Mathematics

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

155 questions
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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 claim : $$g''(x)\geq 0$$ We have :…
Barackouda
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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…
math110
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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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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 : $$\forall x>0,f''(x)f(x)-f'(x)^2\ge0\tag{$\star$}$$ Simple…
Adren
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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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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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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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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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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 $\epsilon>0$…
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 $r_0<r_1$ be positive fixed constants. Let $$a<r_0<r_1<c<b, \tag{1}$$ and let $\lambda \in [0,1]$ satisfy $ \lambda a +(1-\lambda)b=c. $ Set $D(a,b,c)=\lambda…
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 is…
Asaf Shachar
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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 $\sqrt{\frac{a+4bc}{4a+bc}}+\sqrt{\frac{b+4ca}{4b+ac}}+\sqrt{\frac{c+4ab}{4c+ab}}\ge 3,$ 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 $a=b=c=1$ and $abc=0.$ I've tried to use AM-GM but the following…
Tran Ngoc Khuong Trang
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