# Questions tagged [convexity-inequality]

This is useful method for an estimation convex or concave functions on a closed segment.

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### Proving a function in $\mathbb{R}^2$ is convex

Let $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ such that $f(x,y)=\ln(e^{5x}+e^{2y})$ for all $(x,y)\in \mathbb{R}^2$. According to what I've seen (and taught by my teachers), a function $f$ is convex if ...
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### An inequality of convexity

Let $x_1 \ge x_2 \ge \dots \ge x_n \ge 0$ and $y_1 \ge y_2 \ge \dots \ge y_n \ge 0$ such that $$\forall k \in [\![1,n ]\!], \prod_{i=1}^k x_i \le \prod_{i=1}^k y_i$$ (with equality for $k=n$). I want ...
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### Brezis Exercise 3.12: Passing Conjugate into Inequalities

This is Exercise 3.12 from Brezis: Let $E$ be a Banach space and let $x_0 \in E$. Let $\varphi: E \to (-\infty, +\infty]$ be a convex l.s.c. function with $\varphi \not\equiv +\infty$. Show that the ...
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### Why is the negative $l^p$ norm like function convex when $0 < p < 1$?

Let $$\varphi(x) = \begin{cases} -\frac{1}{p}x^p &, \mbox{ if } x \geq 0, \mbox{ where } 0 < p < 1, \\ \infty &, \mbox{ if } x < 0. \end{cases}$$ I want to show convexity of this ...
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### Convex function inequality with eigenvalues of Hessian matrix are bounded above and away from zero [closed]

We have a twice continuously differentiable function $f:\mathbb{R}^d\to\mathbb{R}$ and $\mu,L>0$ constants We have $\mu\cdot E\preccurlyeq\nabla^2f(x)\preccurlyeq L\cdot E$. that means that the ...
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### proof that sin (a + b + c)/3 ≥ 1/3 (sin a + sin b + sin c) [closed]

click here for the question I tried to prove it using the convexity
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### strict convexity attains minimum

exercise: Let $f$ be a strictly convex function defined in an interval $A$. Suppose that there exist distinct points $a$ and $b$ in $A$ such that $f(a)=f(b)$ Show that $f$ attains a minimum. If $f$ ...
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### Why is showing $LHS - RHS \geq 0$ enough to prove convexity of a function?

I understand the definition of a convex function to be the following: $$\alpha f(x) + (1 - \alpha) f(y) \geq f(\alpha x + (1-\alpha) y)$$ Intuitively, this means that any point on a line connecting ...
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### Question about convexity: how do we prove that $\displaystyle \sum_{i=1}^{k}p_{i}b_{i}\geq\prod_{i=1}^{k}b^{p_{i}}_{i}$?

Let $b_{1},b_{2},\ldots,b_{k}$ be nonnegative numbers and $p_{1} + p_{2} + \ldots + p_{k} = 1$ where each $p_{i}$ is positive. Then \begin{align*} \sum_{i=1}^{k}p_{i}b_{i}\geq\prod_{i=1}^{k}b^{p_{i}}_{...
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### Little result using RCF theorem and an inequality by Vasile Cirtoaje

Well , I want to share with you a little result : The function $$f(x)=(x)^{2(1-x)}$$ is concave on $I=[\frac{1}{2},1]$ Moreover Vasile Cirtoaje proved that : $$f(x)+f(1-x)\leq 1\tag{1}$$ So we can ...
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### When does the inequality hold?

I am trying to find a condition on $c$ such that the below inequality holds true $$\frac{1 - e^{-st}}{st} - \frac{1}{st+c} > 0$$ where $s$, $c$ and $t$ are greater than $0$. I tried simpyfing it ...
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