1
vote
1answer
18 views

Help with integral/logarithm inequality

I have to prove the following inequality: $1/(n+1) < \int_n^{n+1} 1/t$ $dt$ $<1/n$ I thought it would be easier to attack this via integration, so I get: $1/(n+1) <$ log $(n+1)-$ ...
2
votes
0answers
36 views

When this inequality true?

If $a$ and $b$ are non-negative integers and $c$ and $d$ are non-negative real numbers, for what values is the following inequality true? $\log((a+b)!) - \log(a!b!) \ge(a+b) \log(c+d) - (a \log(c) ...
1
vote
0answers
82 views

Showing that a logarithmic inequality holds

Given $0 < x_1 < x_2 < x_3 < x_4 < 1$, how can I show that the following inequality holds: $$ \frac{1}{R(x_1, x_3)}+\frac{1}{R(x_2, x_4)}<\frac{1}{R(x_1, x_2)}+\frac{1}{R(x_3, x_4)} ...
10
votes
2answers
242 views

$\lfloor n^{1/2}\rfloor+\cdots+\lfloor n^{1/n}\rfloor=\lfloor \log_2n\rfloor +\cdots+\lfloor \log_nn \rfloor$

Prove that: $\lfloor n^{1/2}\rfloor+\cdots+\lfloor n^{1/n}\rfloor=\lfloor \log_2n\rfloor +\cdots+\lfloor \log_nn \rfloor$, for $n > 1,\, n\in \mathbb{N}$ For example. For $n=2$, we have $\lfloor ...
2
votes
2answers
27 views

Let $x_0 > x_1 > x_2>x_3$ be any positive real numbers . What is the largest value of the real number k such that..

Question : Let $x_0 > x_1 > x_2>x_3$ be any positive real numbers . What is the largest value of the real number k such that $$\log \frac{x_0}{x_1}1993 + \log \frac{x_1}{x_2}1993 +\log ...
2
votes
1answer
162 views

Logarithms melting my brain

So I've got an inequality: $\ln(2x-5) > \ln(7-2x)$ and I attempt to solve by doing the following: $$\frac{\ln(2x)}{\ln(5)} > \frac{\ln(7)}{\ln(2x)}$$ $$\Rightarrow \ln(2x) \cdot \ln(2x) > ...
0
votes
2answers
46 views

Having trouble solving an inequality

I'm a trying to prove a recurrence relation (by substitution) for an algorithm class and I'm shamefully stuck in a rather simple looking inequality. I need to solve this inequality for constant $c$: ...
0
votes
1answer
40 views

How do I simplify and calculate this inequality?

$\log(x^3) > |x-1|$ I can't figure out how to go about solving this inequality, besides this one step: $3\log(x) > |x-1|$
11
votes
5answers
2k views

Why is it trivial that $\left(1+\frac{2\ln3}{3}\right)^{-3/2}\leq\frac{2}{3}$?

Can someone tell me why $$\left(1+\dfrac{2\ln3}{3}\right)^{-3/2}\leq\dfrac{2}{3}$$ is trivial because for me its not and I will need to do the calculation to see it.
5
votes
6answers
228 views

Prove that for all $x>0$, $1+2\ln x\leq x^2$

Prove that for all $x>0$, $$1+2\ln x\leq x^2$$ How can one prove that?
2
votes
2answers
59 views

Why is $x\log(x)$ convex?

Why is $x\log(x)$ convex? According to the definition it must hold: $(tx+(1-t)y)\log(tx+(1-t)y)\le tx\log(x)+(1-t)y\log(y)$ for all positive $x,y$ and $t\in[0,1]$ edit: It is allowed to ...
2
votes
1answer
80 views

Is anyway to prove this: $\prod_{k=1}^{n}(a_{k})< (1/n^n)*(\sum_{k=1}^{n}(\sqrt{1+a_{k}*a_{k+1}}))^n$

$$ \prod\limits_{k=1}^{n}a_{k} < {1 \over n^{n}}\left(\,\sum_{k = 1}^{n}\,\sqrt{1+a_{k}\,a_{k+1}\,}\,\right)^n $$ ak and n are positive real number greater than 0. EDIT: a_{k+1} becomes a_{1} ...
0
votes
1answer
59 views

Solve the inequality $2^{\left( x^{3}-x\right) } < 1$

$2^{\left( x^{3}-x\right) } < 1$ Let $2^{\left( x^{3}-x\right) }-1=f\left( x\right)$ To find the values for which $f(x)<0$ I let $f(x)=0$: $2^{\left( x^{3}-x\right) }-1=0$ $2^{\left( ...
1
vote
1answer
84 views

How to prove the following inequality of logarithm?

Let $x,y,z\in\mathbb{C}.$ Suppose $$z=\frac{1}{2}(xy\pm\sqrt{x^2y^2-4(x^2+y^2)} ).$$ Show that $$log^+|z|\leq log^+|x|+log^+|y|+log 2.$$ Where $log^+\phi=max\{0,log\phi\}.$ Here we are also ...
2
votes
3answers
57 views

How to solve the inequality $n! \le n^{n-2}$?

The inequality is $n! \le n^{n-2}$. I used Stirling's approximation for factorials and my answer was $n \le (e(2\pi)^{-1/2})^{2/5}$ but this doesn't seem right. Any help would be much appreciated.
0
votes
1answer
61 views

From $ \sum^\infty_{\lfloor \log n \rfloor + 1}n/{2^r} $ to $ \sum^\infty_{r=0}1/2^r $?

$$ E[h] = E[\sum^\infty_{r=1}I_r] = \sum^\infty_{r=1}E[I_r] $$ $$ = \sum^{ \lfloor \log n \rfloor}_{r=1}E[I_r] + \sum^\infty_{\lfloor \log n \rfloor + 1}E[I_r] $$ $$ \leq \sum^{ \lfloor \log n ...
3
votes
2answers
165 views

Prove that $\log^25 + \log^27 > \log12$.

Prove that $\log^25 + \log^27 > \log12$. What I tried so far: $\log^25 + \log^27 > \log3 + \log4$ $(\log5 + \log7)^2 - 2 \cdot \log5 \cdot\log7 > \log3 + \log4$ But it seems that I'm not ...
6
votes
6answers
208 views

Inequality, what is wrong with $\log(-1) = - \log(-1)$?

Can anyone tell me what is wrong with the following line of argument: $$ \log(-1) = \log(-1) - \log(1) = - \bigg( \log(1) - \log(-1) \bigg) = - \log \Big( \frac{1}{-1} \Big) = - \log(-1) $$ ...
1
vote
2answers
40 views

Logarithm problem : Prove that $log_{3^2} \frac{1}{2} > 0$

Logarithm problem : Prove that $log_{3^2} \frac{1}{2} > 0$ My approach : $log_{3^2} \frac{1}{2} > 0$ $\Rightarrow \frac{1}{2} log_3 \frac{1}{2} >0$ $\Rightarrow \frac{1}{2} [ log_3 1 ...
0
votes
3answers
55 views

Solving inequality having log

I am struggling to solve this inequality involving logarithm. How to find out values of $n$ for which below inequality holds good: $${\log_2n \over n} >{ 1 \over 8}$$
2
votes
2answers
77 views

Inequality $C\lceil\log{n}\rceil! \geq n^k$

I've been struggling to prove there exist $C$ for $n, n_{0}, \forall k >0 \in \mathbb{R}$ such that $\forall n > n_{0}$: \begin{equation}C\lceil\log{n}\rceil! \geq n^k\end{equation} As you ...
3
votes
3answers
104 views

Inequality with logarithms

How do I show that $$ \frac{1}{n-1}\geq \ln \left ( \frac{n}{n-1} \right ) $$ for $ n>1 $? As far as I can tell, exponentiating both sides with base $e$ won't help, because then I get a nasty ...
1
vote
2answers
61 views

determine x in $x\log_\frac{1}{10}(x^2+x+1)>0$

I wanted to know, how can i determine the values of x for which $x\log_\frac{1}{10}(x^2+x+1)>0$ going to the question, we must have $x>0$ and $\log_\frac{1}{10}(x^2+x+1)>0$ or both must ...
4
votes
1answer
488 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 ...
3
votes
2answers
55 views

Logarithm inequality for vectors

I am trying to prove the following result. Let $d$ be a vector in $\mathbf{R}^{n}$ with $\|d\|_{\infty} < 1$. Then, $$ \sum_{i=1}^{n} \log(1 + d_{i}) \geq \mathbf{1}^{T} d - \frac{\|d\|_{2}^{2}}{2 ...
8
votes
4answers
710 views

Prove that $\log _5 7 < \sqrt 2.$

Prove that $\log _5 7 < \sqrt 2.$ Trial : Here $\log _5 7 < \sqrt 2 \implies 5^\sqrt 2 <7.$ But I don't know how to prove this. Please help.
0
votes
2answers
59 views

Why does this inequality hold: $4n+2\le4n\log{n}+2n\log{n}$

Why is the following true? (I came across this in an algorithm analysis book but this inequality is not related to algorithm analysis) $$ 4n+2\le4n\log{n}+2n\log{n} $$
6
votes
6answers
244 views

Elegant way to solve $n\log_2(n) \le 10^6$

I'm studying Tomas Cormen Algorithms book and solve tasks listed after each chapter. I'm curious about task 1-1. that is right after Chapter #1. The question is: what is the best way to solve: ...
0
votes
1answer
48 views

Looking for suggestions on how to proceed with showing that:

for $x \ge 2863:$ $$\ln\left(\left\lfloor\frac{x}{6}\right\rfloor!\right) < \sum_{k=5}^{\infty}-\mu(k)\ln\left(\left\lfloor\frac{x}{k}\right\rfloor!\right)$$ I've written a java application which ...
17
votes
1answer
291 views

$x^3-3x-3=0$, prove that $10^x<127$

$x$ is the real root of the equation $$3x^3-5x+8=0,\tag 1$$ prove that $$e^x>\frac{40}{237}.$$ I find this inequality in a very accidental way,I think it's very difficult,because the actual value ...
0
votes
0answers
78 views

Using Stirling's Formula to approximate a difference of logarithms of factorials in the same way as Jitsuro Nagura.

In Jitsuro Nagura's classic proof of a prime existing between $x$ and $\frac{6x}{5}$, he uses Stirling's formula to show that: $$T\left(x\right) - T\left(\frac{x}{2}\right) - ...
0
votes
0answers
49 views

Do these inequalities regarding the gamma function and factorials work?

I am seeing if it is possible to generalize lemma 1 in Nagura's proof that there is always a prime between $x$ and $\frac{6x}{5}$. In a previous question, I asked whether the following inequality is ...
2
votes
0answers
60 views

Trying to generalize an inequality from Jitsuro Nagura: Does this work?

I am investigating the generality of Lemma 1 in Nagura's proof that there is always a prime between $x$ and $\frac{6x}{5}$. In Lemma 1, Nagura establishes when $n > 1$, $x \ge 1$: ...
0
votes
1answer
56 views

How do you evaluate an inequality that involves logarithms of factorials?

For $x > 1$, $n > 2$ with $2 \mid x+1$ and $n \mid x+1$, does it then follow that: $$\log(\lfloor\frac{x}{2}\rfloor!) - \log(\lfloor\frac{x}{n}\rfloor!) \le \log(\lfloor\frac{x+1}{2}\rfloor!) - ...
47
votes
4answers
2k views

Prove $\left(\dfrac{2}{5}\right)^{\frac{2}{5}}<\ln{2}$

Inadvertently, I find this interesting inequality,But this problem have nice solution? prove that $$\ln{2}>(\dfrac{2}{5})^{\frac{2}{5}}$$ This problem have nice solution? Thank you. ago,I find ...
0
votes
0answers
33 views

Does it follow that if $\{\frac{x}{2}\} \ge \frac{1}{2} + \frac{\{x\}}{2}$, $\log(\lfloor\frac{x}{2}\rfloor!) \le \log\Gamma(\frac{x+1}{2})$?

The answer seems to be yes. Here's my reasoning. Let $\{x\} = x - \lfloor{x}\rfloor$ Assume $\{\frac{x}{2}\} \ge \frac{1}{2} + \frac{\{x\}}{2}$ $$\log(\lfloor\frac{x}{2}\rfloor!) = ...
5
votes
1answer
117 views

Understanding a famous proof by Jitsuro Nagura: Need help understanding one step in the main theorem

I am going through the proof by Jitsuro Nagura which shows that there is always a prime between $x$ and $\frac{6x}{5}$ where $x \ge 25$. Nagura uses the following definitions: $$\vartheta(x) = ...
8
votes
1answer
222 views

Is it possible to generalize Ramanujan's lower bound for factorials that he used in his proof of Bertrand's Postulate?

I am starting to feel more confident in my understanding of Ramanujan's proof of Bertrand's postulate. I hope that I am not getting overconfident. In particular, Ramanujan's does the following ...
1
vote
1answer
35 views

Sufficient conditions for an inequality with a log

I need to find sufficient conditions so that $x \geq \frac{1}{a-\ln{x}}$ for $a>1$ and $x > 0$. Is there a way to get a tight solution to the problem?
4
votes
2answers
64 views

Given $1<a<b<c$ prove $\log_a\log_ab+\log_b\log_bc+\log_c\log_ca>0.$

Given $1<a<b<c$ prove $$ \log_a\log_ab+\log_b\log_bc+\log_c\log_ca>0. $$ How to approach problems like this? I tried usual transformations but no help. I guess I have to use ...
3
votes
2answers
1k views

Intuition behind logarithm inequality

One of fundamental inequalities on logarithm is: $$ 1 - \frac1x \leq \log x \leq x-1 \quad\text{for all $x > 0$},$$ which you may prefer write in the form of $$ \frac{x}{1+x} \leq \log{(1+x)} \leq ...
1
vote
2answers
151 views

Proof of inequality involving logarithms

How could we show that $$\left|\log\left( \left({1 + \frac{1}{n}}\right)^{n + \frac{1}{2}}\cdot \frac{1}{e}\right)\right| \leq \left|\log\left( \left({1 - \frac{1}{n}}\right)^{n - \frac{1}{2}}\cdot ...
1
vote
1answer
72 views

How to prove: for some $c>0,x>2 , c,x\in \mathbb R , \, \int_2^x \frac{\mathrm dt}{\log t}-\frac{x}{\log x} \leq \frac{cx}{(\log x)^2}$

How to prove: for some $c>0,x>2 , c,x\in \mathbb R$ $$ \int_2^x \frac{\mathrm dt}{\log t}-\frac{x}{\log x} \leq \frac{cx}{(\log x)^2}$$ I have tried my textbook, notes and also tried to find ...
1
vote
1answer
78 views

A simple inequality with logarithms and exponential

I want to prove that for $k>0$: $ 2^k \geq \frac{-1}{\log_2(1-\frac{1}{2^k})}$ I've plotted both functions and it seems to be the case for k>0. In fact, it would also be nice to see that: $ ...
2
votes
2answers
117 views

About the use of Stirling approximation

How to prove this inequality: $$\ln \Gamma \left( x \right)-2\ln \Gamma \left( \frac{x+1}{2} \right)>\frac{2x}{3}$$ Sry I forgot to mention that $x>300$
3
votes
3answers
265 views

Solving inequality involving logarithms

I must be doing something wrong. I want to solve the following, where n is a positive integer, and p is a real number between 0 and 1. $$(1-p)^n \le 0.4$$ So I take the log on both sides: ...
5
votes
3answers
178 views

Why is $\int\limits_{1}^{n} \log x \,dx \le \sum\limits_{x = 1}^{n}\log x$?

It has been a long time since I studied integrals, so this question may sound stupid. I was going through this wiki page, and came across the following inequality: $$\int_{1}^{n} \log x \,dx \le ...
4
votes
2answers
177 views

I am trying to solve the inequality $\log_{\log{\sqrt{9-x^2}}} x^2 <0$

I am trying to solve the inequality $$\log_{\log{\sqrt{9-x^2}}} x^2 <0.$$ I got $\mathrm{S.S}=(-\sqrt8 ,-1)\cup( 1,\sqrt8)$, but a friend got $\mathrm{S.S}=(-1,1)- \{0\}$. Please, what is ...
1
vote
2answers
68 views

Explanation of this inequality

Is there a graphic visualization of $\sum_{k=1}^{n} 1/k \, \, \leq \, \, \,1 \, + \, \int_1^n \! (1/x) \, \mathrm{d} x$ as intuitive as the integral test ? I can't see why the inequality is true. I ...
2
votes
1answer
76 views

Simple inequality help

I need a function $f(x)$ that satisfies the properties bellow for all integers $k$ $$ \frac{\log(k+1)}{k+1}-\log\left(1+\frac 1 k\right)+f(k+1)-f(k)<0 \ $$ $$ \lim_{k \rightarrow \infty} f(k)=0 $$ ...