Articles of functional inequalities

Show that $|\sin{a}-\sin{b}| \le |a-b| $ for all $a$ and $b$

I’ve recently been going over the mean value and intermediate value theorems, however I’m not sure where to start on this.

Find function such that $\sum y_i\le\sum x_i\Rightarrow\sum f(y_i)\le\sum f(x_i)$

What kind of a function $f$ must be to satisfy the following? If $\sum_{i=1}^{n} y_i \leq \sum_{j=1}^{n} x_j$, where $x_j, y_i \in [0,1],\forall i,j$ then $$ \sum_{i=1}^{n} f(y_i) \leq \sum_{j=1}^{n} f(x_j).$$ Any help would be appreciated. Thanks in advance! Preferably $f$ must be convex and increasing. $f$ is linear from the answer given by the […]

$\frac{1}{2}(\frac{b_1}{a_1}-\frac{b_n}{a_n})^2(\sum_{1}^{n}{a_i^2 }) ^2 \ge (\sum_{1}^{n}{a_i^2 }) (\sum_{1}^{n}{b_i^2 })-(\sum_{1}^{n}{a_ib_i })^2$

Let $a_1, a_2,….,a_n, b_1, b_2,…,b_n$, let $\frac{b_1}{a_1} = max \{\frac{b_i}{a_i}, i=1,2, \cdots n \}$ , $\frac{b_n}{a_n} = min \{\frac{b_i}{a_i}, i=1,2, \cdots n \}$ show that: $$\frac{1}{2}(\frac{b_1}{a_1}-\frac{b_n}{a_n})^2(\sum_{1}^{n}{a_i^2 }) ^2 \ge (\sum_{1}^{n}{a_i^2 }) (\sum_{1}^{n}{b_i^2 })-(\sum_{1}^{n}{a_ib_i })^2$$

Is the derivative of a function bigger or equal to $e^x$ will always be bigger or equal to the function ?!

It seems to be the case, but i don’t have a proof. Given the function $f$ such that $f(x) \geq e^x$, is it true that $f'(x) \geq f(x)$?! I was experimenting with wolfram and it appears that $\frac{f'(x)}{f(x)} \geq 1$ whenever $f$ is bigger or equal to $\exp(x)$. Note : as suggested in the comments, […]

If $xy+xz+yz=3$ so $\sum\limits_{cyc}\left(x^2y+x^2z+2\sqrt{xyz(x^3+3x)}\right)\geq2xyz\sum\limits_{cyc}(x^2+2)$

Prove that for any set of three positive real $x, y, z$ such that $xy+yz+zx=3$ $x^2(y+z)+y^2(x+z)+z^2(x+y)+2\sqrt {xyz}\left(\sqrt{x^3+3x}+\sqrt{y^3+3x}+\sqrt{z^3+3x}\right)\ge$ $\ge 2xyz(x^2+y^2+z^2+6)$ Reasoning I know that the minimum value is with $x=y=z$ We have to find an equation equivalent to the date where we subtract and divide by xyz and don’t appear other variables.

Strictly Monotonic increasing, Symmetric ,doubling Function $F:\to $

*strong text****Is this symmetric, doubling,**strictly monotone increasing function $F:[0,1]\to[0,1]$ below, continuous and $F(x)=x$ over the unit interval? Let $F: [0,1]\to [0,1],F(1)=1,F(\frac{1}{2})=\frac{1}{2}\, F(0)=0\, \,:\text{be a strictly increasing Function}$ that in addition satisfies, $(2)$halving/doubling and $(3)$ symmetry, inverse symmetry; oddness/left relfection right symmetry around $\frac{1}{2};F(1-x)+F(x)=1$ . $$(1)F:[0,1]\to [0,1]\,,,F(1)=1,F(\frac{1}{2})=\frac{1}{2}, F(0)=0$$ $$(2).(x)∈[0,1];∀(n)∈Z;[F(\frac{1}{2^{n}}\times x)=\frac{1}{2^{n}}\times F(x)]$$ $$(3)\text{ (symmetric,and in-jective/bi-conditional symmetric)}\,\forall (x,y)\in\text{dom(F)=[0,1]}\,[x+y=1] […]

Show that $|f'(x)| \le \frac{2M_0}{h} +\frac{hM_2}{2}$ and $M_1 \le 2\sqrt{M_0M_2}$

Let $f$ be a twice derivable function and $M_i =\sup_{x \in \mathbb{R}} |f^{(i)}(x)|$ and $|M_0|, |M_2|<\infty $. Preferably using the Taylor series on the interval $[x,x+h]$ show the following inequalities: (a) $$|f'(x)| \le \frac{2M_0}{h} +\frac{hM_2}{2}$$ and (b) $$M_1 \le 2\sqrt{M_0M_2}$$ Any hints and/or a solution would be appreciated. P.S. I know that $(b) => (a)$ […]

Are there (known) bounds to the following arithmetic / number-theoretic expression?

I apologize in advance if this is something that is already well-known in the literature, but I would like to ask nonetheless (for the benefit of those who likewise do not know): Are there (known) lower and upper bounds to the following arithmetic / number-theoretic expression: $$\frac{I(x^2)}{I(x)} = \frac{\frac{\sigma_1(x^2)}{x^2}}{\frac{\sigma_1(x)}{x}}$$ where $x \in \mathbb{N}$, $\sigma_1(x)$ is […]

Does $1 + \frac{1}{x} + \frac{1}{x^2}$ have a global minimum, where $x \in \mathbb{R}$?

Does the function $$f(x) = 1 + \frac{1}{x} + \frac{1}{x^2},$$ where $x \in {\mathbb{R} \setminus \{0\}}$, have a global minimum? I tried asking WolframAlpha, but it appears to give an inconsistent result.

Question about equivalent norms on $W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega)$.

Assume $\mathbb{‎‎H}=W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega)$ with the norm induced from inner product $$\langle u,v\rangle_{‎\mathbb{‎‎H}}=\int_\Omega \Delta u \,\Delta v\, dx$$ for any $u \in W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega)$ it is well-known that we have inequality (Rellich inequality) $$ ‎\Lambda_N ‎\int_{\Omega}‎\frac{u^2}{|x|^4}\mathrm{d}x ‎\leq ‎\int_\Omega ‎|\Delta u|^2 \, ‎\mathrm{d}x‎ $$ where $‎\Lambda_N=(‎\frac{N^2(N-4)^2}{16})‎$ is optimal constant and also it is known that […]