Questions on proving, manipulating and applying inequalities.

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1
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1answer
35 views

Where i am going wrong in solving the inequality?

If $\cos x \left(\cos x+\frac12\right) >0$ then where should $x$ lie in the interval $(0,\pi)$ What I tried When i made two cases i got correct answer but when i used wavy-curve method. I am not ...
2
votes
0answers
63 views

Prove $\sum_{cyc}\frac{x(y-z)}{(2x+y)^2} +\frac13 \cdot \frac{x^2+y^2+z^2}{xy+yz+zx} \geqslant \frac13$ for positive $x,y,z$

$x,y,z > 0$, prove $$\sum_{\text{cyc}}\frac{x(y-z)}{(2x+y)^2} +\frac13 \cdot \frac{x^2+y^2+z^2}{xy+yz+zx} \geqslant \frac13$$ While this inequality can be proved by brute force, the elegant ...
0
votes
0answers
34 views

Prove $\sum_{cyc} \frac{5x+4y}{5x+4z} \leqslant \sum_{cyc} \frac{x+z}{x+y}$ for positive $x,y,z$

Let $x,y,z >0$, prove $$\sum_{cyc} \frac{5x+4y}{5x+4z} \leqslant \sum_{cyc} \frac{x+z}{x+y}$$ where $$\sum_{cyc} \frac{5x+4y}{5x+4z}=\frac{5x+4y}{5x+4z}+\frac{5y+4z}{5y+4x}+\frac{5z+4x}{5z+4y}$$ I ...
-3
votes
4answers
64 views

Prove that, if $a>c$ and $b>d$, thus $ab>cd$ [closed]

I would like to ask you a question: how could I prove that, if $a>c$ and $b>d$, thus $ab>cd$? Thank you for help. P.s. I forgot to tell you that $a>0, b>0, c>0, d>0.$
1
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3answers
168 views

Arcsin estimation

How do I prove that $$\arcsin (x)>\frac{3}{1+2\sqrt{1-x^2}}\text{ ?}$$ We received this example while we are learning integration so it must have something to do with it. But I can't seem to ...
1
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1answer
85 views

Let $a_1,…,a_{100}$ be non-negative numbers such that $a_1^2+…+a_{100}^2=1$ Prove that $a_1^2.a_2+…+a_{100}^2.a_1\le \frac{12}{25}$

Let $a_1,...,a_{100}$ be non-negative numbers such that $a_1^2+...+a_{100}^2=1$ Prove that $a_1^2.a_2+...+a_{100}^2.a_1\le \frac{12}{25}$ I was thinking about Cauchy Schwarz, but $4$ th powers make ...
0
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1answer
89 views

About the solution to “Finding the range of $y= \sqrt x + \sqrt{3-x}”$

I was reading the solution of "Find the range of $y = \sqrt{x} + \sqrt{3 -x}$" and I had some points of confusion about the solution posted in the OP. I wrote here my interpretation of the solution. ...
2
votes
1answer
15 views

Why $\left\lceil{ \frac{n}{1 +\Delta (G)}}\right\rceil \ge \gamma (G)$?

a dominating set for a graph $G = (V, E)$ is a subset $D$ of $V$ such that every vertex not in$ D$ is adjacent to at least one member of $D$. The domination number $γ(G)$ is the number of vertices in ...
4
votes
0answers
55 views

Hilbert's Inequality - improved???

Assume for convenience that $a_n\ge0$ (this also clarifies why certain inequalities below are in fact stronger than certain other inequalities below). Of the various inequalities Hilbert proved, I'm ...
5
votes
1answer
40 views

Norms inequality in a sequence space

Let $1 \leq p<q \leq \infty$ (p an q are not related) Let $\Phi$ be the vector space of all sequences with at most finitely many nonzero elements, meaning $\Phi=\{\{x_n\}_{n=1}^\infty|$ there is ...
3
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0answers
34 views

Surface area of Convex bodies contained in one another

Suppose we have two compact convex bodies one contained in the other in $\mathbb{R}^n$, $C\subset D\subset \mathbb{R}^n$. Does it follow that the ($n-1$ dimensional) surface area of $C$ is less than ...
0
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2answers
45 views

Prove this inequality in complex domain (5)

Let $z_{1},z_{2},z_{3}\in C$ show that $$|z_{1}+z_{2}+z_{3}|^2+|(z_{1}-z_{2})(z_{1}-z_{3})|+|(z_{2}-z_{3})(z_{2}-z_{1})| +|(z_{3}-z_{1})(z_{3}-z_{2})|\le 3(|z_{1}|^2+|z_{2}|^2+|z_{3}|^2)$$ Iif ...
1
vote
1answer
42 views

Why $Z_n$ is normally distributed?

We know $\epsilon_n \sim N(0,1)$, and $$Z_n = \frac {\mu_n^T(I-M_n)\epsilon_n} {\sqrt {\mu_n^T(I-M_n)\mu_n}},$$ where $M_n=X_n(X_n^TX_n)^{-1} X_n^T$, $\mu_n=X_n\beta_n$. Why $Z_n \sim N(0,1)$ ?? ...
0
votes
0answers
20 views

Demonstration involving inequality of traces of product of psd matrix

Let, $ \forall i \in [1, N]: P_i \in \mathbb{R}^{n \times n}, P_i \succ 0, w_i \in \mathbb{R}, \bar{P} = \sum_{i=1}^N w_i P_i$. Then, I want to demonstrate that $ \sum_{i=1}^N w_i ...
0
votes
4answers
36 views

Another elementary log inequality

I came across this as a part of an answer to a exercise: $$(n + \frac{1}{2}) \log(1 + \frac{1}{n}) - 1\gt 0$$ for $n\gt 0$, where $n$ is a real. How do you prove this? I tried some Taylor ...
3
votes
1answer
46 views

Cyclic Inequality in 3 variables

How can I prove the following inequality $$\frac{2a}{1+b^2}+\frac{2b}{1+c^2}+\frac{2c}{1+a^2}\geq 3, \forall\ a,b,c>0, a+b+c=3.$$ I tried Cauchy inequality, AM-GM, but I don't get anything ...
0
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0answers
36 views

Is there a name for these inequalities? Where can I look them up?

Consider the operators $A,B,C$ on Hilbert space $\mathcal H$: Show that: $$ \left \vert \left \vert AB \right \vert \right \vert \le \left \vert \left \vert A\right \vert \right \vert \left \vert ...
0
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2answers
151 views

Can some inequalities help to pin down an unique solution in a linear system of equations with infinite solutions?

I need to discuss the number of solutions of the following system of equations. Any help would be very appreciated. Consider the known parameters $a_1,...,a_4;d_1,d_2,d_3$ such that $0< a_i< ...
1
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0answers
43 views

Minimum variance, fixed mean , discrete random variable

Consider the ordered set $\mathcal{S}$ $=$ $\{0,a_i,a_2,\ldots,a_n\}$, where $a_i$ are all stricly positive real numbers and $a_i< a_{i+1}$ forall indices i. What is the random variable $X$ which ...
3
votes
1answer
57 views

Nice Inequality

I'm solving this inequality trying to use some changes of variable (for example $u=\frac{bc}{a}$, $v=\frac{ac}{b}$, $w=\frac{ab}{c}$), but I couldn't simplify the expression. The inequality is: For ...
-1
votes
2answers
39 views

What is the greatest (numerical) lower bound for $2x/(x + 1)$, if $x \geq C$ where $C > 0$ is a constant?

Question What is the greatest (numerical) lower bound for $2x/(x + 1)$, if $x \geq C$ where $C > 0$ is a constant? My Attempt The function $$f(x) = \dfrac{2x}{x + 1}$$ has first derivative ...
4
votes
1answer
190 views

Can we use matrix to solve this inequality?

Let $$f(x)=\begin{cases} 1&0\le x\le 1\\ 0&\rm{others} \end{cases}$$ Let $x_{i},a_{i}(i=1,2,\cdots,n)$ be positive real numbers, show that: ...
0
votes
1answer
21 views

Determine the largest and smallest values of Cov(X, Y)

Suppose $X$ ~ $Normal(0, 100)$ and $Y$ ~ $Binomial(80, 0.25)$ Determine (with explanation) the largest and smallest values of Cov(X, Y). The Cauchy-Schwarz inequality gives: $\sqrt{Var(X)Var(Y)} ...
1
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1answer
26 views

Is $(n-1)+\max\limits_{1\leqslant i\leqslant n} x_i\geqslant \left(1-\left(1-\frac{1}{n}\right)^n\right)\sum\limits_{i=1}^{n}x_i$?

Let $n>0$ and $x_i>0$ for all $i\in \{1,\ldots,n\}$ be integer numbers. I would like to compare $$(n-1)+\max\limits_{1\leqslant i\leqslant n} x_i,$$ and ...
1
vote
0answers
17 views

Trying to find the asymptotic behaviour of an inequality involving integers

Let $m,q,v$ be integers with $m\geq 2$, and $v|q-1$. A certain result that I have which is not important for this question, holds when $$q^{\frac{m}{2}-2}(q-mv)\geq v^{m-1}. \quad (1)$$ I would like ...
2
votes
2answers
71 views

How can I show $1-\frac{1}{x}+x^{1-\frac{1}{x}}<x$ for real $x>1$?

Denote $$f(x):=1-\frac{1}{x}+x^{1-\frac{1}{x}}$$ How can I prove that $f(x)<x$ holds for every real $x>1$ ? Wolfram gives the taylor series ...
1
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1answer
27 views

Minimize or maximize the powers

I came up with this problem and I could not find a proof. Basically the problem is, suppose positive numbers $a_i$, $i=1,2,\ldots,N$ satisfy $$\sum_{i=1}^Na_i=1$$ then for $p>0$ when the expression ...
0
votes
1answer
30 views

Comparing entropies $H((f(X,Y), g(X,Y)))$ and $H ((f(X,Y),g(X,Z)))$

Let X,Y,Z be three independent uniform distributions on $\{0,1\}^n$; $f, g:\{0,1\}^n\times\{0,1\}^n\rightarrow\{0,1\}$ be two boolean functions. Is it true that $$H((f(X,Y), g(X,Y)))\leq H ...
0
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2answers
27 views

Help finishing proof for $\sum_{i=1}^{n-1} (-1)^{i+1}i! \leq \frac{(2n)!}{2}$

I need help finishing this proof. I've come to a point where I don't know how to continue. I need to prove that the following inequality is true for all positive integers $n$. $$\sum_{i=1}^{n-1} ...
1
vote
0answers
57 views

Prove that $(xy+xz+yz)^2 \leq (2x^2 + y^2)(2z^2 + y^2)$

What is the easiest way way to prove that $$(xy+xz+yz)^2 <= (2x^2 + y^2)(2z^2 + y^2)$$ holds for real $x$, $y$, $z$? I've solved it using some "advanced" mathematics, but I would like to know ...
1
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0answers
33 views

Does this inequality $e^{H_n}\ln[H_n(H_n+0.5)]-e^{H_n-1}(H_n+1)\le 2n$ Hold?

Valid for all values of $n\ge 1$ Does this inequality hold? $$e^{H_n}\ln[H_n(H_n+0.5)]-e^{H_n-1}(H_n+1)\le 2n$$ Where $H_n=\sum_{i=1}^{n}\frac{1}{i}$ Let $A=H_n$ I simplified to ...
0
votes
0answers
64 views

Why can't this be done? Or can it?

I was writing an answer to this question here From AM-HM $$\frac1{1+x}+\frac1{1+y}+\frac1{1+z}=2$$ $$\frac{1+x+1+y+1+z}{3}\ge \frac{3}{\frac1{1+x}+\frac1{1+y}+\frac1{1+z}}$$ $$\implies ...
3
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1answer
61 views

Prove inequality $8xyz<1$ for positive $x,y,z$

Positive real $x,y,z$ such that $$\frac1{1+x}+\frac1{1+y}+\frac1{1+z}=2.$$ Prove inequality $$8xyz<1$$ My work so far: I tried rearrangement and AM-GM but fail.
0
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0answers
20 views

Solution to supremum inequality

I am currently learning for an exam. The task is to find a $k$ that fulfills the following condition: $$\sup_{\omega\in [-\pi,\pi]} \lVert 1 - G_c(z)L(z) \rVert _{z=e^{i\omega}} < 1$$ where ...
1
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0answers
15 views

Terminology for asserting truth of equality/inequality based on symbolic equalities/inequalities

This may seem silly, but I am curious about algorithms used to computationally assert the truthiness (true, false, or unknown) of symbolic statements subject to a set of inequality constraints, for ...
1
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1answer
24 views

Bernoulli's Inequality when $-2≤x<-1$

Why is it that Bernoulli inequality $(1+x)^r>1+rx$ is said to be true for every integer $r≥0$ and every real $x≥-1$; why the range $-2≤x<-1$ is not included? It seems that, by induction (or ...
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5answers
96 views

Prove $\ln^2(x)>\ln(x+1)\cdot\ln(x-1)$ for $x>2$ [closed]

Could anybody please help prove the following: $\ln^2(x)>\ln(x+1)\cdot\ln(x-1)$, for $x>2$.
0
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0answers
16 views

Inequality with supremum of a specific integral

I'm reading Kai Diethelm's "The analysis of fractional differential equations" and there's a part I haven't been able to figure out. It's from page 93. Let $p >0.$ Why does the following ...
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1answer
24 views

$\text{supp}(f) = \text{supp} (g) , \|f\|_{L^{2}(\mathbb R)} \leq \|g\|_{L^{2}(\mathbb R)} \implies \|f\|_{L^{2}([0,1])} \leq \|g\|_{L^{2}([0,1])}$?

Let $f,g \in L^{2}(\mathbb R)$. Suppose that $\int_{\mathbb R} |f(t)|^{2} dt \leq \int_{\mathbb R} |g(t)|^{2} dt.$ We also assume that support of $f$ and support of $g$ are equal. My Question ...
1
vote
1answer
41 views

complex inequality with 4

Show that if $z\in C$ and $|z|=1$ show that $$\sum_{k=1}^{n}(n-k+1)|1+z^k|\ge \lfloor\dfrac{n}{2}\rfloor\left(n-\lfloor\dfrac{n}{2}\rfloor\right)|1-z|$$ (2008 Romanian mathematical competitions) for ...
2
votes
0answers
47 views

Generalized weighted mean inequality

Let ${p_{1}},{p_{2}},\ldots,{p_{n}}$ and ${a_{1}},{a_{2}},\ldots,{a_{n}}$ be positive real numbers and let $r$ be a real number. Then for $r\ne0$ , we define ...
0
votes
2answers
49 views

Cauchy-Schwarz inequality on complex numbers

I need a proof for the Cauchy-Schwarz inequality on complex numbers, i.e. $|{(a_1b_1 + a_2b_2+...+a_nb_n)}|^2\leq(|a^2_1|+|a^2_2|+...+|a^2_n|) (|b^2_1|+|b^2_2|+...+|b^2_n|)$, where each ...
3
votes
0answers
93 views

prove $\frac{1}{\sqrt[4]{a^3(a+b^2)}}+\frac{1}{\sqrt[4]{b^3(b+c^2)}}+\frac{1}{\sqrt[4]{c^3(c+a^2)}} \geqslant \frac{3}{\sqrt[4]{2}}$

$a,b,c >0$ and $abc=1$, prove $$\frac{1}{\sqrt[4]{a^3(a+b^2)}}+\frac{1}{\sqrt[4]{b^3(b+c^2)}}+\frac{1}{\sqrt[4]{c^3(c+a^2)}} \geqslant \frac{3}{\sqrt[4]{2}}$$ 1. I tried rearrangement and AM-GM but ...
2
votes
1answer
115 views

Unconventional Inequality $ \frac{x^x}{|x-y|}+\frac{y^y}{|y-z|}+\frac{z^z}{|z-x|} > \frac72$

$x,y,z >0$, and $x \neq y \neq z$, prove $$ \frac{x^x}{|x-y|}+\frac{y^y}{|y-z|}+\frac{z^z}{|z-x|} > \frac72$$ I never see this kind of inequality in any textbook yet. No idea whatsoever to ...
1
vote
1answer
46 views

prove that $(ab+bc+ca)\bigg(\frac{a}{b(a^2+2b^2)}+\frac{b}{c(b^2+2c^2)}+\frac{c}{a(c^2+2a^2)}\bigg) \ge 3$

For $a, b, c$ positive reals prove that $$(ab+bc+ca)\bigg(\frac{a}{b(a^2+2b^2)}+\frac{b}{c(b^2+2c^2)}+\frac{c}{a(c^2+2a^2)}\bigg) \ge 3$$ I used the Cauchy-Swartz inequality in the LHS so I was ...
1
vote
1answer
73 views

Verifying an inequality

I have a problem proving an inequality regarding probabilities. You may prefer to skip to the definitions and the inequality right away without reading the paragraph below. Suppose there are $n$ ...
1
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0answers
54 views

Is it true for $a,b,c$ positive reals that $\frac{a}{\sqrt{a^2+2b^2}}+\frac{b}{\sqrt{b^2+2c^2}}+\frac{c}{\sqrt{c^2+2a^2}} \ge \sqrt3$

Is it true for $a,b,c$ positive reals that $$\frac{a}{\sqrt{a^2+2b^2}}+\frac{b}{\sqrt{b^2+2c^2}}+\frac{c}{\sqrt{c^2+2a^2}} \ge \sqrt3$$ My thoughts: The LHS is equal to ...
3
votes
1answer
93 views

How to prove that $(x^2+y)(y^2+z)(z^2+x)+2xyz \leqslant 10$, where $x,y,z \in R^+$ and $x^2+y^2+z^2+xyz=4$

$x,y,z \in R^+$ and $x^2+y^2+z^2+xyz=4$, prove $$(x^2+y)(y^2+z)(z^2+x)+2xyz \leqslant 10$$ I try several trig substitutions but feel hopeless with the cyclic term here. The condition ...
0
votes
3answers
37 views

How to prove that : $\sum_{j = 1}^{s} \sqrt{\alpha_j} \leq \sqrt{s} * \sqrt{\sum_{j = 1}^{s} \alpha_j}$

$\sum_{j = 1}^{s} \sqrt{\alpha_j} \leq \sqrt{s} * \sqrt{\sum_{j = 1}^{s} \alpha_j}$ How would I go about proving this? I think Cauchy - Schwarz might be useful, but I can't quite get it to work. ...
7
votes
2answers
223 views

$x,y,z \geqslant 0$ and $x^2+y^2+z^2+xyz=4$, prove $x^{\frac85}+y^{\frac85}+z^{\frac85} \geqslant 3$

$x,y,z \geqslant 0$ and $x^2+y^2+z^2+xyz=4$, prove $$x^{\frac85}+y^{\frac85}+z^{\frac85} \geqslant 3$$ 1) The equality occurs only at $x,y,z=1$. Let's assume $F=x^n+y^n+z^n$, I noticed that $n=1$ then ...