Questions tagged [convexity-inequality]

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

Let $f$ is a convex function on $[a,b]$. Prove that: $$\max_{[a,b]}f=\max(f(a),f(b)).$$
Let $f$ is a concave function on $[a,b]$. Prove that: $$\min_{[a,b]}f=\min(f(a),f(b))$$

147 questions

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1 answer

If $f$ is convex, show that $f(x)/x$ is non-decreasing in $x$

A proof for deducing Lypaunov's inequality seems to be centered on showing that if $f$ is convex on $0< a \le x \le b$, then $f(x)/x$ must be non-decreasing on the interval. I am not aware that $f$ needs to be differentiable. Starting from a…

convexity-inequality

asked Feb 17 '21 at 01:00

Gregory

3,641

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Showing that a function $\varphi$ is convex if $\varphi\left(\int_{0}^{1}f(x) d\lambda(x)\right) \leq \int_{0}^{1}\varphi(f(x))d\lambda(x)$

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, Lebesgue-measurable function $f: \mathbb{R} \to…

convexity-inequality

asked Jan 23 '24 at 11:09

MathGeek

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Proof that $x^\frac{1}{p}$ is concave without derivative $(p>1)$

I need to show that $f(x_1,...,x_n)= \sum_{i=1}^{n} x_i^\frac{1}{p}$ is concave (for $x_i>0$ and $p>1$) without using any derivatives. I have attempted to prove that the secant line lies below the curve for the one-variable function $x \in…

convexity-inequality

asked Sep 26 '21 at 11:11

Padawan

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1 answer

Convexity of $ (a+b)^{1/n}$

How to prove that $(a+b)^{1/n} \le a^{1/n}+b^{1/n}$ by the convexity of $(a+b)^{1/n}$

convexity-inequality

asked Nov 29 '20 at 14:44

sara

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0 answers

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 \begin{equation} f ( (1- \lambda) a + \lambda b ) \leq ( 1- \lambda) f(a) + \lambda…

convexity-inequality

asked Jul 18 '23 at 17:23

herbhofsterd

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2 answers

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 it satisfies $(\forall x,y\in \mathbb{R}^2, 0\leq…

convexity-inequality

asked May 21 '22 at 16:13

Aishgadol

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1 answer

comparing two concave functions with different inequalities

I am deriving some bounds involving two functions, which are $f(x)=\log(x)$ and $f(x)=-x\times\log(x)$. I found in several books that these two functions are concave. I have draw them down using python to quickly check. For that reason, by applying…

convexity-inequality

asked Mar 06 '19 at 15:18