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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Equivalent definitions of convexityy

Let $K\subset \mathbb{R}^n$ open and convex, $f: K \rightarrow \mathbb{R}$. Show $$(i) \text{ for } f\in C^2: f \text{ convex} \iff \sum_{i,j=1}^n\partial_i\partial_jf(x)\xi_i\xi_j\geq 0 \forall \xi \...
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Correct my sketch of proof about the convexity of the "natural" power tower on $[1,\infty)$

Hi I want to show the following fact : Problem : Let $x\geq 1$ and $n\geq 1$ a natural number and define: $$f(x)={}^{2n}x=\underbrace{x^{x^{⋰^{x}}}}_{2n\text { times}}$$ Then we have : $$f''(x)\ge 0$$ ...
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Sufficient condition for convexity involving an average of slopes

Suppose $h(x)\ge0$ is increasing and concave for all $x\ge0$. For $\Delta>0$, let $$ f(x)=\frac{h(x+\Delta)-h(x)}{\Delta}. $$ I feel that $f$ is a convex function, i.e. $$ f(tx+(1-t)y)\le t f(x)+(1-...
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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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Is there a conclusion about convexity for n-dimensions

There is a proof in the exercises (3.1) of "Convex Optimization": Suppose $f: R \to R$ is convex, and $a, b \in \text{dom} f$ with $a < b$. We can prove: $$f(x) \le f(a) * (b−x)/(b−a) + ...
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Convexity of a function defined on $W^{1,p}(]0,r[)$

Consider the continuous, non-decreasing function $$\beta:\mathbb{R} \rightarrow \mathbb{R},$$ where $\beta(0)=0.$ Define $$g(s)=\int_0^s \beta(u)du,$$ where $s$ is a positive constant. Let now $A=\{w \...
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Concavity of symmetric function.

I would like to check the concavity of a smooth function $f(x_1,...,x_n)$ that is symmetric under any permutation of its parameters. This is for an optimization problem. In principle, I would have to ...
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Convexity, concavity doubts

Let $f$ be a numerical function, $f:D\to\mathbb{R}$, and we want to algebraically determine if $f$ is convex. According to the definition, $f$ is convex if and only if $\forall~x_1,x_2\in D$ the line ...
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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 ...
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Prove continuous $f$ is convex if and only if $\frac{f(x_2) - f(x_1)}{x_2 - x_1} \leq \frac{f(x_3) - f(x_1)}{x_3 - x_1}$ for $x_1 < x_2 < x_3$

Prove continuous $f$ is convex if and only if for $x_1 < x_2 < x_3$ $$S_{1, 2} = \dfrac{f(x_2) - f(x_1)}{x_2 - x_1} \leq \dfrac{f(x_3) - f(x_1)}{x_3 - x_1} = S_{1, 3}$$ I get the intuitive idea ...
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how to obtain the strong convexity inequality

I was playing around with the strong convexity definition and got stuck at some point. I was wondering if someone could kindly help me out. We say that function $f$ is strongly convex if $1) f(x) \geq ...
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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. $abc=1$ ...
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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 ...
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$\dfrac{e^t + e^{-t}}{2} \leq e^{ \frac{t^2}{2} } $ [duplicate]

We want to prove that $ \forall t, \dfrac{e^t + e^{-t}}{2} \leq e^{ \frac{t^2}{2} } $ I thought to a convexity inequality. If $\psi$ is convex, $\dfrac{ \psi(a)+\psi(b) }{2} \leq \psi( \frac{ a+b }{...
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Concavity/ Convexity of power means function?

Let $a_i > 0$ forall $1 \leq i \leq n$ and let $M(x) := \big(\frac{\sum_{i = 1}^n a_i^x}{n} \big)^{\frac{1}{x}}$ be the power means function. It is well known that the power means function is non-...
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Strongly convex functions (equivalence and intuition)

I trying to prove and understand the equivalence of definitions for a $\gamma$-strongly convex function. I am aware that a function $f:\mathbb{R}^n\mapsto\mathbb{R}$ is strongly convex of modulus $\...
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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 $f$ ...
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Condition of concavity for a function [closed]

I have a continuous function $f:\mathbb{R}^n\to\mathbb{R}$ such that $\forall x,y\in\mathbb{R}^n$ I have that $f(x)\ge\cfrac{1}{2}f(x-y)+\cfrac{1}{2}f(x+y)$. How can I prove that $\forall x,y\in\...
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Inequality :$\left(\frac{x}{x+1}\right)^{\tan^2(x)}+\left(\frac{1}{1+3}\right)^{\tan^2(1)}+\left(\frac{3}{x+3}\right)^{\tan^2(3)}>\frac{40}{39}$

Claim : Let $\frac{13}{10}\leq x<\frac{\pi}{2}$ $$f(x)=\left(\frac{x}{x+1}\right)^{\tan^2(x)}+\left(\frac{1}{1+3}\right)^{\tan^2(1)}+\left(\frac{3}{x+3}\right)^{\tan^2(3)}>\frac{40}{39}$$ The ...
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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 $(a+b)^{1/n}$
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Prove that :$f(a^2)+f(b^2)+f(c^2)\leq f\left(\frac{1}{4}\right)+f\left(\frac{1}{4}\right)+f(0)$ where $f(x)=\sqrt{\frac{1+\sqrt{1+x}}{x^x}}$

Hi it's a problem found by myself : Let $0<x<1$ then define : $$f(x)=\sqrt{\frac{1+\sqrt{1+x}}{x^x}}$$ Then let $a,b,c>0$ such that $a+b+c=1$ then we have : $$f(a^2)+f(b^2)+f(c^2)\leq f\left(\...
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Prove this refinement of Nesbitt's inequality based on another

Let $a,b,c\in[1,2]$ such that $a,b$ are constants then prove : $$f(c)=\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{b+a}\geq h(c)=(c-1)\frac{g(2)-g(1)}{2-1}+g(1)\geq g(c)=\sqrt{\frac{9}{4}+\frac{3}{2}\frac{(a-...
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4 votes
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General Inequality for Convex Functions

Suppose we have a convex function $f:[-1,1]\to \mathbb{R}$ with $f(-1)=f(1)=0$ and $\{t_i\}_{i=1}^k$ is a sequence of numbers such that $k\ge 2$ and $$ \sum_{i=1}^k \frac{t_i}{f(t_i)+f(-t_i)}=0\ \ \ \ ...
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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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How to prove this inequality on given set?

Let $p,q\in\{1,2,\dots,n\}$ and $pq\ne1$. Then prove that $$\frac{p+q-2}{pq}\le 1-\frac{1}{n}.$$ Moreover equality holds iff $(p,q)=(1,n)$ or $(n,1).$ My attempt: $$\frac{p+q-2}{pq}=\frac{1}{p}(1-\...
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1 answer
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Uniqueness of a point where the chord coincides with the tangent

Let $F:[0,1] \to [0,\infty)$ be a $C^2$ function satisfying $F(1)=0, F'(1)=0$, which is strictly strictly decreasing on $[0,1]$. Suppose that for some $a \in(0,1)$, $F'' < 0$ on $[0,a)$ and $F'' &...
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Expectation of random matrix

Assume $Q$ is a positive definite random matrix such that $0 < \lambda_{\min}(Q)....\leq \lambda_{\max}(Q) \leq 1$ holds. I want to show that \begin{align} E\left[\frac{\lambda_{\min}(Q)}{\lambda_{\...
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Does the convex envelope inherit monotonicity properties?

Let $F:[0,\infty) \to [0,\infty)$ be a $C^1$ function satisfying $F(1)=0$, which is strictly increasing on $[1,\infty)$, and strictly decreasing on $[0,1]$. Suppose also that $F|_{(1-\epsilon,1+\...
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Given four real numbers $a,b,c,d$ so that $1\leq a\leq b\leq c\leq d\leq 3$. Prove that $a^2+b^2+c^2+d^2\leq ab+ac+ad+bc+bd+cd.$

Given four real numbers $a, b, c, d$ so that $1\leq a\leq b\leq c\leq d\leq 3$. Prove that $$a^{2}+ b^{2}+ c^{2}+ d^{2}\leq ab+ ac+ ad+ bc+ bd+ cd$$ My solution $$3a- d\geq 0$$ $$\begin{align}\...
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Does this strong convexity estimate hold?

Let $F:[0,\infty) \to [0,\infty)$ be a $C^2$ strictly convex function, and let $r_0<r_1$ be positive fixed constants. Let $$a<r_0<r_1<c<b, \tag{1}$$ and let $\lambda \in [0,1]$ satisfy $...
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Limit of a convex function

I would need a check on the following exercise: Let $f:\mathbb{R} \rightarrow \mathbb{R}$ a convex function. Prove that $\lim_{x \rightarrow \infty} f(x)$ and $\lim_{x \rightarrow - \infty} f(x)$ ...
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Show that Lambert's function is concave using mid-point concavity

All is in the title so we want to show that : Let $a,b\geq e$ then we have $$\operatorname{W}(a)+\operatorname{W}(b)\leq 2\operatorname{W}\Big(\frac{a+b}{2}\Big)$$ The trick is to put : $a=x\ln(x)$ $...
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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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2 votes
3 answers
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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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2 votes
1 answer
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Prove that $\Gamma(\operatorname{W}(x))$ is convex $\forall x>0$

Background : At the begining I was studing a function wich increases slowly and maybe have some property useful in number theory .Particulary I have found : Let $0<x\,$ define the function : $$f(x)...
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4 votes
1 answer
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Does this bound hold beyond the domain where the function is convex?

Let $F:(0,\infty) \to [0,\infty)$ be a continuous function satisfying $F(1)=0$, which is strictly increasing on $[1,\infty)$, and strictly decreasing on $(0,1]$. Suppose also that $F|_{(1-\epsilon,1+...
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4 votes
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Does local convexity imply global convexity around a minimum?

This is a follow-up of this question. Let $F:(0,\infty) \to [0,\infty)$ be a continuous function satisfying $F(1)=0$, which is strictly increasing on $[1,\infty)$, and strictly decreasing on $(0,1]$. ...
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minimizing a concave function

After decoupling a big optimization, the inner one is as follow: $min_X \quad log_2 \left(det(A+BX\right))$ $s.t. norm(X)\leq \gamma,$ where $A\in \mathbb{C}^{n\times n}$ and $B\in \mathbb{C}^{n\times ...
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2 answers
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Does convexity around a point imply the function is above the tangent at that point?

Let $\phi:\mathbb [0,\infty) \to [0,\infty)$ be a $C^2$ function, and let $c>0$ be a constant. Suppose that for any $x_1,x_2>0, \alpha \in [0,1]$ satisfying $\alpha x_1 + (1- \alpha)x_2 =c$, we ...
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Prove that the tangent $A$ and $B$ are perpendicular .

Let $f(x)=x^x$ and $g(x)=\Big(\frac{1}{x}\Big)^{x}$ .Let $A$ be the tangent of $f(x)$ at $x=1$ and $B$ the tangent of $g(x)$ at $x=1$ . Then A and B are perpendicular . Proof without first derivative ...
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1 vote
1 answer
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Does strict convexity plus asymptotic affinity imply bounded mean?

Let $F:[0,\infty) \to [0,\infty)$ be a $C^2$ strictly convex function, with $F''$ everywhere positive. Let $\lambda_n \in [0,1],a_n\le c_0<b_n \in [0,\infty)$ satisfy $$ \lambda_n a_n +(1-\lambda_n)...
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4 votes
2 answers
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A Hamiltonian inequality using uniform convexity and Taylor's formula (Evans PDE, §3.3.3, Lemma 4, (36))

In the proof of Lemma 4 (Semiconcavity again) in section 3.3.3 of the book "Partial differential equations" written by Lawrence C. Evans, there's a claim We note first using Taylor's ...
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