# inequality

1. ### Inequality

Prove that, for x, y, z positive, \frac{3+x^4+y^3+z^2}{1+2x^3+3y^2+6z}+\frac{3+y^4+z^3+x^2}{1+2y^3+3z^2+6x}+\frac{3+z^4+x^3+y^2}{1+2z^3+3x^2+6y}>=\frac{3}{2}
2. ### inequality 2

Prove the following inequality ,where a,b,c are positive Nos: \frac{1}{b(a+b)}+\frac{1}{c(b+c)}+\frac{1}{a(c+a)}\geq\frac{27}{2(a+b+c)^2}
3. ### inequality

prove that: (a+b+c)\geq\frac{(\sqrt a+\sqrt b+\sqrt c)^2}{3} I tried by expanding the square
4. ### Inequality

$$\left| \sin \left( \sum_{k=1}^n x_k \right) \right| \le \sum_{k=1}^n \sin (x_k)$$ $\; \; \; \; \; (0 \le x_k \le \pi ; \; \; k=1,2,...,n)$

pls help:)
6. ### Cannot prove inequality

Prove that \: a_n =\frac{|\sin(n)|}{n} +\frac{|\sin(n+1)|}{n+1}>\frac{1}{6}\; ,n\in \mathbb{N}.
7. ### Inequality #2 without calculus

Given x^y < y^x , where x>y \geq \lambda >0, find the range of \lambda . Without calculus!
8. ### Prove inequality without calculus

Prove that a^b < b^a , for a>b\geq 3.
9. ### Hard inequality

Given a,b,c>=1 and a+b+c=9. Prove that (âˆša+âˆšb+âˆšc)^2>=ab+bc+ca.
10. ### Prove inequality

If 0<a_1 < a_2 < a_3 <... <a_n < \frac{\pi}{2} . Prove that \tan(a_1 ) <\frac{sin(a_1) +sin(a_2 ) +sin(a_3)+...+sin(a_n )}{cos(a_1 ) +cos(a_2 )+cos(a_3 )+...+cos(a_n )}<\tan(a_n ) .
11. ### Lotka-Volterra with periodic coefficients differential inequality

I would like to ask which transformation should I use to recive from this Lotka-Volterra inequality xâ€²=x(aâˆ’bxâˆ’cy)<x(aâˆ’bx), this differential form x(t)\leq \frac{x(T)e^{A(t)}}{1+x(T)\int \limits_{T}^{t}e^{A(s)}b(s)ds}, \text{where} A(t)=\int_T^t a(s)ds.
12. ### Probability inequality problem

Suppose A and B are independent and identically distributed. For the purposes of this problem, x and y are not capped at 1, but are definitely at least 0. Pr(A)<x Pr(B)<y Pr(AUB)<? I know Pr(AUB)<x+y, but is it possible to make the right side any lower? What is the lowest it can be given the...
13. ### Integral inequality

Compare e with : \int_{0}^{-\infty } e^{e^x } \cdot e^x dx .
14. ### Proof for a challenging inequality

I have something I believe to be true, but I'm uncertain, so I'm looking for a proof. For positive real numbers a,b,c,d Prove that if a>=b and c<=d, then a/c <= b/d
15. ### Inequality similiar to harmonic sum

For which values of n the inequality holds true : 1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n^4 }=\sum_{j=1}^{n^4 }j^{-1} >2n.
16. ### Prove inequality

Prove: e^{n^2} \left(1^1 \cdot 2^2 \cdot 3^3 \cdot ...\cdot n^n \right)\leq n^{n^2 + n + 6}.
17. ### Inequality with sqrt

Prove that \sqrt{1} +\sqrt{2} +...+\sqrt{n} \geq n.
18. ### Solve inequality

For which values of N : \; \sum_{j=1}^{N} j^{-1} >\frac{N}{\ln(N+1)}.
19. ### Prove inequality without calculator

Prove inequality: \sqrt{2} +\sqrt{3} >\pi .
20. ### Help : Deriving an inequality related to Lucas Sequence

Hi, I am studying a paper by Yann Bugeaud (click here), on page 13 there is an inequality (16) as given below- $\Lambda:=\left|\left(\frac{\alpha(\gamma-\delta)}{\gamma(\alpha-\beta)}\right)\left(\frac{\gamma^{s}}{\alpha^{r}}\right)^{-d}-1\right| \ll \alpha^{-\eta r d}$ which is obtained from...