# Tagged Questions

Questions on proving, manipulating and applying inequalities.

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### Sharp (Reverse) Harmonic-Arithmetic Mean Bounds

Let $\mathbf{x} =$ {$x_{i}$} be a set of $n$ positive reals. In every good book on inequalities, one finds the classical result \begin{eqnarray} AM(\mathbf{x}) \geq GM(\mathbf{x}) \geq HM(\mathbf{x}),...
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### On some iterated inequalities and $x \geq 5$

Let $x \in \mathbb{N}$. Suppose that I have a function $f:\mathbb{N}\rightarrow\mathbb{Q}$, with initial bounds $$2 - \frac{2}{x_0} < f(x_0) = \frac{2{x_0}}{x_0 + 1} \leq 2 - \frac{5}{3x_0}.$$ ...
Let $f$ be a non-negative function on $\mathbb R^d$ satisfying the following: (1) There exists a non-increasing function $g$ on $(0,\infty)$ such that (1-i) $C_1^{-1} g(|x|) \le f(x) \le C_1 g(|x|)... 0answers 44 views ### If$L > 1$is an odd almost perfect number with$\omega(L)=6$, then$L$must be divisible by$3$. Edited July 15 2016 Let$\mathbb{N}$denote the set of positive integers. Let$\sigma = \sigma_{1}$denote the (classical) sum-of-divisors function. Let$I(x) = \dfrac{\sigma(x)}{x}$denote the ... 0answers 56 views ### olympiad-type inequality Prove that for any$x_1,\dots,x_n>0$$${\root{n}\of{\prod _{k=1}^{n}\ \sum_{t=1}^{k}\ \frac{1}{t^2\cdot\sqrt[t]{x_1\cdot\ldots\cdot x_t}} }} \ \cdot\ \sum _{k=1}^{n}\frac{\sum_{j=1}^{k}\sum_{i=1}^... 0answers 53 views ### Generalization of Jensen's inequality to multivariate functions Is there a generalization of Jensen's inequality for convex multivariate functions? By convex, let's say f is a multivariate function defined on the convex set A, and for all x,y \in A and \... 0answers 76 views ### Find all integers p>q\ge 0, such that for all real x,y \in [0;1] following inequality holds [px]+[py]\ge [qx+y]+[x+qy] Find all integers p>q\ge 0, such that for all real x,y \in [0;1] following inequality holds$$\lfloor px \rfloor + \lfloor py\rfloor \ge \lfloor qx+y\rfloor+\lfloor x+qy \rfloor$$I used x\... 0answers 48 views ### Partitioning a set of integers (with Alice and Bob) Let d_1,\ldots,d_n \in \mathbb{N}_{\ge 2} (not necessarily distinct) be given. Define D:=\operatorname{lcm}(d_1,\ldots,d_n) and d:=\sum_{i=1}^n d_i . (1) Alice claims that whenever \... 0answers 35 views ### Finding a maximum with some constraints I would like to maximize the term l_1b_1+l_2b_2+l_3b_3-2 such that the following conditions hold: 1>l_1>l_2>l_3>0 , l_1,l_2,l_3 \in \mathbb{Q} , b_1,b_2,b_3 \in \mathbb{N} ... 0answers 31 views ### Find the best constant C_{n} such this complex inequality nd we can consider this problem In general? if |z_{1}|=|z_{2}|=\cdots=|z_{n}|=1 if there exist complex z(|z|=1) such$$\sum_{i=1}^{n}\dfrac{1}{||z-z_{i}||^2}\le C_{n}$$find the best C_{n}?$$... 0answers 41 views ### Prove this by inequality with four variables inequality Let$a,b,c,d>0$show that $$\color{blue}{\left(\dfrac{a^2}{b}+\dfrac{b^2}{c}+\dfrac{c^2}{d}+\dfrac{d^2}{a}\right)^2\ge 4(a^2+b^2+c^2+d^2)+\dfrac{8}{3}[(a-b)^2+(a-c)^2+(a-d)^2+(b-c)^2+(b-d)^2+(c-d)^... 0answers 40 views ### Prove \sum_{cyc} \frac{(a+2b+3c)^2}{(2a+b)^2} \geqslant 5 + 7 \cdot \frac{a^2+b^2+c^2}{ab+bc+ca} for positive a,b,c For a,b,c >0, prove$$\sum_{cyc} \frac{(a+2b+3c)^2}{(2a+b)^2} \geqslant 5 + 7 \cdot \frac{a^2+b^2+c^2}{ab+bc+ca}$$My notation$$\sum_{cyc}a^2b= a^2b+b^2c+c^2a$$What I try: 1. Using C-S ... 0answers 67 views ### Prove \sum_{cyc}\frac{x(y-z)}{(2x+y)^2} +\frac13 \cdot \frac{x^2+y^2+z^2}{xy+yz+zx} \geqslant \frac13 for positive x,y,z x,y,z > 0, prove$$\sum_{\text{cyc}}\frac{x(y-z)}{(2x+y)^2} +\frac13 \cdot \frac{x^2+y^2+z^2}{xy+yz+zx} \geqslant \frac13$$While this inequality can be proved by brute force, the elegant ... 0answers 55 views ### Generalized weighted mean inequality Let {p_{1}},{p_{2}},\ldots,{p_{n}} and {a_{1}},{a_{2}},\ldots,{a_{n}} be positive real numbers and let r be a real number. Then for r\ne0 , we define {M_{r}}(a,p)={\left({\... 0answers 60 views ### Inequality involving fourth powers . I have been into inequalities lately and I am stuck with this. I used a famous inequality at first \frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a} \ge 3 (\frac{a^4+b^4+c^4}{3})^{\frac{1}{4}}. From this I ... 0answers 41 views ### Estimating n! as e \left(\frac ne \right)^n \le n! \le ne \left(\frac ne \right)^n I'm told that for n \geq 2,$$\sum_{k=1}^{n-1} f(k) \leq \int_1^n f(x) \, dx \leq \sum_{k=2}^n f(k)$$I am then asked to consider \ln n! = \sum_{k=1}^n \ln k and show that for n \geq 2$$n! \... 0answers 123 views ### When is the inequality$\beta(a_1+b_1, a_2+b_2)\ge \beta(a_1, a_2)\beta(b_1, b_2)$true? Let$\beta(a, b) = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}$. Does there exist some general condition on$a_1, a_2, b_1, b_2\in \mathbb{N}^+$such that $$\beta(a_1+b_1, a_2+b_2)\ge \beta(a_1, a_2)\beta(... 0answers 35 views ### Poincaré-like inequality Let \Omega\subset\mathbb{R}^3 be an open bounded set. Let \partial\Omega=\Gamma^1\cup\Gamma^2, with \Gamma^1\cap\Gamma^2=\emptyset. We denote as \Gamma^1_j, j=1,\dots,p_{\Gamma^1}+1, the ... 0answers 35 views ### \mathbb{P}[X_1(k^\ast)] \leq \left( \frac{e}{k^\ast} \right)^{k^\ast} \frac{1}{1-e/k^\ast} \leq n^{-2} inequality is used to prove the theorem In the book Randomized Algorithms from Motwani and Raghavan, it is stated in page 44 that$$\mathbb{P}[X_1(k^\ast)] \leq \left( \frac{e}{k^\ast} \right)^{k^\ast} \frac{1}{1-e/k^\ast} \leq n^{-2}.$$... 0answers 36 views ### Equality in Hardy's inequality via Hölder's I'm working on Exercise 3.14 in Rudin's Real and Complex Analysis. I was able to answer part (a): that for real p satisfying 1<p<\infty, for every function f in L^p(0,\infty), when F ... 0answers 16 views ### Plot complex inequalities |z^2| > Im(z)^2 I need to draw the set of all complex numbers, which satisfy the following inequality: |z^2| > Im(z)^2 This is what I've already done: |z^2| > Im(z)^2 |z|^2 > Im(z)^2 - use z = a +... 0answers 29 views ### How does this inequality imply this one? I am having a little trouble understanding this part of a proof. There is an integral \text{J}_{n} = \int_0^{\frac{\pi}{2}} x^2\cos^{2n}x dx Now, \text{J}_0 = \frac{\pi ^3}{24} The part of ... 0answers 16 views ### Requesting for details of Grun's result on odd perfect numbers This post is an offshoot of this earlier MSE question. According to this Wikipedia page: Grun 1952 proved that the smallest prime factor p_1 (of an odd perfect number N) is < \frac{2\... 0answers 49 views ### Is this trigonometric expression always strictly positive? Let us define a function f(k,n) by $$f(k,n)=n \left (\cos\frac{k\pi}{n}\right) \left(1-\cos\frac{k\pi}{n}\right) - \sin \frac{k\pi}{n}$$ Where \frac{k}{n} is ... 0answers 38 views ### Cases in which a certain inequality holds true. I previously asked this question on this forum, and have been demonstrated counterexamples to the claim that |a| > |b| implies \big|\frac{b+b^{2}}{a+a^{2}}\big| < 1, which I had previously ... 0answers 33 views ### Finite sequence created by reducing n with each prime under n ends in 0? Given n a fixed integer we constuct the following sequence: a_0=n, a_i=\lfloor \frac{a_{i-1}(p_i-1)}{p_i}\rfloor. For what values of n do we have a_{\pi(n)}=0? Computer calculation shows ... 0answers 63 views ### Elementary-Looking Inequality on n Complex Numbers Let z_1,z_2,\ldots,z_n be complex numbers. Is it true that \displaystyle\sum_{1\le i,j\le n} |z_i+z_j| \ge \displaystyle\sum_{1\le i,j\le n} |z_i-z_j|? I know the inequality holds for reals and ... 0answers 29 views ### Choosing three integers to satisfy an equation under a specific condition Find three integers (a,b,c) such that: x*a + y*b + z*c = a + b only when x = 1, y = 1, z = 0 where x, y and z can be chosen as any non-negative integers. For example, choosing a = 1; b = ... 0answers 58 views ### I am trying to show an inequality involving the product of three inner product terms Define the inner product \langle\cdot,\cdot\rangle for continuous functions defined on [0,1] as:$$\langle\,f\mid g\rangle=\int_{0}^{1}f(x)g(x)e^{\rho x}dx,$$where \rho is a real number. I ... 0answers 37 views ### Prove that (n-1)!S_m\geq (n-m)!m!P_m. If a_1, a_2,\cdots a_n be all positive rationals such that S_n=a_1^m+a_2^m+\cdots +a_n^m, P_m=\sum a_1a_2\cdots a_m (the sum of products m taken m at a time). Prove that$$(n-1)!S_m\geq (n-m)!m!... 0answers 37 views ### Proving$n\cdot\bigg(\sum_{i=1}^{n}|x_{i}|^{2}\bigg) \leqslant (n\cdot|x_{n}|)^{2}$May this be proved by induction? Let$x \in \mathcal{R}^{n}$, where$|x_{n}| \geqslant |x_{i}|$,$i \neq n$. In other words,$|x_{n}|$is the maximal element of$x$. Then, $$n\cdot\bigg(\... 0answers 34 views ### Doob's L^p inequality - Case p = 1 I have found in the wikipedia page following generalisation of Doob's so-called L^p inequality, for general nonnegative submartinagles X_s:$$E[\sup_{0 \le s \le T} X_s] \le \frac{e(1 + E[X_T\log^... 0answers 47 views ### Solving Higher Order Polynomial Inequalities in Two Variables Could someone please point me to a solver (software) and also some techniques to solve equations of the kind shown below? $$x^{3}y+ay^{3}+by^{2}+cy+dx+exy+fxy^{2}\geq0$$ Here,$x,y$are the ... 0answers 30 views ### Inequality of expansion of$\| \ \ \|$The following is from a proof of a paper Distributed subgradient Methods for Multi-Agent Optimization, Nedic & Ozdaglar - Lemma 5 Note:$y(k+1)=y(...
Define $N_c=[\dfrac{1}{2}n\log n+cn]$ where $[.]$ denotes the greatest integer function, and $c$ is any arbitrary fixed real constant. Also, let $M={n\choose 2}$. Then prove, for large $n$, the ...