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 Increase of $(a+\delta)_m - (a)_m$ Given Decrease of $\frac{(a+\delta)_m}{(a)_m}$.

The question I'm struggling with is as follows: Let $a>0$ and $\delta > 0$ (fixed). Suppose $ a \mapsto \frac{(a+\delta)_{m}}{(a)_{m}}$ is decreasing then prove that $ a \mapsto (a+\delta)_{m} ...
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A new type of convexity related to the exponential function

I am new to the study of (undergraduate) convexity and I have recently come across a new definition which generalizes to the classic concept of convex function and is the following. Let be $I\subset \...
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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 ...
MANJUNATHA M R's user avatar
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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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Log-convexity of a function defined by an integral (Normal Mills ratio)

Let's define $f(x)$, for all $x>0$ by : $$f(x)=e^{x^2/2}\int_x^{+\infty}e^{-t^2/2}dt$$ I would like to prove that $f$ is log-convex, which is equivalent to the following condition : $$\forall x>...
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Confused of one step of proofing convexity of a function related to LP

I read about this question about proving that the function of the optimal value of an LP is concave. Proof that $\xi^*(b)$ is concave for arbitrary b in a linear program @Max who asked this question ...
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Convex and Preinvex functions

While I'm reading some research papers on convexity and preinvex functions (which is a generalization of convexity) I noticed that sometimes authors assume that the domain of a convex (preinvex) ...
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Prove $\sqrt{\frac{a+4bc}{4a+bc}}+\sqrt{\frac{b+4ca}{4b+ac}}+\sqrt{\frac{c+4ab}{4c+ab}}\ge 3,$ when $a+b+c=3.$

Problem. If $a,b,c\ge 0: ab+bc+ca>0$ and $a+b+c=3,$ prove that$$\sqrt{\frac{a+4bc}{4a+bc}}+\sqrt{\frac{b+4ca}{4b+ac}}+\sqrt{\frac{c+4ab}{4c+ab}}\ge 3.$$ It was here. Equality holds at $a=b=c=1$ ...
TATA box's user avatar
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Convergence of blackbox sequence

I have a sequence $x_0 = 0$, $x_{n+1} = f(x_n)$ and I know that: $f(x)$ is stricty convex on $[0,1]$ $f(x)$ is increasing on $[0,1]$ $f(1) = 1$ if need be, I can further assume that $f(x)$ is ...
Peter Paul's user avatar
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Proving that a given set defined by an inequality is convex

I want to prove that the convex hull of a region defined by the following inequality is convex: $$(y-z)^2 -4x w \geq 0$$, where $x + y + z + w = 1$ and $x, y, z, w \geq 0$. I tried using the Hessian ...
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Proof a set is affine, a cone, a convex set

I just met a problem when proving if a set is affine, a cone or a convex set. $S = \{a∈R^2 | a_1x_1+a_2x_2≤2, x_1^2+x_2^2≤1,∀x∈R^2\}$. Following the definitions, to see if it is affine, I have to ...
asdjkopoi's user avatar
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Convex order stochastic dominance for nonnegative integer valued random variables

Let $X$ and $Y$ be nonnegative integer valued random variables (with same mean). It is customary to define convex order stochastic dominance for such variables (denoted $X<Y$) if $\sum_{j\geq n}\...
xyz's user avatar
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For $x+y+z=3,$ prove $\frac{1}{x^2+4}+\frac{1}{y^2+4}+\frac{1}{z^2+4}\le \frac{3}{5}.$

Let $x,y,z\ge 0: x+y+z=3.$ Prove that$$\frac{1}{x^2+4}+\frac{1}{y^2+4}+\frac{1}{z^2+4}\le \frac{3}{5}.$$ Here is just my thought progress. I set $0\le xy+yz+zx=q\le 3; 0\le r=xyz\le 1.$ After full ...
Anonymous's user avatar
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If $a,b,c >0 : ab+bc+ca=3,$ find maximal value $\sum\dfrac{a\sqrt{a^2+2}}{a^2+3}$

Question If $a,b,c >0 : ab+bc+ca=3,$ find maximal value $$M=\dfrac{a\sqrt{a^2+2}}{a^2+3}+\dfrac{b\sqrt{b^2+2}}{b^2+3}+\dfrac{c\sqrt{c^2+2}}{c^2+3}$$ By $a=b=c=1,$ I try prove $M\le \dfrac{3\sqrt{3}}...
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inequality with sum and product

Let $(x_1,\ldots,x_n) \in (\mathbb{R}_+)^n$. How can I prove that $$2\max(x_1,\ldots,x_n)\left(\frac{1}{n}\sum_1^n x_k - \prod_1^n x_k^{1/n}\right) \ge \frac{1}{n}\sum_1^n\left(x_k - \prod_1^n x_j^{1/...
Eric 's user avatar
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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 \begin{equation} f ( (1- \lambda) a + \lambda b ) \...
herbhofsterd's user avatar
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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 (...
Marlon Brando's user avatar
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Prove by using convexity: $33\sin33^{\circ}+29\sin29^{\circ}+28\sin28^{\circ}>45$

Without using convexity (and without using calculator, of course) we can make the following. Since, $33+29+28=90$, we need to prove that: $$33(\sin33^{\circ}-\sin30^{\circ})>29(\sin30^{\circ}-\...
Michael Rozenberg's user avatar
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How to prove $\left(\frac{e^n-1}{n}\right)^{2n+1} \leq \frac{(e-1)(e^2-1)...(e^{2n}-1)}{(2n)!}$?

I am trying to prove the following inequality: $$\left(\frac{e^n-1}{n}\right)^{2n+1} \leq \frac{(e-1)(e^2-1)...(e^{2n}-1)}{(2n)!}$$ For the base case $n=1$ $$0 \leq e-1$$ Lets assume the inequality ...
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Difference between majorization and weak majorization.

I am trying to understand the concept of weak majorization and majorization. Following the definitions in the book of Marshall et. al I have formulated the following reasoning. I suppose I am wrong ...
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A question on a derivation from Hajek's book.

Concerning this draft here: http://hajek.ece.illinois.edu/Papers/randomprocJuly14.pdf on page 71 in the proof of Proposition 2.17, the inequalities (2.8) and (2.9) I don't understand how to derive ...
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Can we rewrite a nonlinear constraint as a multiplication of real and integer as convex?

I am having quite an issue with an optimization that I need to solve. I have the constraint of the form r1 == r2*i1 where r1 ...
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Does the $\log$-$\exp$ analogous of Minkowski integral inequality hold true?

Suppose that $X$ is a $\mathcal{X}$-valued random variable and $Y$ is a $\mathcal{Y}$-valued random variable. Assume that $X$ and $Y$ are independent of each other. Suppose that $f: \mathcal{X} \times ...
Bob's user avatar
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Proving that $2\left(a+b\right)\ln{\left(\frac{a+b}{2}\right)}\geq\left(a+1\right)\ln{\left(a\right)}+\left(b+1\right)\ln{\left(b\right)}$ [duplicate]

Prove: $2\left(a+b\right)\ln{\left(\frac{a+b}{2}\right)}\geq\left(a+1\right)\ln{\left(a\right)}+\left(b+1\right)\ln{\left(b\right)}$ Graph of the two-variable function A diagram was drawn to show ...
Barry Alen's user avatar
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3 answers
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Prove that $ (a+b)^{2n} \leq 2^{2n-1}(a^{2n}+b^{2n})$ [duplicate]

I have to prove that for all $(a;b) \in \mathbb{R}^2$, and for all $n \in \mathbb{N}$ we have: $$ (a+b)^{2n} \leq 2^{2n-1}(a^{2n}+b^{2n})$$ without using induction. I tried to use the convexity of $x^...
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"Flower" petal set in convex analysis: when it is non empty and show that is convex

I am trying to solve an exercise of convex analysis. I think it's easy but I am already stuck. Let $a,b \in R^n$ with $a \neq b$. For any $\delta \in (0, +\infty)$ we define the flower petal ...
NotNow11's user avatar
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$3$-var inequality: $\frac{bc}{\sqrt{a}+3}+\frac{ca}{\sqrt{b}+3}+\frac{ab}{\sqrt{c}+3} \leq \frac{3}{4}$ for $a+b+c=3$.

Problem: Let $a,b,c$ be positive numbers satisfied $a+b+c=3$. Prove that $$\dfrac{bc}{\sqrt{a}+3}+\dfrac{ca}{\sqrt{b}+3}+\dfrac{ab}{\sqrt{c}+3} \leq \dfrac{3}{4}$$ I've tried U.C.T method but it doesn'...
ilovemath's user avatar
2 votes
0 answers
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Strong-convexity parameter dimension dependency for $1/2$-Tsallis-entropy with respect to $\|\cdot\|_1$ on the unit simplex.

Let $K \in \mathbb{N}$ and let $f:(0,1)^K \to \mathbb{R}$ be the function $x \mapsto - 2 \sum_{k=1}^K \sqrt{x_k}$ (that, apart from constants, it is the $1/2$-Tsallis entropy). I'm trying to figure ...
Bob's user avatar
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Upper bound on integer coordinates on a function graph

Let $n\geq 2$ and $f\in \mathcal{C}^2([|1,n|],\mathbb{R})$ such that $|f’|$ has a maximum strictly less than $1$ and $f’’$ has a minimum strictly positive. Show that the number of integer coordinates ...
brandon's user avatar
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1 answer
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Magnitude Of Spherical Simplex Centroid Is Decreasing

Let $\sigma$ be the uniform measure on $\mathbb{S}^{d-1}\subset \mathbb{R}^d$. For any region $R\subset \mathbb{S}^{d-1}$, let $X_R$ be a random variable which is uniformly distributed across $R$. We ...
Aaron Goldsmith's user avatar
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is the mix of convex and linear functions always convex function?

I want to prove that the following composed function $g \circ L$ is always (strictly) convex : \begin{alignat*}{3} &g&&(t&&) && =-\log(1-e^{-t}) \qquad && \text{...
Artashes's user avatar
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2 answers
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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 ...
Aishgadol's user avatar
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1 answer
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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 ...
Bastien Tourand's user avatar
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1 answer
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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 : $$f''(x)\ge 0$$ ...
Miss and Mister cassoulet char's user avatar
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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-...
user377704's user avatar
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1 answer
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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 ...
Mathematics_Beginner's user avatar
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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 ...
Mathematics_Beginner's user avatar
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1 answer
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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 ...
Felix Wilde's user avatar
-5 votes
1 answer
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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
MOUNO ANIMATIONS's user avatar
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1 answer
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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 $f$ ...
emil agazade's user avatar
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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 \...
saea's user avatar
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2 votes
0 answers
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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 ...
Neox's user avatar
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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 ...
Padawan's user avatar
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2 answers
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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 ...
curiouscupcake's user avatar
2 votes
1 answer
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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 ...
ball_jan's user avatar
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4 votes
0 answers
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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 ...
Miss and Mister cassoulet char's user avatar
1 vote
2 answers
167 views

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. $abc=1$ ...
SUWG's user avatar
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20 votes
3 answers
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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 ...
Miss and Mister cassoulet char's user avatar