Questions on proving and manipulating inequalities.

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2answers
148 views

$\sqrt{(a+b-c)(b+c-a)(c+a-b)} \le \frac{3\sqrt{3}abc}{(a+b+c)\sqrt{a+b+c}}$

Suppose $a, b, c$ are the lengths of three triangular edges. Prove that: $$\sqrt{(a+b-c)(b+c-a)(c+a-b)} \le \frac{3\sqrt{3}abc}{(a+b+c)\sqrt{a+b+c}}$$
3
votes
3answers
178 views

Help with Inequality

Given that $x, y, z$ are nonnegative real numbers such that : $$x^2 + y^2 + z^2 + xyz = 4$$ Prove that $0 ≤ xy + yz + zx − xyz ≤ 2$
10
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3answers
134 views

seek a direct method to show $(a_k)\in l^2$

Let $\{a_i\}, i=1, 2, \cdots$ be a nonincreasing sequence of positive numbers, and suppose $\sum_{k=1}^{\infty}\frac{a_k}{\sqrt{k}}<\infty~$, show that $\sum_{k=1}^{\infty}a_k^2<\infty.$ Could ...
1
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2answers
61 views

show that if $a,b,c \in \mathbb{R}^+$

show that if $a,b,c \in \mathbb{R}^+$ different from zero, then: ...
3
votes
3answers
2k views

Inequality with Expectations

Given Random variables $X$ and $Y$ is it true always that; $$E(XY)^2 \le E(X^2)E(Y^2)$$ Is it easy to prove?
2
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1answer
163 views

'Simple" Chernoff bounds

Reading an academic paper I've observed the next claim: A simple Chernoff argument will now show that if an event has a constant probability at every step of occurring and there's independence ...
3
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2answers
76 views

$\sum_{i=1}^{n-1} \left|\dfrac{a_ia_{n-i}}{a_n}\right| \geq C_{2n}^n-1$

Given that the equation $$p(x)=a_0x^n+a_1x^{n-1}+\dots+a_{n-1}x+a_n=0$$ has $n$ distinct positive roots, prove that $$\sum_{i=1}^{n-1} \left|\dfrac{a_ia_{n-i}}{a_n}\right| \geq C_{2n}^n-1$$ I had ...
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1answer
94 views

inequality, preferably using elementary methods

I was trying to prove an inequality with 3 variables, and after simplification, it boiled down to trying to prove $$x^2 y+xz^2 +y^2 z \geq x+y+z$$, where $xyz=1$, and $x,y,z$ are all positive real ...
5
votes
3answers
295 views

Showing that complicated mixed polynomial is always positive

I want to show that $\left(132 q^3-175 q^4+73 q^5-\frac{39 q^6}{4}\right)+\left(-144 q^2+12 q^3+70 q^4-19 q^5\right) r+\left(80 q+200 q^2-243 q^3+100 q^4-\frac{31 q^5}{2}\right) r^2+\left(-208 q+116 ...
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2answers
153 views

Prove by induction that $n! > n^2$

How does one prove by induction that $n! > n^2$ for $n \geq 4$
3
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1answer
92 views

Specific inequality

Let $x,y,z$ be different real numbers . Prove that: $$\frac{x^2y^2+1}{(x-y)^2}+\frac{y^2z^2+1}{(y-z)^2}+\frac{z^2x^2+1}{(x-z)^2} \geq \frac{3}{2}$$
1
vote
1answer
42 views

Maximization of an integer input function

Maximize the value of the function $$ z=\frac{ab+c}{a+b+c}, $$ where $a,b,c$ are natural numbers and are all lesser than 2010 and not necessarily distinct from each other. Please provide a proof, ...
20
votes
4answers
3k views

Purely “algebraic” proof of Young's Inequality

Young's inequality states that if $a, b \geq 0$, $p, q > 0$, and $\frac{1}{p} + \frac{1}{q} = 1$, then $$ab\leq \frac{a^p}{p} + \frac{b^q}{q}$$ (with equality only when $a^p = b^q$). Back when I ...
5
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1answer
242 views

Markov inequality example

If I have $x_1, x_2,\ldots, x_n$ independent NON-identically distributed Bernoulli random variables, how do I show that: $$\mathrm{Pr}\left(\sum_{i=1}^nx_i>\beta\mu\right)\le e^{-g(\beta)\mu}$$ ...
2
votes
1answer
176 views

Proof of a lower bound of the norm of an arbitrary monic polynomial

In my course I have come across the following problem: The Chebyshev polynomial of degree $n$, $T_n(x)$, is defined on $[-1,1]$ by $T_n(x)=\cos n\theta$. Let $q_{n+1}(x)$ be any monic ...
4
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1answer
178 views

Please prove that ${n\choose r} < (n+1)^r$. My induction-based proof is ugly!

I came across this inequality in the University of Missouri Youtube Channel lecture 38: College Algebra - Lecture 38 - The Binominal Theorem The professor asks us 1:01:00 into the video to prove ...
7
votes
2answers
491 views

An inequality from the handbook of mathematical functions (by Abramowitz and Stegun)

Prove that $$\frac{1}{x+\sqrt{x^2+2}}<e^{x^2}\int\limits_x^{\infty}e^{-t^2} \, \text dt \le\frac{1}{x+\sqrt{x^2+\displaystyle\tfrac{4}{\pi}}}, \space (x\ge 0)$$
3
votes
1answer
236 views

Young's inequality without using convexity

I was doing some problems from Rudin's Principles of Mathematical Analysis and came across a problem in which he asks you to prove Hölder's inequality via Young's inequality: If $u$ and $v$ are ...
10
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5answers
506 views

Prove $|a+b|+|a-b| \geq |a|+|b|$

I am fighting with this proof-writing problem for a while. The statement says $$|a+b|+|a-b| \geq |a|+|b|.$$ I know the triangle inequality which says$$|a+b| \leq |a|+|b|.$$ How can I use this ...
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3answers
344 views

Equality of absolute values of complex integrals

It was pretty hard finding a short and precise title. Here is my problem: The equation $$\bigg|\int_\gamma f(z)\text{d}z\bigg|\le\int_\gamma\big|f(z)||\text{d}z|$$ holds if $f$ is integrable (where ...
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2answers
107 views

Confused where and why inequality sign changes when proving probability inequality

"Let A and B be two events in a sample space such that 0 < P(A) < 1. Let A' denote the complement of A. Show that is P(B|A) > P(B), then P(B|A') < P(B)." This was my proof: $$ P(B| A) > ...
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0answers
43 views

Show inequality for modified bilinear form

Let $\Omega_h$ denote to the domain that is bounded by a polygon, and $V_h$ to the space of all $c\in C^0(\Omega)$ such that $v_{|T}$ is linear on any (curved) triangle T and $v=0$ in the vertices of ...
8
votes
2answers
341 views

A difficult symmetric inequality

In my studies of various geometric inequalities I reached an inequality which seems true (numerically) but I cannot prove it. Let $p$, $q$, and $r$ be real numbers from the interval $(0,1)$. Let's ...
0
votes
1answer
491 views

Line does not intersect with curve

the line $y=kx+1$ does not intersect with the graph of $y = x^2-3x+5$ at any points Find the range of possible values for $k$? Can anyone help?
2
votes
1answer
110 views

Prove that for any triple $a_{i},a_{j},a_{k} $ are three edge lengths of some triangle

$ a_{1},a_{2},....,a_{n}(n\geq 3) $ are positive numbers that : $(a_{1}+a_{2}+....+a_{n})^2 >\frac{3n-1}{3}(a_{1}^2+a_{2}^2+....+a_{n}^2)$ Prove that for any triple $a_{i},a_{j},a_{k} $ are three ...
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1answer
291 views

Prove $\sup \left| f'\left( x\right) \right| ^{2}\leqslant 4\sup \left| f\left( x\right) \right| \sup \left| f''\left( x\right) \right| $ [closed]

Let $f\left( x\right)$ be a $C^{2}$ function on $\mathbb{R}$. Show that $$\sup \left| f'\left( x\right) \right| ^{2}\leqslant4\sup \left| f\left( x\right) \right| \sup \left| f''\left( x\right) ...
0
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1answer
95 views

Why this inequality yields at most exponential growth?

Let $\Omega=\mathbf{R}^{n-j}\times\omega$, where $\omega\subset\mathbf{R}^j$ is a smooth bounded domain. Consider a function $u:\overline\Omega\rightarrow\mathbf{R}$ that satisfies $$u(x,y)+k\leq ...
2
votes
2answers
860 views

Show that that $|\sqrt{x}-\sqrt{y}| \le \sqrt{|x-y|}$

In a solution for a test, I came upon the following: we now use $|\sqrt{x}-\sqrt{y}| \le \sqrt{|x-y|}$ (prove). I've been unable to solve this - I've looked at the proof of the triangle inequality, ...
1
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1answer
215 views

Poincaré inequality using $H^1$ seminorm

Does this inequality holds for Poincaré Inequality? $$||v||_{L^2} \leqslant C_p |v|_{H^1}$$ and $$ |v|_{H^1} = ||v'||_{L^2} $$ where $| \dot~ |$ is the semi norm and $||\dot~||$ is the norm. I'm ...
2
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1answer
164 views

Inequality for Gamma functions

Let $k, n ,m \in N$ and such that $0\leq k \leq n \leq m$. When the following ineuality is true? $$ ...
2
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3answers
68 views

Algebraic solution of $x + 3^x < 4$

I solved graphically and found that $x + 3^x < 4$ is true for $x < 1$ but I can't find a way to prove it algebraiclly, any hints will be greatly appreciated!
10
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4answers
375 views

Why is this inequality important?

I'm reading Courant's What is Mathematics? In the beginning, he's showing some proofs, there's a proof about An Important Equality and this important equality is: $$(1+p)^n\geq 1+np$$ The book's ...
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0answers
39 views

Extracting a function from set of inequalities

I have set of inequalities in two dimension space which represent relation between $X$ and $Y$. now I want a function whose input is $X$ and output is $Y$. In other words, I want $F$ such that ...
1
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1answer
411 views

Reverse triangle inequality with exponent $p$

I am trying to prove that for any $p > 1$ and for any real numbers $a,b > 0$, the following inequality holds: $$ | a^{\frac{1}{p}} - b^{\frac{1}{p}} |^p \leq 2^p|a-b| $$ In the case where $p$ ...
3
votes
1answer
206 views

Question on proof in Evans PDE

This is on page 542 of Evans PDE book. The last inequality states that $$\int_{U}{C(|Du|+1)|u|dx} \leq \frac{1}{2}\int_{U}|Du|^2dx + C\int_{U}{|u|^2+1 \ dx}$$ Where is this coming from? I think ...
0
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3answers
88 views

Proof that two inequalities are contradictory

Assuming $0 < q < 1$ and $0 < r < q$, I want to show that: $$q^2(2-r) + q(3r^2 + 8r-4) - 8r^2 > 0 \;\text{(I)}\;\;\text{and}\;\; 3q -r-2 < 0\;\text{(II)}$$ cannot hold at the ...
2
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3answers
221 views

Inequality. $(a^2+bc)(b^2+ca)(c^2+ab) \geq abc(a+b)(b+c)(c+a)$

Let $a,b,c$ be three real positive(strictly) numbers. Prove that: $$(a^2+bc)(b^2+ca)(c^2+ab) \geq abc(a+b)(b+c)(c+a).$$ I tried : $$abc(a+\frac{bc}{a})(b+\frac{ca}{b})(c+\frac{ab}{c})\geq ...
2
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3answers
99 views

How do I solve inequalities of the form $\frac{ax}{b}\geq0$?

I need to find the maximum domain for $f(x) = \sqrt{\frac{4x+13}{(x+5)(2-x)}}$ Therefore, I should solve the inequality $$\frac{4x+13}{(x+5)(2-x)} \ge 0$$ I don't remember how to solve inequalities ...
3
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2answers
345 views

Generalization of Bernoulli's Inequality.

When you're first taking an Introduction to Proofs class, you eventually learn about induction. One of the first induction proofs I did was to prove for $n \geq 1$ that $(1+x)^n \geq 1+nx$. You first ...
3
votes
8answers
744 views

Prove that $e^x\ge x+1$ for all real $x$

Without using differentiation, logarithmic function, rigorously, prove that $$e^x\ge x+1$$ for all real values of $x$.
1
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1answer
29 views

solving absolute equation

Which number can be $x$? $$\vert 1-\vert x-1\vert\vert\lt1$$ I got: $$1-x+1=0 \Longrightarrow \boldsymbol{x_1} = 0$$ $$-1+x-1=0\Longrightarrow \boldsymbol{x_2} = 2$$ What is the method of ...
0
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1answer
66 views

How can I show this inequality?

Let $\lambda B=\{x\in\mathbb{R}^n:\ \|x\|<\lambda\}$. Let $\eta>0$, $r_n\in (0,\eta)$ and $r_n\rightarrow \eta$. Suppose $u$ is a measurable function defined in $\eta B$. How can i show that ...
6
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3answers
483 views

Show $2(x+y+z)-xyz\leq 10$ if $x^2+y^2+z^2=9$

If $x,y,z$ are real and $x^2+y^2+z^2=9$, how can we prove that $2(x+y+z)-xyz\leq 10$? Please provide a solution without the use of calculus. I know the solution in that way.
4
votes
1answer
77 views

inequality with numbers--when its true?

Help me please to understand when the inequality true. Let $n<N,$ where $n, N$ are natural numbers. For which $n$ and $N$ the following is true $$ n^{2n+1}\leq N^{N+1}? $$ Thank you.
0
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1answer
60 views

reference needed for Gamma function

Please help me to find a reference (book) for the following upper bound of Gamma function For $x \geq 1$ $$ \Gamma(x)\leq x^{x-1}. $$ Thank you.
2
votes
2answers
119 views

Proving this trigonometric inequality?

Let $|\theta-\theta_0|\leqslant \frac{\pi}4$. How can I prove that $$2(1-\cos(\theta-\theta_0))\geqslant \frac{|\theta-\theta_0|^2}{2}?$$
2
votes
2answers
81 views

Explanation of this inequality

Is there a graphic visualization of $\sum_{k=1}^{n} 1/k \, \, \leq \, \, \,1 \, + \, \int_1^n \! (1/x) \, \mathrm{d} x$ as intuitive as the integral test ? I can't see why the inequality is true. I ...
2
votes
3answers
189 views

Proving a matrix equality

I have 2 matrices: $A \in R^{nxn}$ is a non-singular matrix and $B \in R^{nxn}$ is a singular matrix. Here is the expression I need to prove: $$||A - B|| \ge ||A^{-1}||^{-1}$$ I dont understand why ...
1
vote
0answers
87 views

A Multiplicative version of McDiarmid’s Inequality like the one of Chernoff-Hoeffding Bounds

McDiarmid's Inequality basically says the following: Let $X_1, X_2, X_3, \ldots, X_n$ denote independent random variables and $f$ is a function of $n$ real arguments. If changing the value of the ...
-1
votes
1answer
57 views

Exercise of functions of a real variable

Let $f, g\colon\mathbb{R}\rightarrow\mathbb{R}$ functions so that for all $\,x, y\in\mathbb{R}$, $$f(x+y)+f(x-y)=2f(x)g(y).$$ Prove that if $f$ is not constant zero function and $|f(x)|\leqslant ...