# Questions tagged [inequality]

Questions on proving, manipulating and applying inequalities. Do not use this tag just because an inequality appears somewhere in your question.

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### Is this inequality related to this integral correct?

Let $\alpha >0, 1/3 >\beta>0, T>0$, $h \in [0,T]$. In [Cerrai, Sandra. "Normal deviations from the averaged motion for some reaction–diffusion equations with fast oscillating ...
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### Is that quantity bounded from below?

Let $p>1, s>0$ and $F$ be a function such that $$F(t)\le \frac{|t|^p}{p} +|t|^{p+p^{\prime}}e^{|t|^{p^{\prime}}}\quad\mbox{ for all } t\in\mathbb{R},$$ where $p^{\prime}$ denotes the conjugate ...
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### How to proof Lip-constant weakens when size of function class approaches to infinity?

Taken from paper "A Universal Law of Robustness via isoperimetry" by Bubeck and Sellke. Theorem 3. Let $\mathcal{F}$ be a class of functions from $\mathbb{R}^{d} \rightarrow \mathbb{R}$ and ...
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### Volume of the piece of an n-ball defined by inequality constraints

Consider $x \in \mathbb{R}^n$, and $B = \{x: ||x|| < r\}$, the n-ball with radius $r$ centered at the origin. Let $V$ be its volume. Further, consider $1 \leq m \leq n$ linear inequality ...
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### Is there a shorter or more trivial way to prove that $x > \cos (x)-\cos (2 x)$ holds for all $x>0$?

I want to prove that the inequality $$x > \cos (x)-\cos (2 x)$$ holds for all $x>0$. My attempt: Since the function on the RHS is periodic, we can find the position of extrema (on first ...
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### Knowing $f(t^\ast)\ge 0$ (and some other information), can we show that $f(t)\ge 0$ at $t<t^\ast$?

I asked a similar question before and had to make several changes so before anyone spends time on answering it, I decided to clarify here. We have constants $\alpha_1,\alpha_2,r_1,r_2,c>0$ and we ...
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### Evaluating the value of |$f'(x)$| [duplicate]

Let $f \in C^2[0,2]$ and $|f(x)| \le 1, |f''(x)|\le1$ for all $x \in[0,1]$ Prove that $|f'(x)|\le2$ for all $x \in [0,2]$ I tried Taylor expansion to evaluate $|f'(x)|$, but it seems to work only ...
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### Inequality with special case of equality at $\infty$

Prove that if $a,b,c,d$ are positive reals we have: $$\frac{a}{\sqrt{a^2+b^2}}+\frac{b}{\sqrt{b^2+c^2}}+\frac{c}{\sqrt{c^2+d^2}}+\frac{d}{\sqrt{d^2+a^2}}\leq3$$ I think that I have found a equality ...
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Here is the following question: Solve: $\sec(2x) \ge\sec(x) , x\in [0,\pi]$\ {$\frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}$} Note: This is part b of a question where in part a, I was asked to solve: $... 4 votes 0 answers 51 views ### Inequality with interesting independent constants Let$b_1,\dots,b_{n-1}$be integers satisfying$0 \le b_i \le n-i$for each$i \in [n-1]$such that$\sum_{i=1}^{n-1} b_i = \alpha \binom{n}{2}$where$\alpha$is constant strictly between$0$and$1$.... 0 votes 0 answers 28 views ### How can i find minimum value of this inequality$a_1,a_2,...,a_n$are 8 distinct positive integers.$b_1,b_2,...,b_n$are another 8 distinct positive integers ($a_i,b_j$are not necessarily y distinct for$i, j = 1, 2, ...8$).Enter the smallest ... 2 votes 1 answer 53 views ### How do we rigorously prove that for$n>1$,$(1+x)^{n-1}<1$for$-1<x<0$? Given$n>1$and $$(1+x)^{n-1}<1$$ Intuitively I can see that for$x \in (-1,0)$, we have$1+x<1$, and if we raise that to any power then it will be smaller than 1. How do we prove this ... 2 votes 0 answers 106 views ### If$\sum_{cyc}\frac{a}{a+1}=1$Show that$abc\le \frac{1}{8}$Given positive reals$a,b$and$c$, such that $$\sum_{cyc}\frac{a}{a+1}=1$$ Show that $$abc\le \frac{1}{8}$$ This problem is fairly easy we can just clear denominators and then use AM-GM. But since I ... -1 votes 0 answers 32 views ### A cyclic inequality with fractions [closed] I want to show$\frac{1}{a+b-ab}+\frac{1}{a-b+ab}+\frac{1}{-a+b+ab}\geq \frac{3}{2}$for$a,b \geq 1$directly. Could you give me an advice? I think we have to note that the denominators are not ... -2 votes 1 answer 20 views ### If$-x>|y|>z$, then how does$x+y$compare to$|y|+z$? If$-x>|y|>z$, then how does$x+y$compare to$|y|+z$? an absolute value is non-negative, so$x<0$and$z\geq 0.$1 vote 1 answer 43 views ### If two functions are close apart can I prove the difference of their empirical loss is also small? I am trying to understand the proof of Theorem 3 in the paper "A Universal Law of Robustness via isoperimetry" by Bubeck and Sellke. Basically there exist atleast one$w_{L,e}$in$\...
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RecallA pair $(q,r)$ is admissible if $q\geq 2, r\geq 2$ and $\frac{2}{q} = N \left( \frac{1}{2} - \frac{1}{r} \right),(q,r,N)\neq(\infty,2,2).$ Strichartz estimates Let $\phi \in L^2(\mathbb R^N),$...