# Questions tagged [inequality]

Questions on proving, manipulating and applying inequalities.

2,847 questions
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### Hilbert's Inequality - improved???

Assume for convenience that $a_n\ge0$ (this also clarifies why certain inequalities below are in fact stronger than certain other inequalities below). Of the various inequalities Hilbert proved, I'm ...
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### How to prove this polynomial inequality?

How can we prove the following? If $\frac{dP_{n}}{dz}|_{z=z_{0}}=0$ then $|P_{n}(z_{0})|<2$ for all $n>1$, where $P_{n}(z)\equiv P_{n-1}^{2}+z$ and $P_{1}\equiv z$ $z$ is in the complex plane. ...
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### I conjecture $3(a^ab^bc^c)^{\frac{1}{a+b+c}}\ge (a^ab^b)^{\frac{1}{a+b}}+(b^bc^c)^{\frac{1}{b+c}}+(c^ca^a)^{\frac{1}{c+a}}$

I Conjecture $$3\left(a^ab^bc^c\right)^{\dfrac{1}{a+b+c}}\ge \left(a^ab^b\right)^{\frac{1}{a+b}}+\left(b^bc^c\right)^{\frac{1}{b+c}}+\left(c^ca^a\right)^{\frac{1}{c+a}}$$ This conjecture is based on ...
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### Heisenberg's inequality

The Heisenberg's inequality in $\Bbb{R}$ reads $$\|f\|_{L^2}^4\leq \int_{\Bbb{R}}x^2f(x)^2dx\int_{\Bbb{R}}\xi^2\hat{f}(\xi)^2d\xi$$ where by $\hat{f}$ we refer to the Fourier transform of $f$. The ...
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### Can one use properties of polynomials in order to generalize the generalized Cauchy-Schwartz inequality?

Sorry about the edits guys, I forgot to add binomial coefficients, I hope I didn't cause any needless confusion. Edit(again): I've been thinking about this a bit and perhaps I should clarify the ...
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### Is there a combinatorial proof to this inequality?

I verified that this inequality: $$\sum_{i=0}^{k-1} \sum_{j=0}^{k+1} {3 k-3\choose i} {3 k+3\choose j} \geq \sum_{i=0}^{k} \sum_{j=0}^{k} {3 k\choose i} {3 k\choose j}$$ holds for all $k$ between 1 ...
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### How to prove or falsify this inequality?

In STEP 2014 Paper II Question 2, an inequality is assumed for candidates to attempting the question about the approximation of $\pi$ $$\int_{0}^{\pi } (f(x))^2 dx \le \int_{0}^{\pi } (f'(x))^2 dx$$...
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### Question about the proof of General Sobolev Inequality in P.D.E. by Evan

I have been reading the chapter of Sobolev Space in Partial Differential Equations by Lawrence C. Evan, and I came across the General Sobolev Inequality stated as follows: Theorem (General Sobolev ...
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### Difficult inequality with $\pi$

My question is about an inequality ,originally I wanted to prove this : If $a,b,c,d >0$, and $a+b+c+d=4$, prove that $$a^{ab}+b^{bc}+c^{cd}+d^{da} \geq \pi.$$ from here My approach is to use this ...
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### Prove that $\frac{a+b+c}{3}\geq\sqrt[53]{\frac{a^4+b^4+c^4}{3}}$

Let $a$, $b$ and $c$ be non-negative numbers such that $(a+b)(a+c)(b+c)=8$. Prove that: $$\frac{a+b+c}{3}\geq\sqrt[53]{\frac{a^4+b^4+c^4}{3}}$$ I think $uvw$ does not help here. My another similar ...
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Dalzell integral The equation $$\int_0^1 \frac{x^4(1-x)^4}{1+x^2}dx=\frac{22}{7}-\pi$$ proves that $\frac{22}{7}-\pi>0$ because the integrand is positive. Some Dalzell-type integrals for $\log(... 0answers 1k views ### My favorite proof of the generalized AM-GM inequality: where it came from? I have already posted (most of) the present question as a (misplaced) answer to a question about understanding a particular proof of the AM-GM inequality. I sincerely hope I am not breaking the code ... 0answers 334 views ### Does the Gagliardo–Nirenberg interpolation inequality hold on compact closed manifolds? The Gagliardo–Nirenberg interpolation inequality on a bounded domain is of the form $$|D^j u|_{L^p} \leq C_1|D^m u|_{L^r}^\alpha|u|_{L^q}^{1-\alpha} + C_2|u|_{L^s}$$ where are there restrictions on ... 0answers 189 views ### Solution set of inequalities in$\mathbb{R}^6$Let$\theta\in (0,1)$fixed. We define$A_1$be the set of all$(a_1,a_2,a_3,a_4,a_5,a_6)\in (0,1)^6$such that the following conditions hold: (1) \quad a_2\le a_1, a_4\le a_3, a_6 \... 0answers 485 views ### Inequality between incomplete beta and gamma functions Let the regularized incomplete beta and gamma functions be defined as usual: $$I_p(z,w) = \frac a {B(z,w)} \int_0^p t^{z-1} (1-t)^{w-1} \,\mathrm dt,$$ \... 0answers 2k views ### Proving the Power Mean Inequality using Chebyshev's sum inequality Question: Can Chebyshev's sum inequality be used to prove the generalized mean inequality, or at least a portion of it? If we let $$P(r) = \sqrt[r]{\frac{x_1^r + x_2^r + \cdots +x_n^r}{n}}$$, we have ... 0answers 270 views ### Solving a system of linear inequalities I have a personal problem I want to solve. I have a system of linear inequalities with 97 unknowns and 150000 ineqaulities, I think formal notation of the problem should be something like this \begin{... 0answers 327 views ### Behaviour at infinity of a function in terms of first and second derivatives In a paper (dealing with spectra of certain Schrodinger operators) I found the following assumption for a function$f\in C^\infty(\mathbb R^n;\mathbb R)$: there exists a constant$C>0$and a ... 0answers 206 views ### Bounding function involving Beta functions Given$\frac{a}{x-1} \leq \frac{b}{y-1} \leq \frac{c}{z-1}$with$a,b,c > 0$and$x,y,z > 1$, I want to show that $$\frac{(\frac{a}{a+b})^{x-1}(\frac{b}{a+b})^{y-1}}{B(x,y)\cdot (x+y-1)} + \frac{... 0answers 53 views ### Are these upper and lower bounds for \frac{x!}{\left\lfloor{x}\right\rfloor!} useful? If so, are they already known? Truncating the infinite series for the derivative of the Digamma function$$ \psi'(x) = \sum_{n=0}^\infty\frac{1}{(x + n)^2} $$after m-1 terms, where m is a positive integer (the case m=2 ... 0answers 116 views ### Proof of a technical fact in the book of Schapire and Freund on boosting I am currently looking at Exercise 10.3, Chapter 10 of the book on Boosting by Schapire and Freund. More precisely, in the middle of the exercise they propose to use, without proof, the technical fact ... 0answers 183 views ### Proving that q(A,S)>0 if A>0, S<0 and p_i(A,S)>0 Consider the polynomials$$p_1=3(56-56A+A^2)-112(-6+A)S-14(-36+A)S^2+112S^3+7S^4,p_2=-112(-6+A)-28(-36+A)S+336S^2+28S^3,p_3=-28(-36+A)+672S+84S^2,p_4=672+168S,p_5=168,q=-168S-... 0answers 120 views ### An annoying optimization problem At first I thought the following problem looked simple, but I've had serious problems pinning it down: Suppose that$\prod_{i=1}^k x_i$is fixed. Then find the minimum value of$$\sum_{i=1}^k (1 - ... 0answers 78 views ### Inequality involving power to fractional part of integer multiples of logarithm of integer to coprime base. For$x \in \mathbb{R}^+$, let$\{x\} = x - \lfloor x \rfloor$denote the fractional part of$x$. Let$k \in \mathbb{N}$. Show that$2^{\{k \log_2(3)\}} < \dfrac{2}{1 + 2^{-k}}$for$k > 1$. ... 0answers 137 views ### Gronwall's inequality for higher order derivatives. Gronwall's inequality says that solutions to the initial value problem$u'(t) \leq \beta(t)u(t)$with$u(0)=u_0$are bounded by solutions to the problem with inequality replaced with equality for$t\...
Von Neumann's trace inequality states that $|tr(AB)| \le \sum_{i=1}^{n} \sigma_i(A) \sigma_i(B)$ where $A, B$ are general square matrices with singular values $(\sigma_i(A)), (\sigma_i(B))$, ...