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Questions tagged [convexity-inequality]

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

3
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2answers
110 views

A constraint that implies convexity

Let $f:\mathbb{R} \to \mathbb{R} $ be a function such that $\forall x<y, \exists z\in(x, y) $ with $(y-x) f(z) \le (y-z) f(x) +(z-x) f(y) $. a) Give an example of a non-convex function $f$ ...
1
vote
1answer
23 views

greatest value of function depends on parameter $k$

Find the greatest value of the function $f(x)=x^4-6kx^2+k^2$on the interval $[-2,1]$ depending on the parameter $k$ My Try: $$f(x)=x^4-6kx^2+9k^2-8k^2$$ $$f(x)=(x^2-k)^2-8k^2$$ from $-2 \...
0
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1answer
8 views

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 ...
6
votes
3answers
243 views

find the maximum and minimum of $\sum_{i=1}^{n} (10x^3_{i}-9x^5_{i})$

Let $x_{i}\ge 0$ such that $$x_{1}+x_{2}+\cdots+x_{n}=1.$$ Find the maximum and minimum of $$f=10\sum_{i=1}^{n}x^3_{i}-9\sum_{i=1}^{n}x^5_{i}.$$ I have proved $n=2$ $$1\le f\le\dfrac{9}{4}$$ see:...
0
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0answers
39 views

Prove inequality that is used to prove convexity

I'm looking for the proof of an inequality such that I can prove the problem below (this is not the problem I need an answer to, it is the inequality later on that I don't get). Problem $$\text{Let } ...
2
votes
1answer
51 views

How to prove this property using convexity?

Suppose that $f:[a,b]\to\mathbb{R}$ be a twice-differentiable function, and that there exists $c\in(a,b)$ such that $f'(c)=\frac{f(b)-f(a)}{b-a}$. Show that if $f''(x)>0$ for all $x\in[a,b]$ ...
2
votes
1answer
63 views

proving inequality through convexity and continuity

Suppose $f$ is a continuous function. Prove $f$ is convex iff this inequality holds: $$ \frac{f(x)+f(y)+f(z)}{3} + f(\frac{x+y+z}{3}) \ge \frac{2}{3}[f(\frac{x+y}{2}) +f(\frac{y+z}{2}) +f(\frac{z+x}{2}...
0
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0answers
33 views

Convexity of $x^{2q}$ implies $\mathbb{E} \big[ \left(X-X^{'}\right)^{2q} \big]\leq 2^{2q-1} \left( \mathbb{E}X^{2q}+ \mathbb{E}X^{'2q} \right) $

This problem is from page 25 of concentration inequalities a nonasymptotic theory of independence $X$ is a random variable and $X^{'}$ is independent copy of $X$ q is integer and $q\geq1$ It is ...
1
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0answers
26 views

Inequality involving Weibull distribution

define R(x) as the weibull survival function $R_1(t)=e^{-\alpha_1 t^{\beta_1}}$ $R_2(t)=e^{-\alpha_2 t^{\beta_2}}$ with $\alpha_1, \alpha_2 > 0 $ and $\beta_1, \beta_2 >1 $ $\phi_1(t)=\int_{...
0
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0answers
30 views

Non-convex objective function + (non-convex constraint function vs. convex constraint function)

In the optimization problem, both objective function and the constraint functions are non-convex. (topology optimization) - objective function: force on structure - constraint functions: material ...
-1
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1answer
20 views

Proof of Convexity by Contradiciton

I have a long and complicated function $f(T)$ whose derivative is not easy to derive. So, I want to prove that $f(T)$ is a convex function over the interval $(0,\infty)$ . I think it is easier by ...
0
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2answers
67 views

Showing that $x \mapsto |x|^p$ is strictly mid-point convex for $2 \leq p < \infty$ [duplicate]

Let $2 \leq p < \infty$ and consider the function $f : \mathbb{R}^n \to \mathbb{R}$ defined by $$f(x) := |x|^p.$$ Then this function is mid-point convex (in fact strictly), i.e. we have that $$f\...
-1
votes
2answers
65 views

Proving $\sin 2x \geq x$, for $x \in \left[0, \frac{\pi}{4}\right]$

I want to prove the following inequality: $$\sin 2x \geq x \;\;\text{for}\;\; x \in \left[0, \frac{\pi}{4}\right]$$ I know that $\sin 2x = 2\sin x\cos x$, and I tried to use the Taylor series, ...
1
vote
1answer
80 views

Convexity of a function (volume-to-area ratio)

I am working on some estimates on the sphere and I come up with the function $h(t):=\frac{\int_0^t \sin^{m}r \,dr}{\sin ^m t}$ on $(0, \pi)$, where $m\ge 1$ is an integer. (This is the volume-to-...
0
votes
1answer
45 views

If $f$ is continuous and $f(\frac{x+y}{2})\leq \frac{f(x)+f(y)}2$ for all $x , y \in (a,b)$, prove that $f$ is convex. [duplicate]

I know a proof which starts with proving the convexity for "dyadic rational" numbers. I would like to know if someone has some other ideas.
-1
votes
3answers
82 views

Maximum value of $\frac{x}{y} + \frac{y}{z} + \frac{z}{x}$

Given $1 \leq x, y, z \leq 3$, then maximum value of $\dfrac{x}{y} + \dfrac{y}{z} + \dfrac{z}{x}$ is ? I failed with trying using AM-GM. Can any one show me a way to solve the problem, thank you
1
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0answers
34 views

Mid-point convexity and convexity [duplicate]

I'm trying to show that a continuous function $f:(a,b) \to \mathbb{R}$ with the mid-point convexity property is convex. Midpoint convexity is $f((x+y)/2) \leq \frac{1}{2}f(x) + \frac{1}{2} f(y)$. ...
0
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0answers
50 views

How to show that if $f$ is a convex function, then the limit $\underset{t \rightarrow 0^+}{lim}\frac{f(x+ty)-f(x)}{t}$ exists?

Let $E$ be a Banach space, and $U \subset E$ an open non empty set. If $f:U\rightarrow \mathbb{R}$ is a convex function, show that this limist exists: $$\partial _y^+f(x) = \underset{t \rightarrow 0^...
0
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0answers
32 views

How to tranform the quasilinear constraint into convex or affine constraint?

I have the following problem: $\max_{z_{i,j,v}} \sum_{i}\sum_{j}\sum_{v} z_{i,j,v}$ $s.t. \sum_{i}\sum_{j} z_{i,j,v} = \max_{i,j} z_{i,j,v}$ $z_{i,j,v}\in[0,1]$ I can prove the constraint is ...
5
votes
1answer
103 views

Proving $\frac{B_1B_2}{A_1A_3}+ \frac{B_2B_3}{A_2A_4}+…+\frac{B_nB_1}{A_nA_2}>1$

Let $A_1A_2 . . . A_n$ be a cyclic convex polygon whose circumcenter is strictly in its interior. Let $B_1, B_2, ..., B_n$ be arbitrary points on the sides $A_1A_2, A_2A_3, ..., A_nA_1$, respectively,...
1
vote
2answers
91 views

how to maximize $f(x,y)=\frac{x+y-2}{xy}$?

How to maximize $f(x,y)=\frac{x+y-2}{xy}$ where $x,y \in \{1,2,\ldots,n\}$? It seems that maximum will occur when $(x,y)=(1,n)$ or $(n,1).$
1
vote
1answer
50 views

Concave function - Generic proof

For any concave function we have: $$\frac{f(0)+f(x)}{2} \leq f\left(\frac{0+x}{2}\right) \Leftrightarrow$$ for any $f(0) \geq 0$, it follows: $$f(x) \leq 2f\left(\frac{x}{2} \right).$$ How can I ...
1
vote
1answer
52 views

In resticted domain , Applying the Cauchy-Schwarz's inequality

I see this problem a few days ago. $$ a, b, c \in [\alpha , \beta] $$ prove that $$ 9 \le (a+b+c)\biggl( \frac {1}{a} + \frac{1} {b} + \frac{1} {c} \biggr)\le \frac{(2 \alpha + \beta )(\alpha + 2\...
0
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0answers
134 views

Estimates for logarithmic integrals

We have Hermite-Hadamard integral inequality which gives upper and lower bounds for integrals of convex functions. My question: Is there any theorem which gives more powerful estimates than Hermite-...
0
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0answers
73 views

Show that $f$ is convex in $I$ such that $f(\frac{x+y}{2})<\frac{1}{2}(f(x)+f(y))$ [duplicate]

$f$ is continuous in $I$ Show that if $f$ satisfies the condition $f(\frac{x+y}{2})<\frac{1}{2}(f(x)+f(y))$ so it's convex What I have to show here is $f(k_1y_1+k_2y_2+..k_ny_n)<k_1f(y_1)+...
0
votes
2answers
157 views

Prove that the function f(x) = (x-3)^2 is convex on [1,5]

A function f is defined to be convex on the closed interval $[1,5]$ if and only if $f(t+(1-t)5) \le tf(1) + (1-t)f(5)$ for any $t$ between $0$ and $1$ inclusive $(0\le t\le 1).$ Please help me prove ...
-1
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1answer
64 views

Does strict quasiconvexity imply continuity?

Strict quasiconvexity definition: $f(tx + (1-t)y) < \max \{f(x), f(y)\}$ for any $x,y$ in the domain and $t \in (0,1)$
0
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1answer
45 views

Convexity with conditional probabilities

Consider the random variables $X$, $Y$, and $Z$ supported on $\mathbb{R}^3$ and with joint distribution $P_{XYZ}$. Can I say anything about $P_{XY|Z}(x,y|z)\log P_{XY|Z}(x,y|z)$ being convex or ...
0
votes
2answers
116 views

show this inequality $3^{\frac{n}{4}}\cdot (3^{\frac{1}{4}}-1)^{8-n}\le 1+n$

let $n\in Z$,and $n\in[0,8]$,show that $$3^{\frac{n}{4}}\cdot (3^{\frac{1}{4}}-1)^{8-n}\le 1+n$$ it seem use Jenson inequality,But How to use it?Thanks.
8
votes
1answer
197 views

What is maximum of $\frac{x^2+y^2+z^2}{xy+xz+yz}$ when $x, y, z \in [1, 2]$?

If we have real numbers $x, y, z \in [1, 2]$ then what is the maximum of $$\frac{x^2+y^2+z^2}{xy+xz+yz}$$ I tried to use substitution $x=\frac{3+\sin X}{2}$, $y=\frac{3+\sin Y}{2}$ and $z=\frac{3+\...
1
vote
4answers
168 views

Inequality for positive real numbers less than $1$: $8(abcd+1)>(a+1)(b+1)(c+1)(d+1)$ [closed]

If $a,b,c,d$ are positive real numbers, each less than 1, prove that the following inequality holds: $$8(abcd+1)>(a+1)(b+1)(c+1)(d+1).$$ I tried using $\text{AM} > \text{GM}$, but I could not ...
4
votes
1answer
63 views

How can I show that $\left|\sin \frac{s}{2}\right| \geq \frac{|s|}{\pi}, s \in [- \pi , \pi]$?

How can I show that $$\left|\sin \frac{s}{2}\right| \geq \frac{|s|}{\pi}$$ $s \in [- \pi , \pi]$, using that $\psi : x \mapsto \sin x$ is a concave function on $[0 , \pi]$? By definition of concave ...
14
votes
2answers
741 views

find the maximum $\frac{\frac{x^2_{1}}{x_{2}}+\frac{x^2_{2}}{x_{3}}+\cdots+\frac{x^2_{n-1}}{x_{n}}+\frac{x^2_{n}}{x_{1}}}{x_{1}+x_{2}+\cdots+x_{n}}$

give the postive intger $n\ge 2$,and postive real numbers $a<b$ if the real numbers such $x_{1},x_{2},\cdots,x_{n}\in[a,b]$ find the maximum of the value $$\dfrac{\frac{x^2_{1}}{x_{2}}+\frac{x^2_{2}...
3
votes
1answer
108 views

Find the minimum value of $\sqrt{x+y}+\sqrt{(1-x)+y}+\sqrt{x+(1-y)}+\sqrt{(1-x)+(1-y)}$

Given that $x,y\in[0,1]$, find the minimum of the following $$f(x,y)=\sqrt{x+y}+\sqrt{(1-x)+y}+\sqrt{x+(1-y)}+\sqrt{(1-x)+(1-y)}$$ I have found $$f(0,0)=f(0,1)=f(1,0)=f(1,1)=2+\sqrt{2}, f(\dfrac{1}{2}...
1
vote
1answer
52 views

Prove the inequality $\frac{1}{1+a+b}+\frac{1}{1+c+b}+\frac{1}{1+a+c}+a+b+c \le 3 + \frac 13 (ab+bc+ac)$

Let $a,b,c \in [0;1]$. Prove that $$\frac{1}{1+a+b}+\frac{1}{1+c+b}+\frac{1}{1+a+c}+a+b+c \le 3 + \frac 13 (ab+bc+ac).$$ I used Buffalo way. Can anyone offer an easier way
8
votes
1answer
234 views

Prove inequality $\frac a{1+bc}+\frac b{1+ac}+\frac c{1+ab}+abc\le \frac52$

For real numbers $a,b,c \in [0,1]$ prove inequality $$\frac a{1+bc}+\frac b{1+ac}+\frac c{1+ab}+abc\le \frac52$$ I tried AM-GM, Buffalo way. I do not know how to solve this problem
3
votes
8answers
15k views

The minimum value of $ f(x) = | x - 1 | + | x - 2 | + | x - 3 | $ is? [duplicate]

I don't get it why my solution is wrong : My solving : $ f(x) = | x - 1 | + | x - 2 | + | x - 3 | $ When $ x\leq 1 $ $ f(x) = | x - 1 | + | x - 2 | + | x - 3 | = 0 $ = $ 6 -3x $ since $ x\...
0
votes
2answers
124 views

If $0\le a,b,c\le 1$, then $\frac a{1+bc}+\frac b{1+ac}+\frac c{1+ab}\le 2$

If a,b,c are real numbers such that $0\le a,b,c\le 1$, then $$\frac a{1+bc}+\frac b{1+ac}+\frac c{1+ab}\le 2$$ I did the following :- Using Jensen's inequality, take $f(x)=\frac {x^2}{x+p},(p=abc)$ ...