Questions on proving, manipulating and applying inequalities.

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Prove $\frac{3}{64}(ab+bc+ca)^3\geq (de)^3+(ef)^3+(fd)^3$ where $a, b, c$ are three sides of and $d, e, f$ three angle bisectors of a triangle.

A triangle has sides $a, b,c$ and angle bisectors $d, e, f$ where each pair of $a$ and $d$, $b$ and $e$, $c$ and $f$ intersect. Prove that $\frac{3}{64}(ab+bc+ca)^3\geq(de)^3+(ef)^3+(fd)^3$. I was ...
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1answer
41 views

Not understanding a cancellation step in an inequality proof from Spivak's Calculus.

have just been reading through and doing the questions in Spivak's Calculus, but am not entirely sure I am understanding a step in one of the proofs given in chapter $5$ (but was given as an exercise ...
1
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1answer
11 views

Inequality in proof of Taylor polynomial for functions of 2 variables

I have this inequality that bothers me: \begin{equation} |g(h,x)|\le\bigl|o(\sqrt{h^2+k^2})\bigr|\cdot(|h|+|k|) \end{equation} where $g(h,k)=f(x+h, y+k)-f(x,y)-df(x,y)-\frac{1}{2}d^2f(x,y)$. So, ...
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3answers
139 views

How to minimize $ab + bc + ca$ given $a^2 + b^2 + c^2 = 1$?

The question is to prove that $ab + bc + ca$ lies in between $-1$ and $1$, given that $a^2 + b^2 + c^2 = 1$. I could prove the maxima by the following approach. I changed the coordinates to spherical ...
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3answers
97 views

Inequalities with quadratics [closed]

$$\frac{12}{x^2 + 2x} < \frac{3}{x^2 + 4x + 4}$$ I am confused. Can someone help me? Update : you can see my work in the comments. i figured out the answer but the answers other people gave ...
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1answer
30 views

Real values of $x$ in $(1+2^x)\cdot (1+8^x)\cdot (1+9^x)^2 = (1+6^x)^4$

$(1)$ Real values of $x$ in $2^x+3^{-x}+4^{-x}+6^x = 4$ $(2)$ Real values of $x$ in $(1+2^x)\cdot (1+8^x)\cdot (1+9^x)^2 = (1+6^x)^4$ My Try: For First one:: Here ...
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1answer
28 views

If $2^{x + 1} < y$, then what is the “smallest” function of $x$ that cannot be an upper bound for $y$?

(This is a follow-up question to this MSE post.) The title says it all. Let $x$ be a positive integer. If $2^{x + 1} < y$, then does there exist a minimum function $f(x)$ that cannot be an ...
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2answers
95 views

How to prove $\sqrt{\frac{bc}{a+bc}}+\sqrt{\frac{ca}{b+ca}}+\sqrt{\frac{ab}{c+ab}}\geq 1$

Let $a,b,c>0,a+b+c=1$, show that $$\sqrt{\dfrac{bc}{a+bc}}+\sqrt{\dfrac{ca}{b+ca}}+\sqrt{\dfrac{ab}{c+ab}} \geq 1$$ I tried $$\dfrac{bc}{a+bc}=\dfrac{bc}{a(a+b+c)+bc}=\dfrac{bc}{(a+b)(a+c)}$$ It ...
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1answer
36 views

If $2^{x + 1} < y$, then what is the largest polynomial in $x$ that cannot be an upper bound for $y$?

Update: I have posted a follow-up question here. The title says it all. If $2^{x + 1} < y$, then what is the largest polynomial in $x$ (of maximum possible degree) that cannot be an upper ...
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1answer
29 views

which of the following is NOT a possible value of $(e^{f})''(0)$??

Let $f$ be an analytic function on $\bar{𝐷} = \{z \in \mathbb{C}: |z| \le 1\}$. Assume that $|𝑓(𝑧)| ≤ 1$ for each $z\in \bar{D}$. Then, which of the following is NOT a possible value of ...
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2answers
56 views

Prove $|x|^2$ = $x^2$

My first attempt at this proof divided into 2 cases, one where $x^2$ is greater than or equal to 0, and another where $x^2$ is less than 0. For the first case, I said that the definition of absolute ...
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1answer
27 views

Under what conditions is the implication $I(x) < I(y) \implies x < y$ true?

Let $\sigma(x)$ be the sum of the divisors of $x$, and denote the abundancy index of $x$ by $$I(x) = \frac{\sigma(x)}{x}.$$ My question is: Under what conditions on $x$ and $y$ is the implication ...
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3answers
68 views

How to go upon proving $\frac{x+y}2 \ge \sqrt{xy}$? [duplicate]

I'm trying to prove this but am having some difficulty. For any $x,y\in\mathbb R$ such that $x\ge 0$ and $y\ge 0$ we have $$\frac{x+y}2 \ge \sqrt{xy}.$$ So far what I have gotten to is ...
5
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1answer
65 views

What is so special about the Schwarz Inequality?

I am studying Spivak's Calculus and the first two problem sets have rather lengthy,but very interesting, work-throughs of three proofs for the Schwarz Inequality: $$\sum_{i=1}^{n} x_iy_i ...
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1answer
32 views

Solution set to $n^{2p-1}\leq 2^{n(1-p)}$

For what positive, real values of $p$ and positive, integer values of $n$ does the inequality $$ n^{2p-1}\leq 2^{n(1-p)} $$ hold? I tried using logarithms but got nowhere. It comes from me trying to ...
3
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1answer
36 views

Application of Doob's inequality

Suppose that $X_n$ is a martingale with $X_0 = 0$ and $EX^2_n < \infty$. Show that $$P\left(\max_{1\leq m \leq n} {X_m} \geq \lambda\right) \leq \frac{EX^2_n}{EX^2_n+\lambda^2}$$ by using ...
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2answers
66 views

Prove that the exponential $\exp z$ is not zero for any $z \in \Bbb C$

How can the following been proved? $$ \exp(z)\neq0, z\in\mathbb{C} $$ I tried it a few times, but i failed. Probably it is extremly simple. If a draw the unit circle and then a complex number ...
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4answers
229 views

Which is bigger: $(n!)^{n!}$ or $(n^{n})!$? [closed]

To be honest I haven't spent a whole lot of time thinking about this other than the drive back home, and I won't have much time to think about it for a while due to shit-happening. So i thought I'd ...
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1answer
32 views

Inequality in triangle.

If $a,b,c$ are sides of a triangle prove that- $$\frac a{c+a-b}+\frac b{a+b-c}+\frac c{b+c-a}\geq3$$ I am having problem in approaching the problem as the sides are not mentioned as ...
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5answers
65 views

$\left|x\right| < \left|\tan(x)\right|$ close to $0$

I was trying to prove this inequality $\left|x\right| < \left|\tan(x)\right|$ in a neighborhood of $0$. I tried splitting into the four cases opening the modulus but still wasnt able to solve it. I ...
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3answers
45 views

Finding an Upper Bound on This Inequality

I came across this problem that seems a bit peculiar. Take $$\sum_{1 \leq i < j \leq n} |x_i-x_j|,$$ where $x_1,...,x_n \in [1,35].$ I want to figure out the maximum possible value of this sum in ...
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2answers
51 views

Prove $\log(x)^{10} < x$ (for $x > 10^{10}$)

I need to prove that $\log(x)^{10} < x$ for $\ x>10^{10}$ It's pretty clearly true to me, but I need a good proof of it. I tried induction, and got stuck there.
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1answer
35 views

How do we get the inequality?

Proposition: If $A \in \mathbb{R}^{n \times n}$ a symmetric matrix then $||A||= \sup \{ ||Ax||_2: ||x||_2=1\}= \sup \{ |\langle x, Ax \rangle|: ||x||_2=1\}$. Proof: It suffices to show that $||A|| ...
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2answers
41 views

Simple Inequality of Complex Numbers, $\left| \frac{a-b}{1-\overline{a}b} \right| <1$

Exercise from Ahlfor's Complex: Given $a,b \in \mathbb C$, with $|a| <1$, $|b|<1$, prove: $$\left|\frac{a-b}{1-\overline{a}b}\right| <1.$$ My argument: Lemma: If $\alpha, \beta \in ...
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56 views

Proving this inequality: $\sum_i (t x_i - (1-t) y_i)^2 \geq (t(1-t))^2 \sum_i (x_i - y_i)^2$

I have the following inequality, which has been verified extensively by numerical simulation: We are given two positive sequences, both summing up to 1, i.e., $\forall i\in \{1, ..., n\}: \sum_{i} ...
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2answers
122 views

Prove $(a^2+b^2+c^2)(a+b-c)(b+c-a)(a+c-b)\leq abc(ab+bc+ac)$

Let $a,b,c$ are $3$ edge of a triangle. Prove $(a^2+b^2+c^2)(a+b-c)(b+c-a)(a+c-b)\leq abc(ab+bc+ac)$. My try: I suppose $c=\min\{a,b,c\}$ but I don't know what next.
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37 views

prove that $-1 \le \frac{ac+bd}{\sqrt{a^2+b^2}\sqrt{c^2+d^2}} \le 1$

For the real numbers $a, b, c, d$ prove that $$-1 \le \dfrac{ac+bd}{\sqrt{a^2+b^2}\sqrt{c^2+d^2}} \le 1$$ Actually if we let $\vec{u} = (a, b)$ and $\vec{v} = (c, d)$ then by dot product we got ...
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0answers
12 views

Sufficient condition for upper semicontinuous functions

My question might be fundamental but I'm glad if you give some help since I don't find any idea. Let $X$ be a bounded and closed subset of $\mathbb{R}$. A function $f:X\to\mathbb{R}$ is called to be ...
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1answer
38 views

Proving inequality [closed]

I have been assigned a problem to solve in my probability class and I'm having a hard time starting it. $\frac{1}{1+x^2}\frac{1}{x}e^{\frac{-x^2}{2}}\leq \int^{\infty}_x e^{\frac{-t^2}{2}}dt\leq ...
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2answers
47 views

Does taking limits change “$<$” into “$\le$”

I saw this in my notes and didn't think much of it at the time of writing, but now I wanted to make sure this was/wasn't some property of limits no one ever told me about. Let's say: $L$ is a limit ...
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0answers
12 views

Upper estimate between an original function and its sup-convolution under a limitation

My setting maybe look rather special but I'm glad if you give some answers. Let $f:[0,1]\to\mathbb{R}$ be a bounded, upper semicontinuous function and $f^{\varepsilon}:[0,1]\to\mathbb{R}$ be $f$'s ...
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2answers
38 views

Find infimum of this set

I need to find infimum of this set $$ \left\{ \frac a {b+c} + \frac b {a+c} + \frac c {a+b} : a,b,c \in \mathbb R^+\right\}$$ Some hint would be helpful
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3answers
30 views

Solution for $\frac{8(9^n)}{8(9^n)+2(4^n)} \gt 0.99$

I'm currently trying to come up with a solution for $\frac{8(9^n)}{8(9^n)+2(4^n)} \gt 0.99$ I don't understand how i could evaluate the n within to find the exact value of n at all, and would ...
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0answers
15 views

Upper bound of $\left|\sum_{n,k} c_{n,k}\, \alpha_k\right|^2$.

We have a double finite series, such that: $$\sum_{n,k} |c_{n,k}|^2=1$$ What can we conclude about upper bound of $\left|\sum_{n,k} c_{n,k}\, \alpha_k\right|^2$ for $|\alpha_k|<M$? I have tried to ...
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1answer
31 views

generalised log sum inequality

The log sum inequality states that $ \sum_i a_i\ln\frac{a_i}{b_i}\geq a\ln{\frac{a}{b}}$ where $a=\sum_i a_i$ and $b=\sum_i b$. Is there a generalisation (with whatever conditions) that extends it ...
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2answers
73 views

Prove $a+b \leq 1$ with two disjoint squares in a larger square.

Two disjoint squares are located inside a square of side $1$. If the lengths of the sides of the two squares are $a$ and $b$, prove $a+b \leq 1$. I thought about setting up a coordinate axes with ...
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2answers
21 views

Random Point within a Circle of radius $n$

Consider $(X_1,X_2)$ be a random point chosen inside a circle of radius $n$, with center at the origin and thus $X_1,X_2$ have the joint density function $$p(x_1,x_2) = \left\{\begin{array}{cc} ...
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5answers
59 views

Proof for $e^{\frac{1}{n+1}}-1-\frac{1}{n}\leq0$

I'm looking for a proof of $e^{\frac{1}{n+1}}-1-\frac{1}{n}\leq0$, optionally $\ln(n)+\frac{1}{n+1}\leq \ln(n+1)$
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2answers
108 views

Prove: $a^2\left ( \frac{b}{c}-1 \right )+b^2\left ( \frac{c}{a}-1 \right )+c^2\left ( \frac{a}{b}-1 \right ) \geq 0$.

Let $a,b,c$ are $3$ edge of a triangle. Prove: $a^2\left ( \frac{b}{c}-1 \right )+b^2\left ( \frac{c}{a}-1 \right )+c^2\left ( \frac{a}{b}-1 \right ) \geq 0$. Can it not do a brute force?
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1answer
33 views

Use Jensen's Inequality to prove $\sin{A}+\sin{B}+\sin{C} \leq \dfrac{3\sqrt{3}}{2}$ [duplicate]

Prove that for any triangle $ABC$, have $\sin{A}+\sin{B}+\sin{C} \leq \dfrac{3\sqrt{3}}{2}$. I know that we have to use Jensen's inequality here, but I am not sure how to apply it. We know that ...
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2answers
38 views

Bound on sum of combinations

I came across the following inequality $\sum_{i=0}^D \binom N i \le N^D+1$. I am not sure how to prove this. I tried to do it by induction on $D$, and started with observing the values of sum for ...
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2answers
54 views

Prove $\frac{1\cdot 3\cdots(2n-1)}{2\cdot 4 \cdots(2n)}<\frac{1}{\sqrt{2n+1}}$, $n\ge 1$

Prove $\frac{1\cdot 3\cdots(2n-1)}{2\cdot 4 \cdots(2n)}<\frac{1}{\sqrt{2n+1}}$, $n\ge 1$. I begin by letting $n=1$ then $\frac{1}{2}<\frac{1}{\sqrt{3}}$. Then assume $\frac{1\cdot ...
3
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1answer
29 views

Given $0<x<1$ and $0 \le z \le 1-x$, why is $\sqrt{z+x}-(1-\sqrt{x})\sqrt{\frac{z}{1-x}} \le \sqrt{x}$?

This seems true, but I have so far failed to prove it. Any clever estimates I could use?
3
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2answers
61 views

Prove that $\frac{a}{b+2c+3d}+\frac{b}{c+2d+3a}+\frac{c}{d+2a+3b}+\frac{d}{a+2b+3c} \geq \frac{2}{3}.$

Let $a,b,c,d$ be positive real numbers. Prove that $$\dfrac{a}{b+2c+3d}+\dfrac{b}{c+2d+3a}+\dfrac{c}{d+2a+3b}+\dfrac{d}{a+2b+3c} \geq \dfrac{2}{3}.$$ I was thinking of trying $a \geq b \geq c$ ...
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4answers
46 views

Help in proving an inequality with logarithms

I want to show that: $(n+1)(\log(n+1)-\log(n)) > 1$, if $n>0$. I have tried exponentiating it and I got $( (n+1)/n )^{n+1} > e.$ Then I tried to show that the limit of $( (n+1)/n ...
5
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1answer
85 views

Prove that one of the elements can't be in the interval $(0,1)$

Let be $a,b,c\in\mathbb{R}$ so that the sum of two of them is never equal to $1$. Prove that atleast on of $\frac{ab}{a+b-1},\frac{bc}{b+c-1},\frac{ca}{c+a-1}$ can't be in the interval $(0,1)$. I ...
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1answer
50 views

How to prove that the inequality holds for any nonzero x?

The inequality is given by $$x^{H}(\Phi(x))^{-1}x-x^{H}\dfrac{aa^H}{a^H\Phi(x)a}x\ge0,\text{ for any }x\ne0,$$ where $\Phi(x)$ is positive definite and is a function of $x$, $a$ can be any nonzero ...
2
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3answers
224 views

Prove with use of derivative [closed]

How to prove this inequality using derivative ? For each $x>4$ , $$\displaystyle \sqrt[3]{x} - \sqrt[3]{4} < \sqrt[3]{x-4} $$
4
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3answers
68 views

Proving $\frac{1}{\sqrt{1-x}} \le e^x$ on $[0,1/2]$.

Is there a simple way to prove $$\frac{1}{\sqrt{1-x}} \le e^x$$ on $x \in [0,1/2]$? Some of my observations from plots, etc.: Equality is attained at $x=0$ and near $x=0.8$. The derivative is ...
2
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2answers
43 views

Some Algebra Problem}

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 ...