# Tagged Questions

Questions on proving, manipulating and applying inequalities.

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### Combinatorical problem

$k$ is a natural constant.Determine $x,y,z$ knowing that $\binom{z+k}{x+y} + \binom{z}{x} \le k$ and $2x+y \le z$.
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### Israel tst 2011 geometrical inequality

Inside an equilateral triangle of area $S$ lies a point, whose distances to the vertices are $x,y, z$. Prove that $xy + yz + zx \geq \frac{4}{\sqrt{3}} S$ I haven't got any idea yet. But I guess ...
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### Inequality with $log$ and $d^x$

Please help me to solve this inequality $Log[d] <\frac{(-1 + d^a) (d^a - d^b) b} {d^{a}(-1 + d^b) a (a - b)}$ with $0<\delta<1$, $a\geqq1$, $b>a$ and thus $b>1$. $a$ and $b$ are ...
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### How to solve $(2p_1^2-2p_1+1)^n \le 2^{-10}$ where $p_1 = 1-(1-(1/n))^N$.

Let $$S_{n,N}=(2p_1^2-2p_1+1)^n$$ and $p_1 = 1-(1-(1/n))^N$. I would like to solve $S_{n,N} \leq 2^{-10}$ for $n$. This seems hard to do exactly but is there a good approximation one can find? We ...
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### How tight is this trace inequality?

I would like to know how tight the following trace inequalities are for real symmetric $A$ and real symmetric $B \succeq 0$ $$\mbox{trace} (AB) \leq \lambda_{\max} (A) \cdot \mbox{trace} (B)$$ or ...
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### Prove the inequality between integral and summation of multiplicative inverse

I want to prove the following inequality: $$\ln(n) = \int\limits_1^n{ \frac{1}{x} dx } \geq \sum_{x = 1}^{n}{\frac{1}{x + 1}} = \sum_{x = 1}^{n}{\frac{1}{x}} - 1$$ I ask this question as I'm ...
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### Meaning of $Ax \leq b$

I continue to come across $Ax \leq b$ or $Ax= b$ in optimization problem, but I am having trouble interpreting the meaning of this. Does this have a similar meaning to the following (Cramer's Rule) ...
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### Find values of $m$ and $n$ such that $m \leqslant 6 \sin x+ \cos (2x) -1\leqslant n$ [on hold]

$$m \leqslant 6 \sin x+ \cos (2x) -1 \leqslant n$$ I have no clue how to do it. please help.
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### Solution of Inequality $\displaystyle \frac{1}{x-6}\le 3$

Solve the inequality: $\displaystyle \frac{1}{x-6}\le 3$ solution: \begin{align*}\frac{1}{x-6}& \le 3 \\ x-6& \le \frac{1}{3} \\x& \le 6+\frac{1}{3}\\ x&\le19/3\end{align*} but, ...
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### Implication of exponential growth: how is it deduced?

Let $L$ be a differentiable function defined on $\mathbb{R}\times\Omega$ with $\Omega\subseteq\mathbb{R}^n$. I will say it has exponential growth if for all $O\subset\subset\Omega$ open there exists a ...
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### Mathematical Induction Inequality problem [on hold]

I am trying to solve the following problem with mathematical induction: $$\forall n>1,\qquad \frac{1}{2^2}+\frac{1}{3^2}+\ldots+\frac{1}{n^2}<\frac{n-1}{n}$$ but since I am new to the concept ...
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### Does Gaussian convolution respects order?

Assume that we have two continuous integrable functions $f,g \in L^1(\mathbb{R})$ such that, for some $x_0 \in \mathbb{R}$, we have, $$f(x_0) \leq g(x_0) \; \; \; \; (1).$$ Now let us define the ...
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### how to proceed next in this logarithmic inequality?

The question is $$\frac{1}{\log_4{\left(\frac{x+1}{x+2}\right)}}<\frac{1}{\log_4{(x+3)}}$$ I did the first step for defining the arguments of both sides and got $x\in(-3,-2)\cup (-1,\infty)$ ...
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### Find the maximum of the $k$ such $0\le x^2(3-2x)(2x^k+(3-2x)^k)\le 3$

Find $k_{\max}$,such $$0\le x^2(3-2x)(2x^k+(3-2x)^k)\le 3,0\le x\le 1$$ since $$x^2(3-2x)>0\Longrightarrow 2x^k+(3-2x)^k\ge 0$$ it is clear for $k\in R$ and other case it's not easy to solve
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### On real part of the complex number $(1+i)z^2$

Find the set of points belonging to the coordinate plane $xy$, for which the real part of the complex number $(1+i)z^2$ is positive. My solution:- Lets start with letting $z=r\cdot e^{i\theta}$. ...
Define $a_n=\left ( 1+\frac{1}{n} \right )^n$ for $n\geq 1$. I want to show that it is increasing. First, we have $$\frac{a_{n+1}}{a_n}=\left ( \frac{1+\frac{1}{n+1}}{1+\frac{1}{n}} \right )^n\left ( ... 1answer 35 views ### Express c and d in terms of m where c and d are zeroes of f where m > -2 Let$$f(x) = x^2 - mx -(6m^2+25m+25)$$where m > - 2 It can be shown that f(x) has two zeroes. Suppose we have c,d \in \mathbb R s.t. c < d and f(c) = f(d) = 0, express c and d ... 1answer 69 views ### Circles in complex plane. Find the real value of a for which there is at least one complex number satisfying |z+4i|=\sqrt{a^2-12a+28} and |z-4\sqrt{3}|\lt a. My solutions:- Graphical solution:- |z+4i|=\sqrt{a^2-... 0answers 14 views ### Proof using positive (semi)definite matrices and a sharp matrix inequality Take symmetric and real matrices F, f and f' such that F \geq f and F>f'. Here F \geq f means that F-f is positive semi-definite, and F>f' means that F-f' is positive definite. I ... 1answer 15 views ### Determining Bounds to calculate mass Let E be the solid region defined by the inequalities x \ge 0, 0\le z \le \sqrt(x^2 + y^2), x^2 + y^2 + z^2 \le 4 Suppose that E has mass density \mu(x,y,z) = xz. Calculate the ... 2answers 39 views ### How to prove \prod_{i=1}^{n}(x-4i+2)(x-4i+1)>\prod_{i=1}^{n}(x-4i+3)(x-4i) for all x\in\mathbb{R}? I would like to prove that for n\in\mathbb{N} we have f_n(x):=\prod_{r=1}^{n}(x-4r+2)(x-4r+1)>\prod_{r=1}^{n}(x-4r+3)(x-4r)=:g_n(x) for all x\in\mathbb{R} (actually it would suffice for n ... 1answer 21 views ### Upper and lower bound of the ratio of summation Consider x_1,x_2,x_3,....,x_n\in \mathbb{N}^+ What is the upperbound and lowerbound of the following expression R=\frac{\sum_{i=1}^{n-1}(x_i + x_{i+1})}{\sum_{i=1}^{n}x_i} Here is my trail. ... 1answer 21 views ### Having trouble proving Inequality [duplicate] I am having trouble proving this inequality: 2ab\leq a^2+b^2 I can transpose the equation and change around signs. But I am not sure If I need to use k+1 here or just prove the inequality. In ... 1answer 70 views ### Upper bound of x_1x_2x_3+x_2x_3x_4+\dots+x_{n}x_1x_2 Let n\geq 3 be a positive integer and let x_i's be non-negative real numbers with x_1+x_2+\dots+x_n=1. What is the maximum of x_1x_2x_3+x_2x_3x_4+\dots+x_{n}x_1x_2? If the sum were symmetric ... 2answers 50 views ### Which quantity is greater? [on hold] A. -0.1 or B. -0.10101010101 This is actually an evaluation of an expression when plugging certain values in the GRE. I plugged in the value -0.1 and arrived at my doubt 1answer 42 views ### a inequality similar to geometric means Let a, b be two positive constants. We sure have$$ a^2+b^2\geq 2ab $$My question: would it be possible to have an inequality like$$ a^2+b^2\geq Ca^{2+\epsilon}b^{1-\eta} $$where C, \epsilon... 1answer 30 views ### Show that for all z \in \overline{D}(0;1), (3-e)|z| \leq |e^z - 1|\leq |z|(e-1) Show that for all z \in \overline{D}(0;1), (3-e)|z| \leq |e^z - 1|\leq |z|(e-1) I think I'm supposed to use the following chain of inequalities$$|e^z -1|\leq e^{|z|}-1 \leq |z|e^{|z|}$$But ... 2answers 71 views ### Dominance between two functions Let two functions f(z) and g(z) with z\in[0,c] with c a constant such that c<b. I'd like to check whether f(z)-g(z)>0. I've tried to set f(z) to its minimal value and g(z) to its ... 3answers 139 views ### Prove the Inequality \frac{1}{1-x}-\frac{x(3-x)(2-x)(13x^4-50x^3+89x^2-84x+36)}{4(1-x)(2x(1-x))^2}<1 Can anyone suggest any hints to prove the following inequality:$$\frac{1}{1-x} - \frac{x(3-x)(2-x)(13x^4 - 50x^3 + 89x^2 - 84x + 36)}{4(1-x)(2x(1-x))^2} < 1,$$for all x \in (0,1)? 3answers 41 views ### Show d_2(f,g) = \sqrt{\int\limits_0^1 |f(x) - g(x)|^2 dx} is a metric on C[0,1] I am surprised that this question hasn't been asked on here I need to show that$$d_2(f,g) = \sqrt{\int\limits_0^1 |f(x) - g(x)|^2 dx}$$is a metric on C[0,1] Proof: As usual, positive ... 0answers 32 views ### Solve Equation with max integer [closed] Solve please \dfrac{\left[\sqrt{x-[x ]}\right]}{(x+3)(x+4)}\ \geq0 edit 1answer 32 views ### Solving inequality of two independent exponentially distributed RVs I have huge problems solving following excersice: There are two molecules. The decay of the molecules is exponentially distributed with \alpha_1 = 1 (for molecule 1) and \alpha_2 = 2 (for ... 1answer 38 views ### Using CS inequality to find maximum of a function I am trying to us Cauchy-Schwarz inequality to find the maximum of:$$|(a^2)(b^2)(a-b)+(b^2)(c^2)(b-c)+(c^2)(a^2)(c-a)|$$Where a, b, and c are real numbers, and a+b+c=0 and a^2+b^2+c^2=2. ... 1answer 69 views ### Find maximum value of a function [closed] a, b, and c are real numbers, and a+b+c=0 and a^2+b^2+c^2=2. I need help finding the maximum value of:$$\big|a^2b^2(a-b)+b^2c^2(b-c)+c^2a^2(c-a)\big| To be honest, I don't know where ...
For any $0 \leq x \leq y \leq 1$, define $f(y;x):=\frac{y^2}{2}-\frac{2 y^3}{3}+\frac{y^4}{4} - \frac{x^2}{2} + \frac{x^3}{3}$ and \$g(y;x):=\frac{y^2}{3}-\frac{2 y^3}{4}+\frac{y^4}{5} - \frac{x^2}{3} +...