Articles of inequality

If $N = q^k n^2$ is an odd perfect number with Euler prime $q$, and $k=1$, does it follow that $\frac{\sigma(n^2)}{n^2} \geq 2 – \frac{5}{3q}$?

Let $\sigma=\sigma_{1}$ be the classical sum-of-divisors function. If $N = q^k n^2$ is an odd perfect number with Euler prime $q$ (i.e., $q$ satisfies $q \equiv k \equiv 1 \pmod 4$), and $k=1$, does it follow that $$\frac{\sigma(n^2)}{n^2} \geq 2 – \frac{5}{3q}?$$ Note that, since the Euler prime $q$ satisfies $q \equiv 1 \pmod 4$, […]

Is $x^t$ subadditive for $t \in $?

I feel like I have seen this many times, that $(x+y)^t \leq x^t + y^t$ when $t \in [0,1]$, but I don’t remember ever proving it. I feel like it surely must be true. Can someone lead me in the right direction on this one?

An inequality of integrals

Let $f\in C^1([a,b])$ with $f(a)=0$. How can I show that there exists a positive constant $M$ independent of $f$ such that $\int^b_a|f(x)|^2dx\leq M\int^b_a|f^\prime(x)|^2dx$?

The ratio $\frac{u(z_2)}{u(z_1)}$ for positive harmonic functions is uniformly bounded on compact sets

I want to prove the following: If $E$ is a compact set in a region $\Omega \subset \mathbb C$, prove that there exists a constant $M$, depending only on $E$ and $\Omega$, such that every positive harmonic function $u(z)$ in $\Omega$ satisfies $u(z_2) \leq M u(z_1)$ for any two points $z_1, z_2 \in E$. This […]

Mean Power Inequality

Most of the proofs of mean power inequality are based on jensen’s inequality. Can the mean power inequality be prooved without use of one? Mean Power Inequality:

How prove this $|x_{p}-y_{q}|>0$

let $$x_{1}=\dfrac{1}{8},x_{n+1}=x_{n}+x^2_{n},y_{1}=\dfrac{1}{10},y_{n+1}=y_{n}+y^2_{n}$$ show that: for any $p,q\in N^{+}$ we have $$|x_{p}-y_{q}|>0$$

Prove inequality $n<3^n$ using mathematical induction

Prove that $n<3^n$ where $n \in \mathbb N$, when $ n=1 $, I have proved it’s true. And assumed when $n = p$ , $p<3^p$ is true. Can any body help me in showing that it is true for $n =p+1$

Minimum of $\frac{1}{x+y}+\frac{1}{x+z}-\frac{1}{x+y+z}$

Let $0\leq x,y,z\leq 1$. What is the minimum of $$F(x,y,z)=\frac{1}{x+y}+\frac{1}{x+z}-\frac{1}{x+y+z}?$$ We have $F(1,1,1)=2/3$, $F(1,1,0)=F(1,0,1)=F(1,0,0)=1$, and $F(0,1,1)=3/2$. Is the minimum $2/3$?

the solution set of $\left | \frac{2x – 3}{2x + 3} \right |< 1$

what is the solution set of $\left | \frac{2x – 3}{2x + 3} \right |< 1$ ? I solved it by first assuming: $-1 < \frac{2x – 3}{2x + 3 } < 1$ ended with: $x > 0 > -3/2$ Is that a correct approach? And how to derive the solution set from the last […]

Prove $\frac{a}{1+b^2}+\frac{b}{1+c^2}+\frac{c}{1+d^2}+\frac{d}{1+a^2} \ge 2$ if $a+b+c+d=4$

if $a,b,c,d$ are positive real numbers that $a+b+c+d=4$,Prove:$$\frac{a}{1+b^2}+\frac{b}{1+c^2}+\frac{c}{1+d^2}+\frac{d}{1+a^2} \ge 2$$ Additional info:I’m looking for solutions and hint that using Cauchy-Schwarz and AM-GM because I have background in them. Things I have done: Using AM-GM $$LHS=\sum\limits_{cyc}\left(a-\frac{ab^2}{1+b^2}\right)\ge \sum\limits_{cyc}\left(a-\frac{ab^2}{2b}\right)=\sum\limits_{cyc}a-\frac{1}{2}(ab+bc+cd+da)$$ So it lefts to Prove that $$\sum\limits_{cyc}a-\frac{1}{2}(ab+bc+cd+da)\ge 2$$ If I show that $ab+bc+cd+da \le4$ then the upper inequality will […]