# Tagged Questions

Questions on proving, manipulating and applying inequalities.

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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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### 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 ...
I proposed the conjecture as following: Let $ABC$ be a triangle, let $D$ be a point inside of $ABC$. From $D$ and $ABC$, define $F$, $E$, and $G$ as the points where the internal angle bisectors of $\... 3answers 82 views ### Prove$5a^2+b^2+c^2\geq 4ab+2ac$I saw this question recently: Let$a,b,c$be real numbers. Prove$5a^2+b^2+c^2\geq 4ab+2ac$. I feel like this is something with AM-GM inequality. Can someone help me with it? 3answers 1k views ### Proving two integral inequalities Can anyone help me to prove that these integral inequalities hold? Here$x$is a real value: $$\left| \int_a^b\ f(x) dx \right| \leq \int_a^b\ |f(x)| dx$$ Here$z$is a complex value: $$\left| \... 1answer 48 views ### Meaning of Ax \leq b I continue to come across Ax \leq b or Ax= b in optimization problem, but I am having trouble interpreting the meaning of this. Does this have a similar meaning to the following (Cramer's Rule) ... 3answers 34 views ### Find the \sum_{sym}ab maximum of the value Let a,b,c,d,e\in (0,1) and such$$a+b+c+d+e=1$$find the maximun of the value$$S=ab+ac+ad+ae+bc+bd+be+cd+ce+de$$I Conjecture the maximun is \dfrac{2}{5}?,such a=b=c=d=e=\dfrac{1}{5},so$$S\... 2answers 57 views ### Prove that for any$n \ge 2$,$1\times3\times5\times \dots \times(2n-1)<n^n$without induction Prove that for any$n \ge 2$,$1\times3\times5\times \dots \times(2n-1)<n^n$without induction I asked for a non induction prove but I am stuck in induction prove too. In induction we should prove ... 6answers 2k views ### Olympiad Inequality$\sum_{cyc} \frac{x^4}{8x^3+5y^3} \geqslant \frac{x+y+z}{13}x,y,z >0$, prove $$\frac{x^4}{8x^3+5y^3}+\frac{y^4}{8y^3+5z^3}+\frac{z^4}{8z^3+5x^3} \geqslant \frac{x+y+z}{13}$$ Note: Often Stack Exchange asked to show some work before answering the ... 2answers 63 views ### An inequality in positive real continuous function I proposed my conjecture as follows: Let$f(x)$is a positive real continuous function that is convex on$[m, M]$, let$m \le x_i \le M$, for$i=1,2,...,n$then show that $$\frac{f(x_1)+f(x_2)+.....+... 2answers 25 views ### Find values of m and n such that m \leqslant 6 \sin x+ \cos (2x) -1\leqslant n [on hold]$$m \leqslant 6 \sin x+ \cos (2x) -1 \leqslant n$$I have no clue how to do it. please help. 3answers 44 views ### Show that this inequality doesn't hold Given (a,b,c) \in \mathbb R^3_+ show that atleast one of the real numbers a(1-b), b(1-c) and c(1-a) is less than or equal to 1\4. I tried to show it by contradiction i.e Suppose that$$a(1-... 1answer 32 views ### I'm having trouble understanding what this problem is asking me This is the problem So my problem is that I dont know how to solve it... I have learned about system of inequalities and that kinda stuff, but I never got anything like this. I do not want anyone to ... 10answers 117 views ### A clean proof of$x^2 \geq x$, for any integer$x$I am trying to prove that$x^2 \geq x$for any integer$x$. Since we know that for any number$n$,$n^2 \geq 0$we conclude that if$x \leq 0$the proposition will hold. Next we must prove that the ... 2answers 59 views ### Solve the following using AM-GM inequality The least value of$a \in R$for which$4ax^2 + \frac{1}{x} \ge 1 $for all$x \gt 0 , is Using AM-GM inequality $$\frac{4ax^2 + \frac{1}{2x} + \frac{1}{2x}}{3} \ge \sqrt[3]{a}$$ 4ax^2 + \frac{1}{... 0answers 33 views ### Hölder inequality application to show that f=1 I want to proof that if f \in L^{1}_{\mu}(\mathbb{R}), f > 0 continuous, satisfies (\int_\mathbb{R} f(x)d\mu)^{3} \le \int_\mathbb{R} f(x)^{3sin^{2}(x)}d\mu * (\int_\mathbb{R}f(x)^{\frac 32cos^{... 7answers 93 views ### 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, ... 1answer 139 views ### Prove that: \left( \sum_{i\neq j}a_{i}b_{j} \right)^2 \geq \left( \sum_{i\neq j}a_{i}a_{j} \right) \left( \sum_{i\neq j}b_{i}b_{j} \right) Let a_{1}, \cdots, a_{n}, b_{1}, \cdots, b_{n} be positive real numbers. Prove that: \left( \sum_{i\neq j}a_{i}b_{j} \right)^2 \geq \left( \sum_{i\neq j}a_{i}a_{j} \right) \left( \sum_{i\neq j}... 1answer 28 views ### Implication of exponential growth: how is it deduced? LetL$be a differentiable function defined on$\mathbb{R}\times\Omega$with$\Omega\subseteq\mathbb{R}^n$. I will say it has exponential growth if for all$O\subset\subset\Omega$open there exists a ... 1answer 101 views ### Minimize$P(x,y,z)=(2x+3y)(x+3z)(y+2z)$, when$xyz=1$Find the minimum value of the product$P(x,y,z)=(2x+3y)(x+3z)(y+2z)$, when$xyz=1$and$x,y,z$are positive real numbers. I don't know how to go about this. AM-GM got really messy, and I don't know ... 2answers 38 views ### How to prove that${l \choose a_1,…,a_n}\le n^{l-1} $, when$a_1+…+a_n=l$. In the proof of (Corollary 8 chap. 3 ) in the book "Sobolev Spaces on Domains" by Burenkov the following inequality is used : given$a_1,...,a_n \in \mathbb{N}$such that$a_1+...+a_n=l$, then $${l \... 3answers 35 views ### Does Gaussian convolution respects order? Assume that we have two continuous integrable functions f,g \in L^1(\mathbb{R}) such that, for some x_0 \in \mathbb{R}, we have,$$f(x_0) \leq g(x_0) \; \; \; \; (1).$$Now let us define the ... 2answers 20 views ### Solving the inequality involving modulus Can I change \frac{1}{|x-2|} \le \frac{1}{|2x-3|} to |x-2| ≤ |2x-3| ? If I remembered correctly, I cant change a \lt \frac{ 1}{|b|} to a|b| \lt 1 instead, I have to change it to a-\frac{1}... 0answers 31 views ### Help with Simplying and equation I would like some help simplifying and equation. Contraints C_1 is a positive integer Constant C is also a positive integer Constant x and y are both real real numbers. x\leq 0, y\... 2answers 74 views ### An inequality on the rank of a block matrix Let \mathbb F be a field, and let r_1, r_2, s_1, s_2 be positive integers. Consider the matrix$$X:=\left[\begin{array}{cc} A & B \\ C & D \end{array} \right],$$where A \in \mathbb F^... 1answer 34 views ### Express c and d in terms of m where c and d are zeroes of f where m > -2 Let$$f(x) = x^2 - mx -(6m^2+25m+25)$$where m > - 2 It can be shown that f(x) has two zeroes. Suppose we have c,d \in \mathbb R s.t. c < d and f(c) = f(d) = 0, express c and d ... 3answers 60 views ### Mathematical Induction Inequality problem [on hold] I am trying to solve the following problem with mathematical induction:$$ \forall n>1,\qquad \frac{1}{2^2}+\frac{1}{3^2}+\ldots+\frac{1}{n^2}<\frac{n-1}{n} $$but since I am new to the concept ... 1answer 37 views ### how to proceed next in this logarithmic inequality? The question is$$\frac{1}{\log_4{\left(\frac{x+1}{x+2}\right)}}<\frac{1}{\log_4{(x+3)}}$$I did the first step for defining the arguments of both sides and got x\in(-3,-2)\cup (-1,\infty) ... 2answers 73 views ### On real part of the complex number (1+i)z^2 Find the set of points belonging to the coordinate plane xy, for which the real part of the complex number (1+i)z^2 is positive. My solution:- Lets start with letting z=r\cdot e^{i\theta}. ... 0answers 27 views ### Find the maximum of the k such 0\le x^2(3-2x)(2x^k+(3-2x)^k)\le 3 Find k_{\max},such$$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$$it is clear for k\in R and other case it's not easy to solve 7answers 2k views ### proving \mathrm e <3 Well I am just asking myself if there's a more elegant way of proving$$2<\exp(1)=\mathrm e<3$$than doing it by induction and using the fact of \lim\limits_{n\rightarrow\infty}(1+\frac1n)^n=\... 1answer 68 views ### Circles in complex plane. Find the real value of a for which there is at least one complex number satisfying |z+4i|=\sqrt{a^2-12a+28} and |z-4\sqrt{3}|\lt a. My solutions:- Graphical solution:- |z+4i|=\sqrt{a^2-... 1answer 1k views ### Proof of \lim\sup(a_nb_n)\leq \lim\sup(a_n)\limsup(b_n) Let a_n>0 and b_n\geq 0, then \lim\sup(a_nb_n)\leq \lim\sup(a_n)\limsup(b_n) My attempt at a proof is as follows. Let A_n=\sup\{a_n, a_{n+1},...\}, B_n=\sup\{b_n, b_{n+1},...\}, and ... 5answers 89 views ### Monotonicity of the sequence (a_n), where a_n=\left ( 1+\frac{1}{n} \right )^n Define a_n=\left ( 1+\frac{1}{n} \right )^n for n\geq 1. I want to show that it is increasing. First, we have$$\frac{a_{n+1}}{a_n}=\left ( \frac{1+\frac{1}{n+1}}{1+\frac{1}{n}} \right )^n\left ( ... 0answers 13 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 ... 7answers 1k views ###$x,y,z \geqslant 0$,$x+y^2+z^3=1$, prove$x^2y+y^2z+z^2x < \frac12x,y,z \geqslant 0$,$x+y^2+z^3=1$, prove $$x^2y+y^2z+z^2x < \frac12$$ This inequality has been verified by Mathematica.$\frac12$is not the best bound. I try to do AM-GM for this one but not ... 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 ... 1answer 23 views ### Query about an algebraic inequality involving$(a-b)^p$I want to know if there exists any inequality of the type$(a-b)^p \geq C(a^p -b^p)$or$(a-b)^p \leq C(a^p -b^p),$where$a>0,\, b>0,C>0$is a constant and$0<p<1.$I am aware of ... 4answers 92 views ### Prove$n^{n/2} < n!$if$n \gt 2$[duplicate] Ive been stuck on this question for so long.How do i do it?$n^{n/2} < n!$if$n \gt 2, n \in \mathbb{N}$. Please help guys. 1answer 20 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. ... 2answers 38 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$n$... 2answers 134 views ### Prove that$(a^2+2)(b^2+2)(c^2+2)\geq 3(a+b+c)^2$For the non-negative real numbers$a, b, c$prove that $$(a^2+2)(b^2+2)(c^2+2)\geq 3(a+b+c)^2$$ What I did is applying Holder's inequality in LHS:$$(a^2+(\sqrt{2})^2)(b^2+(\sqrt{2})^2)(c^2+(\sqrt{2})... 0answers 201 views ### Prove that \sqrt{a^2+3b^2}+\sqrt{b^2+3c^2}+\sqrt{c^2+3a^2}\geq6 if (a+b+c)^2(a^2+b^2+c^2)=27 Let a, b and c be non-negative numbers such that (a+b+c)^2(a^2+b^2+c^2)=27. Prove that:$$\sqrt{a^2+3b^2}+\sqrt{b^2+3c^2}+\sqrt{c^2+3a^2}\geq6$$A big problem here around$(a,b,c)=(1.6185...,...
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 ...