Questions on proving, manipulating and applying inequalities.

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0
votes
1answer
15 views

Solving inequalities using sign patterns

I don't want the solution, I'm just confused as to what to do with the $e^x$ in this instance. Solve for $x$ in $e^x(2\cos x-1)≤0$ where $0≤x≤4\pi$. Typically I would just reduce down to $\cos x ...
0
votes
5answers
63 views

help me find inequality

If $a+2b>3$ and $b+3c>5$ then $a+b+c> \hspace{.1cm} ?$
3
votes
3answers
48 views

Proving an Inequality using a Different Method

Is there another way to prove that: If $a,b\geq 0$ and $x,y>0$ $$\frac{a^2}{x} + \frac{b^2}{y} \ge \frac{(a+b)^2}{x+y}$$ using a different method than clearing denominators and reducing to ...
0
votes
0answers
6 views

Muirhead's inequality generalization

Muirhead's inequality is a well-known fact: for all permutations of finite expressions $x_1^{\alpha_1}x_2^{\alpha_2}...x_n^{\alpha_n}$ (if the sets of powers satisfy $(\alpha_1,\alpha_2...\alpha_n) ...
1
vote
3answers
50 views

Prove inequality $x^sy^{1-s} \leq sx + (1-s)y$

Given $s \in (0,1)$, prove $$x^sy^{1-s} \leq sx + (1-s)y$$ for $x,y > 0$ Tried some algebraic manipulations but I'm guessing I need to use some trick. Any suggestions, hints?
0
votes
0answers
16 views

Calculate an upper bound for $\left|\frac{e^{i\alpha-\beta}-e^{-(i\alpha-\beta)}}{i\alpha-\beta}\right|$

Let $\alpha,\beta\in\mathbb R$. Calculate an upper bound for $$\left|\frac{e^{i\alpha-\beta}-e^{-(i\alpha-\beta)}}{i\alpha-\beta}\right|$$ I think that $\cosh$ is involved in the answer, but I can't ...
3
votes
4answers
216 views

Log concavity of binomial coefficients: $ \binom{n}{k}^2 \geq \binom{n}{k-1}\binom{n}{k+1} $

How do we prove that Binomial coefficients are log-concave? A sequence $a_0, \dots, a_n$ is log-concave if $a_k^2 \geq a_{k-1}a_{k+1}$. $$ \binom{n}{k}^2 \geq \binom{n}{k-1}\binom{n}{k+1} $$ If $ n ...
5
votes
1answer
114 views

Cauchy-Schwarz Inequality without using $\langle a x,y\rangle=a\langle x,y\rangle$

Let $V$ be a vector space and define a function $\langle .,.\rangle:V\times V\to\mathbb{C}$ such that $$\begin{align} & \langle x,y\rangle=\overline{\langle y,x\rangle }\,\,\,\forall x,y\in V\\ ...
1
vote
2answers
94 views

how to solve the a,b,c inequality?

$a,b,c>0,a+b+c=3,$ prove that: $$\frac{a^3}{a^2+2b+c}+\frac{b^3}{b^2+2c+a}+\frac{c^3}{c^2+2a+b}\geq\frac{3}{4}$$
1
vote
0answers
27 views

On the Chernoff bound

Recently, I saw the Chernoff bound written as follows. Let $X_1,X_2,\ldots,X_n$ be drawn i.i.d. on alphabet $\mathcal{X}$ and let $f:\mathcal{X}\to [0,1]$ be any function. Let $\mathbb{E}f(X_1) = ...
6
votes
4answers
119 views

Show that $2^{\sqrt{2}}>1+\sqrt{2}$

Given that $\sqrt{2}>1.4$ and $(1+\sqrt{2})^5<99$, I need to show that $2^{\sqrt{2}}>1+\sqrt{2}$ From the given inequalities, I deduce that $(1+\sqrt{2})<\sqrt[5]{99}$ and ...
0
votes
2answers
24 views

Inequality involving floor

Let $x$ be randomly chosen from $\{1,...n\}$. Define $X_{p}$ such that \begin{equation} X_p= \begin{cases} 1, & \text{if}\ p|x, \\ 0, & \text{otherwise.} ...
1
vote
1answer
48 views

Use stirlings approximation to prove inequality.

I have come across this statement in a text on finite elements. I can give you the reference if that will be useful. The text mentions that the inequality follows from Stirling's formula. I can't ...
11
votes
1answer
241 views

How prove this inequality $\sum_{cyc}\frac{x^a\ln{x}}{(x^a+y+z)^2}\ge 0$

Question: let $x,y,z$ be postive numbers,and such $xyz\ge 1$,and such $a$ is real numbers.show that $$\dfrac{x^a\ln{x}}{(x^a+y+z)^2}+\dfrac{y^a\ln{y}}{(y^a+x+z)^2}+\dfrac{z^a\ln{z}}{( ...
10
votes
6answers
2k views

Inequality with square roots: $\sqrt{x^2+1}+\sqrt{y^2+1}\ge \sqrt{5}$

Let $x$ and $y$ be nonnegative real numbers such that $x+y=1$. How do I show that $\sqrt{x^2+1}+\sqrt{y^2+1}\ge \sqrt{5}$? How do I deal with square roots inside the inequality?
1
vote
1answer
18 views

Logarithmically bounded function fulfills $f(n) \le \lceil m \cdot \log_b r \rceil$ for certain numbers $n,m,r$

Let $f : \mathbb N \to \mathbb N$ be a function such that $f(n) \le 1 + \log_b n$ for some base $b$ and all $n$. Now let $n \in \mathbb N$ have the property that $$ \frac{r^m - 1}{r-1} \le n < ...
9
votes
0answers
45 views

Problem with inequality $\min (x_1,x_2,\ldots,x_n)$

let $0\le x_i$, $i=1,2,\ldots,n$, and $a_i=1+(i-1)d$, $d\in[0,2],\forall i\in\{1,2,3,\ldots,n\}$, show that $$(1+a_n)\left(x_1+x_2+\cdots+x_n\right)^2\ge 2n \min(x_1,x_2,\ldots,x_n) \left(\sum_{i=1}^n ...
2
votes
1answer
53 views

Inequality on integrals of continuous functions: $\int_0^1 f^2(x)\,dx \geq \left(\int_{0}^{1} f(x) \,dx\right)^2$

Let $f\colon [0, 1] \to \mathbb{R}$ be a continuous function. How to prove $$\int_0^1 f^2(x)\,dx \geq \left(\int_{0}^{1} f(x) \,dx\right)^2$$ (I'm not getting anything.. any hint is appreciated)
0
votes
0answers
42 views

Solving 3 linear equations with 6 variables, 3 of which are the same across each equation.

I feel like I could hash out the answer on my own but am struggling to think of an elegant way to show it. The equations are as follows: $$3.5x-2.5y-3z=A$$ $$-7.5x+3.75y+5.25z=B$$ ...
3
votes
3answers
177 views

Proving the inequality $\frac{a+b}{2} - \sqrt{ab} \geq \sqrt{\frac{a^2+b^2}{2}} - \frac{a+b}{2}$

Show that for any two positive real numbers $a$ and $b$, $\frac{a+b}{2} - \sqrt{ab} \geq \sqrt{\frac{a^2+b^2}{2}} - \frac{a+b}{2}$ My attempt: $(\sqrt a-\sqrt b)^2\geq0\\\frac{a+b}{2}\geq ...
2
votes
1answer
32 views

An inequality involving supremum and integral

Let $g$ be a positive function defined on $(0,\infty)$. Is the following inequality always true ? $$ \sup_{r<t<\infty}g(t)\leq C\int_{r}^{\infty}g(t)\frac{dt}{t}, $$ where positive constant $C$ ...
0
votes
0answers
21 views

Finding upper critical value with Chebyshev's inequality

Consider $X$ is a Poisson random variable with distribution $X$~$Pois(\theta)$. I define the mean in my hypothesis as $\lambda$ and nominal significance level $\alpha$. Null hypothesis $H_0 : ...
7
votes
2answers
97 views

Name of $|x|^p+|y|^p\le (|x|+|y|)^p$ ($p\ge 1$)?

I checked these What is the difference between square of sum and sum of square? Prove $(|x| + |y|)^p \le |x|^p + |y|^p$ for $x,y \in \mathbb R$ and $p \in (0,1]$. It is easy to see $p$-th power ...
2
votes
2answers
79 views

Cauchy-Schwarz inequality problem

The problems: Prove that $$\frac{\sin^3 a}{\sin b} + \frac{\cos^3 a}{\cos b} \geqslant \sec (a-b),$$ for all $a,b \in \bigl(0,\frac{\pi}{2}\bigr)$. Prove that $$\frac{1}{a+b} + \frac{1}{b+c} ...
1
vote
0answers
19 views
+200

Lower bound for (function of) density of well-behaved random variable

Suppose we have a non-negative random variable $\tilde{\theta}$ such that $\mathbb{E}\tilde{\theta} = a > 0$, with finite variance $\sigma^2$. $\tilde{\theta}$ can take on values from $0$ to ...
3
votes
2answers
36 views

How to show the inequality is strict?

This is an exercise from Rudin's Principles of Mathematical Analysis, Chapter $6$. Suppose $f$ is a real, continuously differentiable function on $[a, b]$, $f(a) = f(b) = 0$, and $$\int_a^b ...
5
votes
4answers
269 views

Minimum value of reciprocal squares

I am bit stuck at a question. The question is : given: $x + y = 1$, $x$ and $y$ both are positive numbers. What will be the minimum value of: $$\left(x + \frac{1}{x}\right)^2 + ...
5
votes
1answer
336 views

inequality $10<2^{2^{\frac {3}{\log_2 \log_2 10}}}$

While working on this question I ended up with $10<2^2{^{\frac {3}{\log_2 \log_2 10}}}$ I am looking for answers using methods similar to this or this or this or this. Alternative original ...
-1
votes
0answers
27 views

Could someone give a detailed (yet elementary) proof for Jensen's inequality?

I want to prove that Suppose there is a function $f:[a,b] \to \mathbb R$, and there are $x_i \in [a,b], w_i \gt 0 $ for $i=1,\dots,n$ such that $\sum_{i=1}^nw_i=1$, then if the function is convex, ...
1
vote
2answers
24 views

A question on inequality and differentiation of logarithms

Show by differentiating that $\ln x$ is a concave function of $x$. Deduce that if $p,q,x,y$ are positive real numbers with ${1\over p}+{1\over q}=1$, then $$xy \lt {x^p\over p}+{y^q\over q}$$ I ...
-1
votes
2answers
28 views

What is the Solution? x ≥8lgx

x ≥ 8lgx I have to find which x satisfy this inequality. I found the points using graph, but I'd like someone to show me how to find it without it.
2
votes
3answers
237 views

Showing that: $(\frac{a}{b+c})^2+(\frac{b}{a+c})^2+(\frac{c}{a+b})^2+\frac{10abc}{(a+b)(b+c)(c+a)}\ge 2$

Let a;b;c>0. Prove: $$\left(\frac{a}{b+c}\right)^2+\left(\frac{b}{a+c}\right)^2+\left(\frac{c}{a+b}\right)^2+\frac{10abc}{(a+b)(b+c)(c+a)}\geq 2$$ I think ...
3
votes
3answers
91 views

Does $\frac{x+y}{2}>\frac{a+b}{2}$ hold?

$a$ and $b$ are two real positive numbers. Given that $x=\sqrt{ab}$ and $y=\sqrt{\frac{a^2+b^2}{2}}$, which one has a higher value, $\frac{x+y}{2}$ or $\frac{a+b}{2}$? We know that ...
2
votes
2answers
53 views

O.I.M. polygon inequality

I am trying to prove an inequality which was used to prepare the Romanian O.I.M. team. I seem to lack ideas on how to tackle this problem. We take a convex polygon $P_1\ldots P_{n+2}$ and consider ...
3
votes
1answer
56 views

Rewriting $|x-10|+|y-5|\leq 7$ so that absolute values disappear - Algebra

Equation 1: $|x-10|+|y-5|\leq 7$ I want to rewrite this equation into equations that do not have the absolute value. $|A|\leq b$ can be written as $A \leq b$ $A \geq -b$ I want to apply the ...
3
votes
3answers
132 views

Prove an inequality.

Prove that $$\displaystyle{(|x_1+y_1|^p + |x_2+y_2|^p +\dots +|x_n+y_n|^p)^{\frac{1}{p}}\leq (|x_1|^p + |x_2|^p +\dots +|x_n|^p)^{\frac{1}{p}}+(|y_1|^p + |y_2|^p +\dots +|y_n|^p)^{\frac{1}{p}}}$$ for ...
1
vote
2answers
42 views

Show $n \cdot \log_b r + \log_b \frac{r}{r-1} \le \lceil n \cdot \log_b r \rceil$

Let $n$ be a natural number and $b, r > 1$ be two natural numbers, then I guess we have $$ n \cdot \log_b r + \log_b \frac{r}{r-1} \le \lceil n \cdot \log_b r \rceil. $$ where $\lceil x \rceil = ...
1
vote
1answer
86 views

Can one use Hölder's inequality or some other method for this?

Suppose that $\sum_{i=1}^{n}\lambda_{i}=1$, where $\lambda_{i}>0$, and $\sum_{i=1}^{n}b_{i}^{2}=1$, where $b_{i}>0$. Does one have $\sqrt{n}\sum_{i=1}^{n}\lambda_{i}b_{i}\le B$ for some constant ...
1
vote
2answers
43 views

Given that x,y,z are real positive numbers such that $(xyz)(x+y+z)=1$ how can I show that $xyz\leq\frac{\sqrt{\sqrt 3}}{3}$?

Given that $x$, $y$, $z$ are real positive numbers such that $xyz(x+y+z)=1$, how can I show that $xyz\leq\frac{\sqrt{\sqrt 3}}{3}$? We have $(xyz)^{1/3}\leq \frac{x+y+z}{3}$ by inequalities between ...
1
vote
0answers
21 views

Tighter upper bounds with ratios of powers of norms

This question arises in concentration or sparsity measures for finite sequences. Given $x\in \mathbb{R}^K$ and $1 \le r < s$, i try to find a tight upper bound for $$\psi_{r,s}(x) = \frac{\sum_1^K ...
1
vote
1answer
363 views

Equality in the Cauchy-Bunyakowsky-Schwarz inequality for a semi-inner product

I am stuck with an exercise that I found in a textbook by Conway. First, I would like to clarify what is meant by a semi-inner product. Definition. Suppose that $\mathscr X$ is a vector space over ...
4
votes
2answers
140 views

When does $\|x+y\|=\|x\|+\|y\|?$

Let $(V_\mathbb R,\langle,\rangle)$ be an inner product space. I'm trying to see for $x,y\in V$ when does $\|x+y\|=\|x\|+\|y\|?$ Let $\|x+y\|=\|x\|+\|y\|$ Squaring both sides, $\langle ...
3
votes
0answers
62 views

Is my proof rigorous? (Archimedes area of parabola)

I am currently reading Apostol's Calculus volume 1 and was revising the part where the area of a parabolic segment is found. I decided to write my own proof similar to the one in the book, which I ...
3
votes
1answer
74 views
+50

Checking logarithm inequality.

Which one of the following is true. $(a.)\ \log_{17} 298=\log_{19} 375 \quad \quad \quad \quad (b.)\ \log_{17} 298<\log_{19} 375\\ (c.)\ \log_{17} 298>\log_{19} 375 \quad \quad ...
6
votes
5answers
250 views

Show that $\frac{xy}{z} + \frac{xz}{y} + \frac{yz}{x} \geq x+y+z $ by considering homogeneity

Well, I'm preparing for an undergrad competition that is held in April and because of that I've been trying to solve the inequalities I find on the internet. I found this problem: $$\displaystyle ...
1
vote
1answer
50 views

Integral inequality with first two moments equal to $1$.

Let $f\in \mathcal{C}^0([0,1],\mathbb{R})$ such that $$ \int_0^1 f(x)\text{d}x = \int_0^1 xf(x)\text{d}x=1.$$ Show that $\int_0^1 f(x)^2 \ge 4$. I tried to use the Cauchy-Schwarz inequality such ...
1
vote
0answers
43 views

Want to prove an inequality of two norms in a Hilbert space

So here is my problem, Let $D:=[-d,d]\times[-d,d]$ and $C_0^{\infty}$(D) be the set of all smooth functions with compact support in $D$ which are zero on the boundary of $D$. Moreover we have the ...
0
votes
1answer
21 views

Is there an upper bound for expectation of product of two measurable function on a random variable?

I wonder if there is an useful upper bound for $\mathbb{E}_{x\sim p(x)}[f(x)g(x)]$ in the following form: $$ \mathbb{E}_{x\sim p(x)}[f(x)g(x)] \leq \mathbb{E}_{x\sim p(x)}[f(x)]\times xxxxxx $$ The ...
10
votes
5answers
5k views
3
votes
0answers
22 views

The Hardy-Littlewood-Sobolev Inequality

Let $f:\mathbb R^n \to \mathbb C$, $n\ge 2$. I saw the line that the inequality $$ \left\| |x|^{-1} * |f|^2 \right\|_{L^\infty} \le C\|f\|_{L^{\frac{2n}{n-1},2}}^2 $$ with some constant $C>0$. Here ...