$ Any ...
25
votes
27answers
5k views

How can I prove that $xy\leq x^2+y^2$?

How can I prove that $xy\leq x^2+y^2$ for all $x,y\in\mathbb{R}$ ?
2
votes
1answer
63 views

Find the probability that equation has two solutions of different signs

I have 3 random variables $\xi_1, \xi_2,\xi_3$ which are independent and uniformly distributed on segments $[-\sqrt{2}, \sqrt{2}], [-\sqrt{3}, \sqrt{3}], [-\sqrt{\pi }, \sqrt{\pi}]$ respectively. I ...
1
vote
1answer
63 views

Harmonic mean: show $\max\{ax,by\} \ge \frac{1}{a+b}(x+y)$, $a,b>1$, $x,y\ge 0$

Let $z=x+y$ with $x,y\ge0$ and $a,b>1$. Show that $$ \max\{ax,by\} \ge \frac{1}{a+b}z. \tag{1} $$ This requires either the use of: (a) the convex function $f(x)=\frac{1}{x}$, (b) the ...
4
votes
4answers
122 views

Find minimum in a constrained two-variable inequation

I would appreciate if somebody could help me with the following problem: Q: find minimum $$9a^2+9b^2+c^2$$ where $a^2+b^2\leq 9, c=\sqrt{9-a^2}\sqrt{9-b^2}-2ab$
2
votes
1answer
94 views

Two inequalities related to norm

We have some difficulties in the following problem: Let $H$ be a real Hilbert space. Find $\alpha>0$ such that $$ \langle\frac{u}{\sqrt{\|u\|}}-\frac{v}{\sqrt{\|v\|}}, u-v\rangle\geq ...