Questions on proving, manipulating and applying inequalities.

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8
votes
8answers
2k views

Which of the numbers is larger: $7^{94}$ or $9^{91} $?

In this problem, I guess b is larger, but not know how to prove it without going to lengthy calculations. It is highly appreciated if anyone can give me a help. Which number is larger ...
1
vote
1answer
39 views

If $\sin(\theta') \le (1+\alpha)\sin(\theta)$ then $\theta' \le (1+\sqrt\alpha)\theta$

I need to show that if $\sin(\theta') \le (1+\alpha)\sin(\theta)$ then $\theta' \le (1+\sqrt\alpha)\theta$, for small $\alpha$ and $\theta$ and $\theta'$ are between $0$ and $π/2$ I tried to use the ...
4
votes
4answers
91 views

Compute the minimum value of $a^n + b^n + c^n$ subject to $a^2 + b^2 + c^2 = 1 $

Assume that $a,b,c$ are non-negative real numbers and $n$ is a natural number $n \ge 3$. What is $f(n)=$ the minimum value of $a^n + b^n + c^n$ ? I find ; $$f(3) = \frac{1}{\sqrt{3}}\qquad ...
1
vote
2answers
109 views

Lipschitz-type estimate… True or false?

I have two parameters $\alpha,\varepsilon>0$ and the following difference: ...
0
votes
1answer
18 views

Inequality insde algorithm design (kleinberg), Disjoint Path

I am reading the "Algorithm Design" (Kleinberg et al). Inside this book, at the chapter 11.6 there is an inequality I have failed to resolve. Considering $\beta=m^{(1/3)}$ and $|I|>=1$ (it is a ...
0
votes
0answers
14 views

Check inequality

Let $x\in \mathbb{R}^d$ and $e_j$ be a basis vector with 1 at the $j$-position (otherwise $0$). Is it true that $\frac{1}{\mid x+e_j\mid^{d-2}}-\frac{1}{\mid x\mid^{d-2}}=O(\mid x\mid^{-d+1})$? Does ...
0
votes
0answers
21 views

Matrix block with Schur Complement

Consider $X\in S^n$, $S^n$ is a space of $n\times n$ symmetric matrices, partitioned as $$X = \begin{bmatrix}A & B \\B^T & C\end{bmatrix}$$ where $A\in S^k$. If det $ A\neq0$, the matrix ...
4
votes
1answer
103 views

Prove ths sum of $\small\sqrt{x^2-2x+16}+\sqrt{y^2-14y+64}+\sqrt{x^2-16x+y^2-14y+\frac{7}{4}xy+64}\ge 11$

Let $x,y\in R$.show that $$\color{crimson}{f(x,y)=\sqrt{x^2-2x+16}+\sqrt{y^2-14y+64} + \sqrt{x^2-16x+y^2-14y+\frac{7}{4}xy+64} \ge 11}$$ Everything I tried has failed so far.use Computer found this ...
1
vote
1answer
74 views

Prove an inequality using complex analysis

If $f:\mathbb{D}\rightarrow\mathbb{D}$ is holomorphic then prove that $$\frac{|f(0)| - |z|}{1 + |f(0)||z|} \leq|f(z)| \leq\frac{|f(0)| + |z|}{1 - |f(0)||z|} $$ I have been wracking my brain for ...
2
votes
2answers
29 views

Find the set of values of x for which $\frac{x+1}{2x-3}<\frac{1}{x-3}$

Here's what I've done: $\frac{x+1}{2x-3}<\frac{1}{x-3}$ $x+1<\frac{2x-3}{x-3}$ $(x+1)(x-3)<2x-3$ $x^2-2x-3<2x-3$ $x^2-4x<0$ $x(x-4)<0$ $0<x<4$ However this clearly ...
0
votes
1answer
15 views

Prove or disprove inequality using other inequality

if $3x^2+2ax+b+5\sin 2x >0$, then prove/disprove that $a^2-3b+15<0$.($x\in \mathbb R$) If disproven find correct inequality relating $a,b$ I don't know where to start. Any hint will be ...
1
vote
1answer
36 views

If $f(x)=2x^2+2x-4$ and $g(x)=x^2-x+2$, Find the number of integral values of $x\in[1,10]$ such that $\sqrt{f(x)}+\sqrt{g(x)}\ge \sqrt{2}$

If $f(x)=2x^2+2x-4$ and $g(x)=x^2-x+2$, Find the number of integral values of $x\in[1,10]$ such that $\sqrt{f(x)}+\sqrt{g(x)}\ge \sqrt{2}$ I tried squaring two times to remove the square root and ...
6
votes
7answers
340 views

Is $202^{303}$ greater or $303^{202}$?

Find without use of calculator which of the two numbers is greater $202^{303}$ or $303^{202}$. I think we have to do this with calculus because I got this question from my calculus book. I tried ...
0
votes
0answers
21 views

Inequality. Smth abot AM-GM, but [duplicate]

but I`m not indubitably sure. AM-GM doesn`t work. Karamata`s inequality doesn`t work too. Prove that inequality holds $$ \sum_{k=1}^n (\frac{a_1+\ldots +a_k}{k})^2 \leq 4 \sum_{k=1}^n a_k^2$$
1
vote
4answers
251 views

The sum of $m$th powers of positive odd integers up to $2n-1$ is at least $n^{m+1}$

Prove that inequality $$\sum_{k=1}^n (2k-1)^m \geq n^{m+1}$$ holds if $n,m$ are integer positive numbers. Karamata`s inequality doesn`t work.
0
votes
3answers
86 views

If $x^{15}-x^{13}+x^{11}-x^9+x^7-x^5+x^3-x=7$, prove that $x^{16}>15$.

"If $x^{15}-x^{13}+x^{11}-x^9+x^7-x^5+x^3-x=7$, prove that $x^{16}>15$." The above problem came on a local question paper. I tried to solve it by factorizing and sum of G.P. , But I was unable to ...
2
votes
1answer
25 views

why is $\left|xy\ log(\left|x\right|+\left|y\right|)\right|\leq\left|(\left|x\right|+\left|y\right|)log(\left|x\right|+\left|y\right|)\right|$?

I should note that this was used by my book in order to show that the limit of $xy\ log(\left|x\right|+\left|y\right|)$ at $(0,0)$ is $0$. After several attempts in vain, I plotted the function ...
-6
votes
1answer
33 views

Simplify and calculate this expression [closed]

Given that $x, y>0$, and $2x+y+\sqrt{5x^2+5y^2}=10$, prove that $x^4y \leq 16$. I heard that you can use the AM-GM inequality to prove this, but I am not sure how to get about it.
0
votes
0answers
10 views

Can an inequality solve a fundamental calculus of variation problem or shall we pull out an inequality from the problem

I don't have any knowledge of calculus of variations except that I know that its about finding the functions which can result in maximum or minimum value of some quantity given some constraints . I ...
-1
votes
2answers
53 views

Proving the inequality. [closed]

Let $a_1, a_2, \ldots , a_n$ be arbitrary positive numbers. Inequality $a_1\cdot a_2 \cdot \ldots \cdot a_k \geq 1$ holds $\forall k=\overline{1,n}$. Prove that $$\frac{1}{1+a_1} + ...
0
votes
2answers
54 views

Show that $\sum _{k=1} ^N \frac 1 {\sqrt {k^2 + 1} + k} > \frac 1 2 \ln \frac {2N+1} 3$, where $N$ is natural number.

Show that for $N = 1,2,3,\dots$ we have $$\sum _{k=1} ^N \frac 1 {\sqrt {k^2 + 1} + k} > \frac 1 2 \ln \frac {2N+1} 3$$ I got this as a calculus homework. I am supposed to show this, but it ...
1
vote
1answer
35 views

Proving an Inequality Using Functions

Let $n$ be a positive integer and $x>0$. Prove the following: $$\dfrac{x^n}{3}\geq \dfrac{1}{x+2}+\dfrac{(3n+1)\ln(x)}{9}$$ So I approached the problem by considering ...
0
votes
0answers
46 views

Solving integral inequalities ( Gronwall) [closed]

I don't know how to solve this inequality $$ v'(t) \leq ct +(v(t))^p, \qquad p >1 \, \quad \mathrm{and}\ \quad , c > 0 $$ With thanks.
1
vote
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
57 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
31 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
62 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
vote
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
vote
1answer
84 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
votes
1answer
85 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
54 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
votes
0answers
33 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
votes
2answers
44 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
votes
0answers
35 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
votes
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
vote
0answers
41 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
55 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
189 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
vote
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
vote
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 ...