Questions on proving and manipulating inequalities.

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9
votes
2answers
590 views

Proof that $t-1-\log t \geq 0$ for $t > 0$

Using basic calculus, I can prove that $f(t)=t-1-\log t \geq 0$ for $t > 0$ by setting the first derivative to zero \begin{align} \frac{df}{dt} = 1 - 1/t = 0 \end{align} And so I have a critical ...
8
votes
1answer
250 views

If $f$ is a positive, monotone decreasing function, prove that $\int_0^1xf(x)^2dx \int_0^1f(x)dx\le \int_0^1f(x)^2dx \int_0^1xf(x)dx$

If $f$ is a positive, monotone decreasing function, prove that $\int_0^1xf(x)^2dx \int_0^1f(x)dx\le \int_0^1f(x)^2dx \int_0^1xf(x)dx$
8
votes
4answers
276 views

How many sequence of integers ($j_1 , j_2 , . . . , j_k$) are there such that $0 ≤ j_1 ≤ j_2 ≤ . . . ≤ j_k ≤ n$?

I need to solve the problem, How many sequence of integers ($j_1 , j_2 , . . . , j_k$) are there such that $0 ≤ j_1 ≤ j_2 ≤ . . . ≤ j_k ≤ n$? I've been given a hint, (Hint: Reduce the ...
7
votes
1answer
167 views

Prove inequality: $\sum \frac{a^4}{a^3+b^3} \ge \frac{a+b+c}{2}$

Prove inequality with $a,b,c >0$ $$\frac{a^4}{a^3+b^3} + \frac{b^4}{b^3+c^3}+\frac{c^4}{c^3+a^3} \ge \frac{a+b+c}{2}$$ I tried the inequality: $\sum \frac {a^4+b^4}{a^3+b^3} \ge \sum ...
6
votes
1answer
108 views

Solving an inequality : $n \geq 3$ , $n^{n} \lt (n!)^{2}$.

I proved this inequality in the following way: Lemma: $r \in \Bbb N, r \geq 3$. We have $r^r \gt (r+1)^{r-1}$. Proof: We apply the AM-GM inequality to the $r$ positive integers where there are ...
5
votes
1answer
93 views

Prove that $\|a\|+\|b\| + \|c\| + \|a+b+c\| \geq \|a+b\| + \|b+c\| + \|c +a\|$ in the plane.

Prove that $\|a\| + \|b\| + \|c\| + \|a+b+c\| \geq \|a+b\| + \|b+c\| + \|c +a\|$ in the plane. Gentle hints only, please! I know that attempting to decompose R.H.S. into $$\alpha a + \beta b + ...
5
votes
3answers
767 views

Derivatives of the Riemann zeta function at $s=0$

It's a curious fact that for $n>0$, $\zeta^{(n)}(0)\approx -n!$. Apostol gave a table for $\frac{\zeta^{(n)}(0)}{n!}$, among other results on $\zeta^{(n)}(0)$ . the sequence : $$\delta_{n}=\left | ...
5
votes
2answers
201 views

Inequality involving sides of a triangle

How can one show that for triangles of sides $a,b,c$ that $$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b} < 2$$ My proof is long winded, which is why I am posting the problem here. Step 1: let ...
4
votes
4answers
211 views

Working with proofs help?

I'm trying to study for my midterm and doing some random practise questions to work with proofs. However I'm stuck on, as the only way I know how to prove it is through plugging in numbers, however as ...
4
votes
1answer
899 views

Liapunov's Inequality for $L_p$ spaces

Let $1 \leq p,q < \infty$ and $0 \leq \lambda \leq 1$. If $r = \lambda p + (1 - \lambda)q$ and $f \in L_p \cap L_q $, then $$||f||_r^r \leq ||f||_p^{\lambda p} ||f||_q^{(1 - \lambda)q} \tag{*}$$ ...
3
votes
2answers
159 views

Find maximum of $P$

Let $$P = \frac{{{x^2}}}{{{x^2} + yz + x + 1}} + \frac{{y + z}}{{x + y + z + 1}} - \frac{{1 + yz}}{9}.$$ Find maximum of $P$ where $x, y,z$ are nonnegative real numbers such that ${x^2} + {y^2} + ...
3
votes
1answer
147 views

little inequality conjecture

proof or disproof for $n\geq2$ even and $x>0$ $$\sum\limits_{i=0}^{n}x^i\geq \frac{(1+2\sum\limits_{i=1}^{ \frac{n}{2} }x^i)^2}{ \frac{n}{2} (x+1)+1}$$ I came up with this little inequality while ...
3
votes
2answers
154 views

Prove that $\frac{\int_0^1xf^2(x) \mathrm{d}x}{\int_0^1 xf(x) \mathrm{d}x}\le\frac{\int_0^1 f^2(x) \mathrm{d}x}{\int_0^1 f(x) \mathrm{d}x}$ [duplicate]

Let $f:[0,1]\rightarrow\mathbb{R_+}$ be a monotone decreasing function. We want to prove that $$\frac{\int_0^1x(f(x))^2 \,\mathrm{d}x}{\int_0^1 xf(x) \,\mathrm{d}x}\le\frac{\int_0^1 (f(x))^2 ...
2
votes
2answers
252 views

Mean Value Theorem: Real Analysis

I need to show that $\dfrac{2}{\pi}<\dfrac{\sin(x)}{x}<1$ for $0<x<\dfrac{\pi}{2}$. I know I need to use the mean value theorem, would I just say that since $f$ is continuous in the ...
2
votes
2answers
543 views

A three variable inequality

Given that $a, b, c > 0$ and $a^2 + b^2 + c^2 = 2$, what is the maximum value of $(a^5 + b^5)(a^5 + c^5)(b^5 + c^5)$? Normally when I encounter a problem like this, I seem to be able to push ...
1
vote
2answers
67 views

Comparison theorem for ODE

Here is something I'm trying to prove: Conjecture: Suppose $f'(x) \leq \phi(f(x), x)$ and $f(a)=\alpha$. Suppose $g'(x)=\phi(g(x),x)$ and $g(a)\geq \alpha$. Then $f(x)\leq g(x)\,\,\forall x$. ...
1
vote
4answers
457 views

Problems with Inequalities

It seems like I am facing some confusion while handling with inequalities,I was doing some tasks where it is asked to find the interval of the variable,after some steps I deduced the the necessary ...
0
votes
1answer
58 views

Does $\neg(x > y)$ imply that $y \geq x$?

Given any arbitrary binary relation $\geq$ defined on some set $S$, we define a new binary relation $>$ on $S$ by: $$ x > y \quad\text{iff}\quad (x \geq y) \wedge \neg(y \geq x) $$ In accordance ...
8
votes
3answers
141 views

Proving the inequality $4\ge a^2b+b^2c+c^2a+abc$

So, a,b,c are non-negative real numbers for which holds that $a+b+c=3$. Prove the following inequality: $$4\ge a^2b+b^2c+c^2a+abc$$ For now I have only tried to write the inequality as ...
7
votes
2answers
251 views

How prove this Stronger AM-GM inequality $\frac{n^2-1}{6}\min_{1\le i<j\le n}\left(\sqrt{a_{i}}-\sqrt{a_{j}}\right)^2\le A_{n}-G_{n}$

let $a_{i}>0,i=1,2,\cdots,n,n\ge 3$,show that $$\dfrac{n^2-1}{6}\min_{1\le i<j\le n}\left(\sqrt{a_{i}}-\sqrt{a_{j}}\right)^2\le\dfrac{a_{1}+a_{2}+\cdots+a_{n}}{n}-\sqrt[n]{a_{1}a_{2}\cdots ...
7
votes
1answer
144 views

Inequality with four positive integers looking for upper bound

Umm. This comes from Diophantine quartic equation in four variables and will finish the most important part if it can be done. Four positive integers $w,x,y,z.$ One equation and two inequalities $$ ...
7
votes
1answer
387 views

Showing that $\log(\log(N+1)) \leq 1+\sum\limits_{p \leq N} \frac{1}{p}$

I can't see how you get this. I want to show that $$\log(\log(N+1)) \leq \sum_{p \leq N} \frac{1}{p}+1$$ Can't see how it follows from this. So you show that $$0 \lt -\log(1-x)-x \lt ...
7
votes
4answers
437 views

Inequality with central binomial coefficients

For every even positive number $N$ we have: $$ {2N \choose N } < 2^N {N \choose N/2 } < 2 {2N \choose N } $$ (Furthermore, $\frac{2^N {N \choose N/2 }}{{2N \choose N }} \to \sqrt{2} $ for ...
7
votes
5answers
818 views

Cauchy-Schwarz inequality and three-letter identities (exercise 1.4 from “The Cauchy-Schwarz Master Class”)

Exercise 1.4 from a great book The Cauchy-Schwarz Master Class asks to prove the following: For all positive $x$, $y$ and $z$, one has $$x+y+z \leq 2 \left(\frac{x^2}{y+z} + \frac{y^2}{x+z} + ...
6
votes
2answers
67 views

How to prove that for $n \in \mathbb{N}$ we have $\sum_{k=2}^n \frac{1}{k}\leq \ln(n) \leq \sum_{k=1}^{n-1} \frac{1}{k}$

Prove that for $n \in \mathbb{N}$ we have $\sum_{k=2}^n \frac{1}{k}\leq \ln(n) \leq \sum_{k=1}^{n-1} \frac{1}{k}$ by using Riemann integral?
6
votes
3answers
149 views

Math Induction Proof: $(1+\frac1n)^n < n$

So I have to prove: For each natural number greater than or equal to 3, $$(1+\frac1n)^n<n$$ My work: Basis step: $n=3$ $$\left(1+\frac13\right)^3<3$$ $$\left(\frac43\right)^3<3$$ ...
5
votes
3answers
90 views

Prove $(x+r_1) \cdots (x+r_n) \geq (x+(r_1 \cdots r_n)^{1/n})^{n}$.

I can show that for $x > 0$ and $r_{i} > 0$ we have $$ \left(\, x + r_{1}\,\right)\ldots\left(\, x + r_{n}\,\right)\ \geq\ \left[\, x + \left(\, r_{1}\ldots r_{n}\,\right)^{1/n}\,\right]^{n}.$$ ...
5
votes
1answer
105 views

Prove that $\frac{1}{\sqrt{1+a^2}}+\frac{1}{\sqrt{1+b^2}}+\frac{1}{\sqrt{1+c^2}}\le\frac{3}{2}$

For positive real numbers with $a+b+c=abc$ prove that $$\frac{1}{\sqrt{1+a^2}}+\frac{1}{\sqrt{1+b^2}}+\frac{1}{\sqrt{1+c^2}}\le\frac{3}{2}$$ I made the substitution $a=\tan(\alpha), b = ...
5
votes
2answers
166 views

If $a$ and $b$ are positive real numbers such that $a+b=1$, prove that $(a+1/a)^2+(b+1/b)^2\ge 25/2$

If $a$ and $b$ are positive real numbers such that $a+b=1$, prove that $$\bigg(a+\dfrac{1}{a}\bigg)^2+\bigg(b+\frac{1}{b}\bigg)^2\ge \dfrac{25}{2}.$$ My work: ...
5
votes
5answers
443 views

How to find the minimum of $a+b+\sqrt{a^2+b^2}$

let $a,b>0$, and such $$\dfrac{2}{a}+\dfrac{1}{b}=1$$ Find this minimum $$a+b+\sqrt{a^2+b^2}$$ My try: since $$2b+a=ab$$ so ...
5
votes
6answers
5k views

Proving by induction: $2^n > n^3 $ for any natural number $n > 9$ [duplicate]

I need to prove that $$ 2^n > n^3\quad \forall n\in \mathbb N, \;n>9.$$ Now that is actually very easy if we prove it for real numbers using calculus. But I need a proof that uses mathematical ...
5
votes
4answers
183 views

Prove that $\frac{n^n}{3^n} < n! < \frac{n^n}{2^n} \forall n \geq 6 $

Prove that $$\frac{n^n}{3^n} < n! < \frac{n^n}{2^n} \forall n \geq 6 $$ I'm trying induction, this is what I have so far: Basecase $(n=6): 64<720<729$ is true. Inductive Case: Assume ...
4
votes
1answer
96 views

Inequality in 4 variables

I came across the following problem in a book. Four real numbers $p,q,r,s$ satisfy $p+q+r+s=9$ and $p^2+q^2+r^2+s^2=21$. Prove that there is a permutation $a,b,c,d$ of $p,q,r,s$ Such that $ab-cd\ge ...
4
votes
2answers
140 views

I have used Cauchy and Jensen. It is not helping me very much. Advice on solving this problem.

Let $a$, $b$ and $c$ be positive real numbers with $abc=1$. Prove that $$ \frac{a^{n+2}}{a^n+(n-1)b^n}+\frac{b^{n+2}}{b^n+(n-1)c^n}+\frac{c^{n+2}}{c^n+(n-1)a^n} \geq \frac{3}{n} $$ for each ...
4
votes
2answers
117 views

How to prove $\int_{-\pi}^\pi (f(x))^2dx\le \int_{-\pi}^\pi (f'(x))^2dx$ [duplicate]

Let $f$ be $C^1$ in $[-\pi, \pi]$ and satisfies $\int_{-\pi}^\pi f(x)dx=0$, periodic boundary condition. Then, prove that $\int_{-\pi}^\pi (f(x))^2dx\le \int_{-\pi}^\pi (f'(x))^2dx$. I try to prove ...
4
votes
3answers
160 views

Prove that $a^2 - b^2 + c^2 - d^2 \ge (a - b + c - d)^2$

In thinking about a base case in this problem, I came up with the following question. Given real numbers $a \ge b \ge c \ge d \ge 0$, prove that the following holds: $a^2 - b^2 + c^2 - d^2 \ge (a - ...
4
votes
1answer
657 views

If $ax^2-bx+c=0$ has two distinct real roots lying in the interval $(0,1)$ $a,b,c$ belongs to natural prove that $\log_5 {abc}\geq2$

If $ax^2-bx+c=0$ has two distinct real roots lying in the interval $(0,1)$ with $a, b, c\in \mathbb N$, prove that $\log_5 {abc}\geq2$. The equations I could form are: 1) $f(0)>0$ and ...
4
votes
1answer
248 views

Proof of an inequality of $L^p$ norms

For a general measure space, we define : $\|f\|_p= \left(\int\vert f\vert^p du\right)^{1/p}$. Let $0 < a < b < c < \infty$ and prove the following: $$ \|f\|_b \leqslant \max\{\|f\|_a, ...
3
votes
1answer
99 views

Prove $a^3+b^3+c^3\geq a^2b+b^2c+c^2a$ [duplicate]

if $a,b,c$ are positive real numbers,Prove:$$a^3+b^3+c^3\geq a^2b+b^2c+c^2a$$ Things I have done so far: I know the fact that $$a^3+b^3+c^3\geq\frac{1}{2}[ab(a+b)+bc(b+c)+ca(c+a)]$$ However i ...
3
votes
2answers
101 views

How prove $(x+\sqrt{x^{2}-1})^{n}+(x-\sqrt{x^{2}-1})^{n}\leq 2(1+n(x-1))^{n}$ for $n\in\mathbb{N}$?

Let $x\ge 1$. How prove that $(x+\sqrt{x^{2}-1})^{n}+(x-\sqrt{x^{2}-1})^{n}\leq 2(1+n(x-1))^{n}$ for $n\in\mathbb{N}$?
3
votes
3answers
141 views

How to understand Cauchy's proof of AM-GM inequality(the last step)

The AM-GM inequality: $$a_1a_2\cdots a_n\leq\left(\frac{a_1+\cdots + a_n}{n}\right)^n$$ the trivial case: $a_1a_2 \leq \left(\frac{a_1+a_2}{2}\right)^2 $ is self-evident. then cauchy use this fact ...
3
votes
4answers
162 views

Integral inequality for continuous function

Let $ f $ be a continuous, real-valued function on $[0, 1] $. Show that $$\int_0^1 \int_0^1 |f (x)+f (y)| dx dy \ge \int_0^1 |f (x)| dx $$ I tried to dissect the square in triangles and use ...
3
votes
3answers
128 views

Less than or equal sign

If I know for two numbers a and b that $${a < b }$$ Then is it correct to say that $$ a \leq b $$ I know that the second statement is true as long as the first one is. It seems OK as it is ...
3
votes
3answers
520 views

Bernoulli's Inequality

I'm asked to used induction to prove Bernoulli's Inequality: If $1+x>0$, then $(1+x)^n\geq 1+nx$ for all $n\in\mathbb{N}$. This what I have so far: Let $n=1$. Then $1+x\geq 1+x$. This is true. Now ...
3
votes
1answer
103 views

Does this inequality have any solutions in $\mathbb{N}$?

Does this (number-theoretic) inequality have any solutions $x \in \mathbb{N}$? $$\frac{\sigma_1(x)}{x} < \frac{2x}{x + 1}$$ Notice that we necessarily have $x > 1$.
3
votes
3answers
203 views

Solve this inequality $\prod_{i=1}^{50} \frac {2i-1}{2i} < \frac {1}{10}$

Prove that $ \frac{1}{2}\cdot \frac{3}{4}\cdot \frac{5}{6}\cdot \frac{7}{8}\cdot \frac{9}{10}\cdot \frac{11}{12}\cdot \frac{13}{14}...\cdot \frac{91}{92}\cdot \frac{93}{94}\cdot ...
3
votes
8answers
949 views

Prove that $e^x\ge x+1$ for all real $x$

Without using differentiation, logarithmic function, rigorously, prove that $$e^x\ge x+1$$ for all real values of $x$.
3
votes
6answers
241 views

Prove $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\geq \frac{9}{a+b+c}$ for $a\gt0$, $b\gt0$, $c\gt0$

Please help me for prove this inequality $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\geq \frac{9}{a+b+c}$, for $a>0$, $b>0$, $c>0$.
2
votes
1answer
60 views

Prove: $ \sum\frac{ab}{a^2+b^2}+\frac{1}{4}(\sum\frac{1}{a})\geq\frac{15}{4} $

Let $a,b,c>0$ such that $a+b+c=1$ Prove: $ \sum\frac{ab}{a^2+b^2}+\frac{1}{4}(\sum\frac{1}{a})\geq\frac{15}{4} $ I don't have any idea. You guy have any idea??
2
votes
1answer
136 views

Maximum of the sum of cube

(1) $-2\leq a_{i} \leq 2$ $~(i=1,2,3,4,5)$ (2) $\displaystyle\sum_{cyclic}a_{i}=0$ then, find the maximum value of $\displaystyle\sum_{cyclic}a_{i}^{3}$ also, can it be generalized as for ...