Mathematics Stack Exchange
Mathematics Stack Exchange

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: $$\min_{[a,b]}f=\min(f(a),f(b))$$

130 questions
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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 $z=\frac{3+\sin Z}{2}$. But the expression became too messy.…
Dedaha
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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
Roman83
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Is this monotonicity property equivalent to convexity?

This is a follow-up of this question. Let $\psi:[0,\infty) \to [0,\infty)$ be a strictly increasing $C^2$ (or $C^{\infty}$) function, satisfying $\psi(0)=0$. Suppose that the function $f(r)=\psi'(r)+\frac{\psi(r)}{r}$ is non-increasing. Must…
Asaf Shachar
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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: wolfarma When $n=3$, How prove that…
math110
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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 $F|_{(1-\epsilon,1+\epsilon)}$ is strictly convex for some…
Asaf Shachar
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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}\Rightarrow d\left ( a+ b+ c \right )- d^{2}= d\left (…
user818748
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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
Asaf Shachar
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Is this property equivalent to convexity?

Let $\psi:[0,\infty) \to [0,\infty)$ be a strictly increasing $C^1$ function, satisfying $\psi(0)=0$. Suppose that for every $r>0$, $$\psi'(r)+\frac{\psi(r)}{r} \le 2\psi'(0). \tag{1}$$ Is it true that $\psi$ is concave? The converse statement…
Asaf Shachar
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Prove the inequality $\sum \limits_{k=1}^n \frac{k+1}{k} \cdot \sum \limits_{k=1}^n \frac{k}{k+1} \le \frac{9}{8}n^2$

Prove that for all $n \in \mathbb{N}$ the inequality $$\sum \limits_{k=1}^n \frac{k+1}{k} \cdot \sum \limits_{k=1}^n \frac{k}{k+1} \le \frac{9}{8}n^2$$ holds. My work. I proved this inequality, but my proof is ugly (it is necessary to check by…
Witold
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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, other than the vertices (that is With $B_i\neq…
Guy Fsone
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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 function, $$ \psi(t \, x + (1 - t) \, y) \geq t…
joseabp91
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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 following claim : $$f''(x)>0$$ Proof for the case $n=1$…
Erik Satie
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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-decreasing. I am interested in the concavity…
Anon
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