# Questions tagged [convexity-inequality]

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

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### Log convexity implies convexity with Artin definition

I'm trying to prove that log convexity implies convexity using Artin's definition of convexity from "The Gamma Function." According to Artin, a real-valued function $f(x)$ defined on an open ...
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### For a fixed positive definite $A$ anb vector $x$, is $B\mapsto x^T B^{1/2} A B^{1/2} x$ always concave?

Let $x\in R^d$ and $A\in R^{d\times d}$ positive definite. Is the map $$B \mapsto x^T B^{1/2} A B^{1/2} x$$ always concave? One known result that gives a little hope is the Lieb inequality (cf. ...
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### Proving the Preposition: Log-Convexity of a Function and the Monotonicity of Ratios

While reading a research paper on log convexity, I encountered a preposition (which is my question). I tried to prove it. I'm not getting any idea how to proceed. The statement is as follows: Suppose ...
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### Showing that a function $\varphi$ is convex if $\varphi\left(\int_{0}^{1}f(x) d\lambda(x)\right) \leq \int_{0}^{1}\varphi(f(x))d\lambda(x)$ [duplicate]

Let $\varphi: \mathbb{R} \to \mathbb{R}$ be a bounded, Borel-measurable function with $$\varphi\biggl(\int_{0}^{1}f(x) d\lambda(x)\biggr) \leq \int_{0}^{1}\varphi(f(x))d\lambda(x)$$ for every bounded, ...
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### Condition for convexity/strict convexity

A function $f$ defined on an interval $A \subseteq \mathbb R$ is said to be convex if for all $a,b \in A$ and all $\lambda \in [0,1]$ it holds that f ( (1- \lambda) a + \lambda b ) \...
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### What is a convex set $S = \mathbb{R}_{++}^2$

I worked on the following task: The function $f(x) = f(x_1, x_2) = x_1^{1/2} \cdot x_2^{1/2}$ is concave on the convex set $S = \mathbb{R}_{++}^2$. I could solve the task with the Hessian matrix (...
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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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### Prove or disprove that the function is convex on a certain interval

Let $-\ln(2)\leq x<0$ and $n\geq 1$ a natural number then define : $$f(x)=\left(0.5+\sum_{k=1}^{2n}e^{k^2x}\right)\left(0.5+\sum_{k=1}^{2n}(-1)^ke^{k^2x}\right)$$ Then it seems we have the ...
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### Prove that: $\frac{1}{1+x+x^2+x^3}+\frac{1}{1+y+y^2+y^3}+\frac{1}{1+z+z^2+z^3}+\frac{1}{1+t+t^2+t^3}\ge1$
Let $xyzt$ st. $xyzt=1$ Prove that: $$\dfrac{1}{1+x+x^2+x^3}+\dfrac{1}{1+y+y^2+y^3}+\dfrac{1}{1+z+z^2+z^3}+\dfrac{1}{1+t+t^2+t^3}\ge1$$ It's look like Vasc inequality (Let $a,b,c>0$ st. $abc=1$ ...
### Prove or disprove that the function $f(x)=x^{x^{x^{x}}}$ is convex on $(0,1)$
Let $0<x<1$ and $f(x)=x^{x^{x^{x}}}$ then we have : Claim : $$f''(x)\geq 0$$ My attempt as a sketch of partial proof : We introduce the function ($0<a<1$): $$g(x)=x^{x^{a^{a}}}$$ Second ...