Articles of functional inequalities

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 […]

Gradient Estimate – Question about Inequality vs. Equality sign in one part

$\DeclareMathOperator{\diam}{diam}\newcommand{\norm}[1]{\lVert#1\rVert}\newcommand{\abs}[1]{\lvert#1\rvert}$For $u \in C^{1}(\overline{\Omega})$, for $\Omega\subset \subset \mathbb{R^{n}}$ a bounded convex set, and for any $1 \leq p \leq q$ such that $\frac{1}{p}-\frac{1}{q}<\frac{1}{n}$, I need to show that $$ \norm{u-\overline{u}_{\Omega}}_{L^{q}} \leq c_{n} \left[ \frac{1+\frac{1}{q} – \frac{1}{p}}{\frac{1}{n}+\frac{1}{q}-\frac{1}{p}}\right]^{1+\frac{1}{q}-\frac{1}{p}} \cdot \frac{(\diam \Omega)^{n}}{\abs\Omega^{1-\frac{1}{n}+\frac{1}{p}-\frac{1}{q}}}\norm{Du}_{L^{p}(\Omega)}, $$ For context, this is a part (c) to a problem where in part (a), we […]

Increasing rearrangement and Hardy-Littlewood inequality

Don’t know how many of you guys are familiar with the theory of rearrangements, but I have a question for you about it. As you can see in Leoni (or in Lieb & Loss), the decreasing and the increasing rearrangement of a measurable function $u:\Omega \to [0,\infty[$ ($\Omega \subseteq \mathbb{R}^N$ measurable and bounded) are respectively […]

Prove that $\int_0^1|f''(x)|dx\ge4.$

Let $f$ be a $C^2$ function on $[0,1]$. $f(0)=f(1)=f'(0)=0,f'(1)=1.$ Prove that $\int_0^1|f”(x)|dx\ge4.$ Also determine all possible $f$ when equality occurs.

Weighted Poincare Inequality

I’m trying to prove a result I found in a paper, and I think I’m being a bit silly. The paper claims the following: By the Poincare inequality on the unit square $\Omega \subset \mathbb{R}^2$ we have that $$\int_{\Omega} f(x)^2 dx \leq C \int_{\Omega}|\nabla f|^2 dx + \left(\int_{\Omega}f(x)dx\right)^2.$$ Thus, if $w(x)$ is a weight satisfying […]

A unsolved puzzle from Number Theory/ Functional inequalities

The function $g:[0,1]\to[0,1]$ is continuously differentiable and increasing. Also, $g(0)=0$ and $g(1)=1$. Continuity and differentiability of higher orders can be assumed if necessary. The proposition on hand is the following: If for all integers $t>0$ and for all $r\in(0,1)$, $g(r^{t+1})>g(r)\cdot g(r^t)$, then for all $p,q\in(0,1)$, $g(pq)\geq g(p)g(q)$.

The Functional Inequality $f(x) \ge x+1$, $f(x)f(y)\le f(x+y)$

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function that satisfies the following conditons. $$f(x)f(y)\le f(x+y)$$ $$f(x)\ge x+1$$ What is $f(x)$? It is not to difficult to find that $f(0)=1$. If $f(x)$ is differentiable, we can further these results so that $f'(0)=1$, and $f(x)=f'(x)$. However, I was not able to go any further than this. […]