Questions on proving and manipulating inequalities.

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231 views

Inequality with Stirling's numbers

I supect that for all $n>k>0$: $k^2\left\{ \begin{array}{c}n\\k\end{array} \right\}^2 +2k\left\{ \begin{array}{c}n\\k\end{array} \right\}\left\{ \begin{array}{c}n\\k-1\end{array} ...
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1answer
79 views

Is the hypergeometric function $F(5/4,3/4; 2, z)$ bounded on $(0,1]$

Consider the classical hypergeometric function $F(5/4,3/4; 2, z)$ for $z\in (0,1]$. Is this bounded by some real number (independent of $z$)? I'm aware of Euler's formula: $$F(5/4,3/4; 2, z) = ...
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2answers
152 views

Proving the inequality: $ x\exp(-x^2/4)(\exp(x)+\exp(-x)) \leq 1000 \exp(-x)$

Doing some tests with Maple I "guessed" the following inequality with exponential function (for $x\geq 0$) $$ x\exp(-x^2/4)(\exp(x)+\exp(-x)) \leq 1000 \exp(-x).$$ Is there an easy proof? Can one ...
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3answers
182 views

Proving the Inequality: $ (x_{1}+x_{2}+…+x_{n})\leq n^{n} x_{1}x_{2} \cdots x_{n}$

I am stuck on proving the following inequality: Let: $x_{1},x_{2},...,x_{n}\geq 0$. Prove that:$(x_{1}+x_{2}+...+x_{n})\leq n^{n} x_{1}x_{2}...x_{n}$ where $n$ is a natural number.
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6answers
2k views

Check my solution to $x^2 + x + 1 > 0$

I spent an hour or so yesterday trying to solve the inequality $x^2 + x + 1 > 0$. Since I'd spent so long on a problem didn't seem like it should be that difficult, I decided I'd call it a day and ...
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0answers
91 views

Inequality with Stirling's numbers of the second kind [duplicate]

Possible Duplicate: Proof strategy - Stirling numbers formula Prove inequality: $ \left\{\begin{array}{c}n\\k-1\end{array}\right\}\left\{\begin{array}{c}n\\k+1\end{array}\right\} \le ...
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1answer
115 views

Clarkson type inequality

Is it true that for $p\in (1,2)$ the following inequalities holds: $$ 2^{p-1} (|x|^p+|y|^p)\leq |x+y|^p+|x-y|^p \leq 2 (|x|^p+|y|^p)$$ for $x, y \in \mathbb{R}$ ? Thanks.
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3answers
197 views

A Cauchy with $\varepsilon$- type inequality for $C^1$ functions

The inequality is the following: For any $\varepsilon >0$ there exists $C_\varepsilon >0$ s.t.: $$\lVert u\rVert_{\infty}\leq \varepsilon\ \lVert u^\prime ...
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4answers
268 views

How many sequence of integers ($j_1 , j_2 , . . . , j_k$) are there such that $0 ≤ j_1 ≤ j_2 ≤ . . . ≤ j_k ≤ n$?

I need to solve the problem, How many sequence of integers ($j_1 , j_2 , . . . , j_k$) are there such that $0 ≤ j_1 ≤ j_2 ≤ . . . ≤ j_k ≤ n$? I've been given a hint, (Hint: Reduce the ...
7
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2answers
232 views

Find the range of the given expression?

If p , q , r denotes the side of the triangle ,then the below expression will always lies between? $$\left(\frac{p}{q+r}\right) + \left(\frac{r}{q+p}\right) + \left(\frac{q}{p+r}\right) $$ I tried ...
4
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1answer
178 views

Mixed $L^p$-norm Inequality on $[0,1]\times\mathbb{R}^2$.

Looking at $(t,x)\in[0,1]\times\mathbb{R}^2$, I came across the statement (for sufficiently smooth) real-valued $f$ that $$ \|f(t,x)\|_{L^\infty_tL^2_x} \lesssim \|f\|_{L^2_tL^2_x}^{1/2}\|\partial_t ...
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0answers
83 views

Markov-like inequality for functionals

Dear fellow mathematicians, The Markov inequality reads, for $(\Omega, \mathcal{F}, \mu)$ being a measure space, and $f$ a real valued function on $\Omega$ (you can also see Stein, Singular Integrals ...
2
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2answers
115 views

What is the least value of the expression?

If p , q , r are all positive numbers. And if p + q + r =1 then what is the least value of $$\left(\frac{1-p}{p}\right) \left(\frac{1-q}{q}\right) \left(\frac{1-r}{r}\right) $$ ? I ...
32
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3answers
1k views

Inequality for expected value

A colleague popped into my office this afternoon and asked me the following question. He told me there is a clever proof when $n=2$. I couldn't do anything with it, so I thought I'd post it here and ...
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3answers
643 views

Find the maximum value of the product xyz?

IF $x , y , z$ are arbitary positive real numbers satisfying the equation $$ 4xy + 6yz + 8xz = 9$$ Find the maximum value of the product $xyz$. I dont know from where to begin . 3 variables and ...
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1answer
595 views

Find the maximum possible value of an expression, subject to a linear constraint

Find the maximum possible value of the expression $$\left(\frac{x^2 + 3y^2 + 9z^2}{1}\right) $$ subject to $x+2y +3z = 12$, where $x,y,z$ are real numbers.
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3answers
2k views

Proof by induction of summation inequality

Prove by induction the summation of $\frac1{2^n}$ is greater than or equal to $1+\frac{n}2$. We start with $$1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\dots+\frac1{2^n}\ge 1+\frac{n}2$$ for all positive ...
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1answer
33 views

If we define the expression $P(x)=x^2$ and an expression $Q(x) = |4x|$, then for how many integer values is $P(x) -Q(x)$ a positive quantity?

If we define the expression $P(x)=x^2$ and an expression $Q(x) = |4x|$, then for how many integer values is $P(x) -Q(x)$ a positive quantity? $ a)2 \quad\quad\quad b) 4 \quad\quad c) 6 ...
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1answer
174 views

Finding the minimum value of $\left(\frac{a + 1}{a}\right)^2 + \left(\frac{b + 1}{b}\right)^2 + \left(\frac{c + 1}{c}\right)^2 $

Find the minimum value of $$\left(\frac{a + 1}{a}\right)^2 + \left(\frac{b + 1}{b}\right)^2 + \left(\frac{c + 1}{c}\right)^2 $$ I tried to expand it and break it into individual terms and use ...
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1answer
111 views

Theta notation from the inequality $c_1lg(n) \leq lg(k) \leq c_2lg(n)$ [duplicate]

Possible Duplicate: tight bounds from a certain inequality Consider the inequality $$ c_1lg(n) \leq lg(k) \leq c_2lg(n),\text{ for } n \geq n_{0} $$ With $c_1,c_2,n_0 > 0$, $lg(k) = ...
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1answer
78 views

Proving or refuting an inequality regarding the variance

I'm trying to prove, or find a counterexample, for the following problem: Let $Y = \{y_i\}_{i=1}^n$ be a set of data, where $y_i \ge 1$ for $i \in \{1,\ldots,n\}$, and let $\alpha$ be a natural ...
4
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2answers
227 views

Bound for the Legendre function of the second kind of degree $1/2$

Let $Q_{1/2}(u)$ be the Legendre function of the second kind of degree $1/2$. One can show that $Q_{1/2}(u) = O(u^{-3/2})$ as $u\to \infty$; see Equation 21 in Section 3.9.2 of Higher transcendental ...
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2answers
932 views

Inequality involving norm of matrix integral

This question seems basic but I could not find an answer. I have seen the inequality $$\left\|\int_a^b x(t) dt \right\| \leq \int_a^b \left\| x(t) \right\| dt $$ where $x(t) \in \mathbb{R}^n$ is a ...
1
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1answer
163 views

Proof on the inequality involving matrix splitting and trace operator

Suppose positive definite matrices $V, B, D\in\mathbb{R}^{n\times n}$ are given, where $D$ only contains diagonal entries of $V$, i.e., $D=diag(V)$, and $X, G\in\mathbb{R}^{n\times 2}$. Could the ...
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2answers
345 views

Bennett's Inequality to Bernstein's Inequality

Bennett's Inequality is stated with a rather unintuitive function, $$ h(u) = (1+u) \log(1+u) - u $$ See here. I have seen in multiple places that Bernstein's Inequality, while slightly weaker, ...
2
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1answer
130 views

Bounding the number of integer solutions of the following inequality

Let $r\geq 1$ be a real number, $-1\leq x\leq 1$ a real number and $y>2$ a real number. We consider this data to be fixed. How can I obtain an upper bound on the number of $(a,b,c,d)\in ...
1
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1answer
310 views

How can I apply Young's inequality here?

I'm reading Fourier Analysis and Nonlinear Partial Differential Equations by Rapha\"el Danchin et al. There are lines on page 9 reads: Using Young's inequality for $\mathbb{Z}$ equipped with the ...
4
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3answers
2k views

lim sup inequality $\limsup ( a_n b_n ) \leq \limsup a_n \limsup b_n $

I´m not sure how to start with this proof, how can I do it? $$ \limsup ( a_n b_n ) \leqslant \limsup a_n \limsup b_n $$ I also have to prove, if $ \lim a_n $ exists then: $$ \limsup ( a_n b_n ) = ...
0
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2answers
492 views

Pretty simple question about running time

What is the smallest value of $n$ such that an algorithm whose running time is $100n^2$ runs faster than an algorithm whose running time is $2^n$ on the same machine?
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3answers
123 views

For $x+y=n$, $y^x < x^y$ if $x<y$.

(updated) I'd like to use this property for my research, but it's somewhat messy to prove. $$\text{For all natural number $x,y$ such that $x+y=n$ and $1<x<y<n$, then $y^x < x^y$}.$$ For ...
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1answer
79 views

Looking for hints of this inequality

I think the following two inequalities are true. However, the proof may not be easy. Does anyone have any hints? Thank you very much! Fix $a>1$. there exists two constants $K_1$ and $K_2$, such ...
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0answers
108 views

Inequalities from power series

Suppose we have a power series $\sum a_n x^n$ with some positive radius of convergence whose coefficients are known. Let $f(x) = \sum a_n x^n$ within the radius of convergence. When truncating the ...
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4answers
267 views

Solving the inequality $\frac{x}{\sqrt{x+12}} - \frac{x-2}{\sqrt{x}} > 0$

I'm having troubles to solve the following inequality.. $$\frac{x}{\sqrt{x+12}} - \frac{x-2}{\sqrt{x}} > 0$$ I know that the result is $x>0$ and $x<4$ but I cannot find a way to the ...
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2answers
257 views

Prove that for every $0<\alpha<\beta<\pi/2$: $\tan\beta/\tan\alpha>\beta/\alpha$

Prove that for every $0<\alpha<\beta<(\pi/2)$: $$\displaystyle\frac{\tan\beta}{\tan\alpha}\gt\frac{\beta}{\alpha}$$ I tried setting a function $f(x) = \tan(x)$ and using the Mean Value ...
2
votes
1answer
204 views

What is the minimum possible value of $(a + b + c)$?

$a$, $b$ and $c$ are real positive numbers satisfying $\dfrac{1}{3} \le ab + bc + ca \le 1$ and $abc \ge \dfrac{1}{27}$ What is the minimum possible value of $(a + b + c)$?
2
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1answer
173 views

Does this inequality hold? proof or counterexample

Does the following inequality $$ \sup_{x\in (0,1)}u^2(x)\leq C_1\int_0^1 x u^2(x)\,\textrm{d}x+C_2\int_0^1 x(u')^2(x)\,\textrm{d}x $$ hold for all $ u\in C^1(0,1)$? If so please give me a ...
4
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1answer
84 views

Is any of the sets a subset of a union of other sets?

I have eleven sets, all of them are subsets of $X:=\{(a,b,c,d)\in[-1,1]^4: a\le b,\text{ and } c\le d\}$: $$\begin{align*} A_1&:=\{(a,b,c,d)\in X: b\ge 0,\ c\le a+b+d\}\\ ...
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2answers
182 views

Result of Chebyshev's Inequality if just more than instead of more than or equals?

This is a question that I happen to think of when looking at the Chebyshev's Inequality. In the inequality, it has this: $$ P(\left| X-\mu \right| \ge k\sigma )\le \frac { 1 }{ { k }^{ 2 } } $$ ...
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3answers
192 views

Inequality relating diameter of the image of a holomorphic function on the unit disk to the derivative at 0. [duplicate]

Possible Duplicate: First derivative bounded by supremum of difference of values in disc Let $f$ be holomorphic in the disk $D_1(0)$ and let $d=\operatorname{diam}(f(D_1(0))$. I want to ...
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3answers
120 views

Is $3^x \lt 1 + 2^x + 3^x \lt 3 \cdot 3^x$ right?

Is $3^x \lt 1 + 2^x + 3^x \lt 3 \cdot 3^x$ right? This is from my lecture notes which is used to solve: But when $x = 0$, $(1 + 2^x + 3^x = 3) \gt (3^0 = 1)$? The thing is how do I choose which ...
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0answers
219 views

Prove the inequality $n! > 2^n$ by induction. [duplicate]

Possible Duplicate: Proof the inequality $n! \geq 2^n$ by induction Prove By Induction that $n!>2^n$ I have to prove the inequality $n! > 2^n$ for all integers $n \geq4$. I am ...
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5answers
591 views

Help understanding proof of generalization of Cauchy-Schwarz Inequality

I'm having trouble with an exercise in the Cauchy Schwarz Master Class by Steele. Exercise 1.3b asks to prove or disprove this generalization of the Cauchy-Schwarz inequality: The following is the ...
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1answer
124 views

Cryptic one-line inequality solution — can you make sense of it?

The following is a snip from the book Olympiads: A Mathematical Olympiad Approach: The obvious approach (for me) is to use the triangle inequality to get $$|a|-|b| \leq |a-b|$$ $$|b|-|a| \leq |b-a| ...
2
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1answer
81 views

A minimization problem for a function involving maximum

Let $a,b,c,d$ be constants in the interval $[-1,1]$. Define $$f(x,y)=\max\{|y-a|,1-b\}+\max\{1-x,1-y\}+\max\{|x-c|,1-d\}$$ for $ -1\le x\le 1, -1\le y\le 1.$ Prove, or disprove, that the minimum value ...
2
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1answer
211 views

Inequality, Vojtěch Jarník Competition 2006

This is the problem from Vojtěch Jarník Competition 2006. Given real numbers $0=x_1,x_2<\dots<x_{2n}<x_{2n+1}=1$ such that $x_{i+1}-x_{i}\leq h$ for $1\leq i \leq 2n$, show that ...
3
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1answer
1k views

What are the regularity conditions for differentiating an inequality?

This looks like a very trivial question, but I could not find an answer on the web or my usual math references. Suppose I have an inequality of the form $f(x) + g(x) \leq 0$ where $f$ and $g$ are, ...
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0answers
182 views

Non-trivial upper bound for binomial sum

I'm trying to upper bound the following sum: $$ \sum\limits_{k=0}^n \begin{pmatrix} n \\ k \end{pmatrix} e^{-\frac{m}{2^k} } $$ where $n>0$ is fixed and $0\leq m \leq 2^n$. A trivial upper ...
3
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1answer
299 views

Concave functions on discrete domain

We are given a positive, non-decreasing function $f$ defined on natural numbers with $f(0) = 0$. $f$ has a submodularity-like property: $f(x+y) \leq f(x) + f(y) $ for all natural numbers $x$ and ...
3
votes
3answers
310 views

Prove $a^ab^bc^c\ge (abc)^{\frac{a+b+c}3}$ for positive numbers.

Prove taht the following inequality holds $$a^ab^bc^c\ge (abc)^{\frac{a+b+c}3}$$ if $a,b,c$ are positive. I'm not sure how to handle these kinds of powers. Are there any "famous" but not so ...
1
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3answers
135 views

Inequality Similar to Triangle Inequality

Can the inequality $$2y\ge|x-y|+|y-z|-|x-z|\quad\, \forall x,y,z\geq0$$ be simplified? It looks similar to the triangle inequality.