Articles of inequality

Preserving the order of a sequence of real numbers

If a have a sequence of real numbers ordered from greatest to smallest $Y_o\geq Y_1 \geq…\geq Y_k$ and I divide each number of the sequence by $\delta<0$, is the order of the sequence flipped, i.e. $\frac{Y_0}{\delta}\leq \frac{Y_1}{\delta}\leq…\leq \frac{Y_k}{\delta}$?

Proving that $|x|^p,p \geq 1$ is convex

I want to show that $|x|^p,p \geq 1$ is convex, for this i have to prove the inequality $|(1-\lambda )x+\lambda y)|^p \leq (1-\lambda)|x|^p+\lambda |y|^p $ for $\lambda \in (0,1)$ Can anyone prove this inequality? I have proved the convexity using the composition of two convex functions, one of which was increasing but I am interested […]

Maximizing $\frac{x(1-f(x))}{3-f(x)}$

Let $f:[0,1]\rightarrow[0,1]$ be a nondecreasing function such that $f(0)=0$ and $f(1)=1$. Let $x_1\in[0,1]$ be the value maximizing $x(1-f(x))$. Let $x_2\in[0,1]$ be the value maximizing $\frac{x(1-f(x))}{3-f(x)}$. Is it always true that $x_1\leq x_2$? Example: $f(x)=x$. Then $x_1=0.5$ and $x_2=3-\sqrt{6}\approx 0.55$, so $x_2>x_1$. It seems to be true because the denominator $3-f(x)$ is a decreasing function in […]

If $9^{x+1} + (t^2 – 4t – 2)3^x + 1 > 0$, then what values can $t$ take?

If $9^{x+1} + (t^2 – 4t – 2)3^x + 1 > 0$, then what values can $t$ take? This is what I have done: Let $y = 3^x$ $$9^{x+1} + (t^2 – 4t – 2)3^x + 1 > 0$$ $$\implies9y^2 + (t^2 – 4t – 2)y + 1 > 0$$ For the LHS to be […]

Proving that one of $a(1-b), b(1-c), c(1-a) \le \frac{1}{4}$

how can a prove that at least one of those is less than or equal to 1/4. $$\forall a,b,c\in \mathbb R^+, \ a(1-b)\leq 1/4 \lor b(1-c) \leq 1/4 \lor c(1-a) \leq 1/4.$$ help please!

When does Schwarz inequality become an equality?

In Spivak Calculus you are asked to prove that in Schwarz inequality, equality holds only when $y_1 = y_2 = 0$ or when there is a number $\lambda$ such that $x_1 = \lambda y_1$ and $x_2 = \lambda y_2$. I can go from $x_1y_1 + x_2y_2 = \sqrt{x_1^2 + x_2^2} + \sqrt{y_1^2 + y_2^2}$ to […]

Suppose $(s_n)$ converges and that $s_n \geq a$ for all but finitely many terms, show $\lim s_n \geq a$

I’ve got a few questions about the problem. Prob :Suppose $(s_n)$ converges and that $s_n \geq a$ for all but finitely many terms, show $\lim s_n \geq a$ The solution here breaks this problem up into two parts. Q1. I don’t understand why is it necessary to consider the finitely many terms that $s_k < […]

How do I show that $\frac xy + \frac yz + \frac zx \ge 1 + \frac {z + x}{x + y} + \frac {x + y}{z + x}$?

Show that $$\frac xy + \frac yz + \frac zx \ge 1 + \frac {z + x}{x + y} + \frac {x + y}{z + x}$$ for $x, y, z \gt 0$. I observed that this is a homogeneous inequality so normalization might work. I tried to set $x = 1$ or $xyz = 1$ […]

Proof without induction of the inequalities related to product $\prod\limits_{k=1}^n \frac{2k-1}{2k}$

How do you prove the following without induction: 1)$\prod\limits_{k=1}^n\left(\frac{2k-1}{2k}\right)^{\frac{1}{n}}>\frac{1}{2}$ 2)$\prod\limits_{k=1}^n \frac{2k-1}{2k}<\frac{1}{\sqrt{2n+1}}$ 3)$\prod\limits_{k=1}^n2k-1<n^n$ I think AM-GM-HM inequality is the way, but am unable to proceed. Any ideas. Thanks beforehand.

An unusual symmetric inequality of trigonometric functions

Given $\sin^2\alpha+\sin^2\beta+\sin^2\gamma=2 $. I have to prove that $ \left| \begin{matrix} \cos\alpha & \cos\beta & \sin\gamma\\\sin\alpha & \cos\beta & \cos\gamma\\\cos\alpha & \sin\beta & \cos\gamma \end{matrix} \right| \leq 2\sqrt2 \sin\alpha \sin\beta \sin\gamma $. I decided to directly expand the determinant, the left becomes $ |2\cos \alpha\cos\beta\cos\gamma-\sin(\alpha+\beta+\gamma)| $. This is quite different from what I encountered before […]