Tagged Questions

1
vote
2answers
34 views

Solving the domain and range of a region satisfying two inequalities?

The question I was provided was: "Find the domain and range of the region satisfied by the following inequalities: i) $y \ge (x-1)^2$ ii)$y \le2x+1$ Any help would be greatly appreciated. Would you ...
2
votes
1answer
31 views

Condition for differential inequality

Let $f(x) = \frac{e^{ - ax}}{1 + {e^{bx}}}$, where $x>0$, $a$ and $b$ are positive constants. Find the condition of $a$ and $b$ so that $$ ( - 1)^nf^{(n)}(x) \ge 0 $$ with all $x>0$ and $n$, ...
2
votes
1answer
38 views

Need help showing the supremum of a function exists.

I was wondering if anyone knows a technique for proving that this function has a supremum less than infinity for $x \in \mathbb{R}$ ,$x \in [-1,1]$ (I am very certain that it does). The function is, ...
0
votes
0answers
84 views

How to prove this problem that involves ceiling function?

Given a process set ${\mathcal {T}} = \left\{ {\tau_i} \right\}_{i=1}^n$ which consists of $n$ processes, where $\tau_i$ denote process $i$. For each process $\tau_i$, there are four parameters ...
5
votes
2answers
95 views

Prove that $-\frac{\sqrt{x}}{1+x}\log{x} \leq \log{2}$ for $0 < x < 1$

Graphically and numerically it is obvious but I'm looking for an analytical reasoning. Just maximizing the left hand side does not yield an analytical expression for the maximum. I also tried some ...
0
votes
3answers
52 views

Where is $f(x) = \log(5x^2-8x-4)+\sqrt{x-1}$ defined?

Find the values of $x$ for which function is defined: $f(x) = \log(5x^2-8x-4)+\sqrt{x-1}$. $ \log(5x^2-8x-4) > 0 \Rightarrow 5x^2-8x-4 > 1 \Rightarrow 5x^2-8x-5 > 0 $ $x = \frac{ 8 \pm ...
1
vote
1answer
39 views

If there is a $T$ such that $V(t)<V(t-T) \ \forall t$, does that imply $V(t) \to 0$?

Let $V(t)$ denote a continuous scalar function $\mathbb{R} \mapsto \mathbb{R}$. Assume that we can find a constant $T \in \mathbb{R}$ such that $V(t)<V(t-T)$ for all $t$. Does that imply that $V(t) ...
4
votes
1answer
60 views

How to show $f(x) \leq 1+\frac{\pi}{4}$ for every $x \geq 1$

Suppose $f$ is a real-valued differentiable function defined on $[ 1,\infty)$ with $f(1)=1$. Suppose , moreover , that $f$ satisfies $$f'(x)=\frac{1}{x^2+f^2(x)}$$ Show that $f(x) \leq ...
2
votes
2answers
54 views

Inequalities of Integer functions

I have the following statement that I'm trying to prove: Assume that $f,g: \mathbb{N} \rightarrow \mathbb{R}^{\ge0}$. If $f(n) \ge g(n)$ then $\lceil f(n) \rceil \ge \lceil g(n) \rceil $. I have a ...
-1
votes
1answer
189 views

Prove $\sup \left| f'\left( x\right) \right| ^{2}\leqslant 4\sup \left| f\left( x\right) \right| \sup \left| f''\left( x\right) \right| $

Let $f\left( x\right)$ be a $C^{2}$ function on $\mathbb{R}$. Show that $$\sup \left| f'\left( x\right) \right| ^{2}\leqslant4\sup \left| f\left( x\right) \right| \sup \left| f''\left( x\right) ...
0
votes
0answers
35 views

Extracting a function from set of inequalities

I have set of inequalities in two dimension space which represent relation between $X$ and $Y$. now I want a function whose input is $X$ and output is $Y$. In other words, I want $F$ such that ...
-1
votes
1answer
47 views

Exercise of functions of a real variable

Let $f, g\colon\mathbb{R}\rightarrow\mathbb{R}$ functions so that for all $\,x, y\in\mathbb{R}$, $$f(x+y)+f(x-y)=2f(x)g(y).$$ Prove that if $f$ is not constant zero function and $|f(x)|\leqslant ...
2
votes
1answer
64 views

Simple inequality help

I need a function $f(x)$ that satisfies the properties bellow for all integers $k$ $$ \frac{\log(k+1)}{k+1}-\log\left(1+\frac 1 k\right)+f(k+1)-f(k)<0 \ $$ $$ \lim_{k \rightarrow \infty} f(k)=0 $$ ...
2
votes
2answers
50 views

Ceiling function inequality

In class, we used the fact that $\lceil{a + b \rceil} \geq \lceil{a}\rceil + \lfloor{b}\rfloor$. However, we weren't given a proof of this statement. I am interested to see how this works. Can anyone ...
1
vote
2answers
56 views

Smooth function on $\mathbb R$ whose small increments are not controlled by the first derivative at infinity

I need some help in finding a (as simple as possible) smooth function $f:\mathbb R \rightarrow \mathbb R$ which does NOT satisfy the following: There exist a constant $C>0$, a compact ...
1
vote
1answer
42 views

how to find arrange the following functions in increasing or decreasing order?

I have the following three functions $f_1(x) = \frac{1}{4} (8-3x + \sqrt{(x-2) (5x-14)}) (1-x)$ $f_2(x) = \frac{1}{8} (12-4x + \sqrt{2} \sqrt{(5x-14)(x-3)} + \sqrt{2} \sqrt{(x-2)(x-3)} )(1-x)$ ...
2
votes
1answer
95 views

Upper bound for $\Gamma(x+y)$

Let $x, y \geq 1$ be two real numbers. I am wondering if one can get an upper bound for $\Gamma(x+y)$ in terms of $\Gamma(x)\Gamma(y)$? Any references or ideas are very appreciated. Thank you.
3
votes
1answer
122 views

Minimizing a function over two variables

Given two natural numbers $i$ and $p$ such that $0 < i \leqslant 2^p$, let $$ \psi(p,i) := p - \alpha + 1 - \frac{1}{2^p}\left((2^p+i)\lg(2^p+i) - i\lg i - i + \alpha - \frac{2^p}{i+1} - ...
0
votes
1answer
44 views

Mathematical function for four corelated attributes

I have $4$ attributes $A,B,C,D$ each of them takes value between $[0,1]$ The more $A$ and $B$, the more the function value is. The more $C$ and $D$, the less the function value is. if C or D equals ...
0
votes
2answers
60 views

A function that maps inequalities to $-1$, $0$, or $1$.

The Python computer language has a built-in operation cmp(a,b) that returns $-1$, $0$ or $1$, if $a<b$, $a=b$ or $a>b$, respectively. I'd like to know if ...
2
votes
2answers
55 views

Geometric intuition for the inequality $(f(y) - c) ( y - d ) \geq (f(d) - c) ( f^{-1}(c) - d )$

Good day to everyone. I am interested in the geometric intuition for the following statement: Let $f:\mathbb{R} \mapsto \mathbb{R}$ be a monotonically increasing, invertible function and $c,d \in ...
0
votes
1answer
59 views

Functional Inequality question where $\int^x_0\frac1{f'(t)}dt = \int^x_02f(t)dt $ ,$ 0 \leq x \leq 1$ and $f(0) = 0$

$$\int^x_0\frac1{f'(t)}dt = \int^x_02f(t)dt \tag{1}$$ where $0 \leq x \leq 1$ and $f(0) = 0$ I need to prove that $$f(\frac1{\sqrt{2}})> \frac1{\sqrt{2}}$$ $$f(\tan (x))> \tan(x) > x , x ...
3
votes
1answer
99 views

Hypergeometric functions inequality

Let $_2F_1(a,b;c,z)$ be the (Gauss) hypergeometric function, and $m$ and $n$ positive integers. From a simple plot it looks like $_2F_1(m+n,1,m+1,\frac{m}{m+n})>\frac{m}{n} ...
1
vote
1answer
80 views

Greater than zero?

I need to show that $$\sum_{i=k^*}^K\binom{K}{i}a^{i-1}(1-a)^{K-i-1}(i-aK)>0$$ given $K\geq k^*$, $0<a<1$ and $K$, $k^*\in\mathbb{Z^+/1}$. I did some computer simulation and saw that it ...
1
vote
1answer
110 views

Inequality holds?

Can anyone prove that $$ \frac{\sum\limits_{i=1}^{k*} a_i i (x+\epsilon)^{(i-1)}}{\sum\limits_{i=1}^{k*} a_i (x+\epsilon)^{i}+k^*-1}>\frac{\sum\limits_{i=1}^{k*} a_i i ...
1
vote
4answers
62 views

Does this inequality hold

Let $f(x)$ be a positive function on $[0,\infty)$ such that $f(x) \leq 100 x^2$. I want to bound $f(x) - f(x-1)$ from above. Of course, we have $$f(x) - f(x-1) \leq f(x) \leq 100 x^2.$$ This is not ...
-2
votes
2answers
142 views

Simple ceiling function problem [closed]

Prove that $\lceil4n/3\rceil\le 4\lceil n/3\rceil$ for all integers $n$. Try to generalize this result to something where something other than 4 and 3 are used.
0
votes
1answer
64 views

Is the hypergeometric function $F(5/4,3/4; 2, z)$ bounded on $(0,1]$

Consider the classical hypergeometric function $F(5/4,3/4; 2, z)$ for $z\in (0,1]$. Is this bounded by some real number (independent of $z$)? I'm aware of Euler's formula: $$F(5/4,3/4; 2, z) = ...
4
votes
2answers
143 views

Bound for the Legendre function of the second kind of degree $1/2$

Let $Q_{1/2}(u)$ be the Legendre function of the second kind of degree $1/2$. One can show that $Q_{1/2}(u) = O(u^{-3/2})$ as $u\to \infty$; see Equation 21 in Section 3.9.2 of Higher transcendental ...
1
vote
2answers
138 views

How to define the sign of a function

$$y=\arctan\frac{x+1}{x-3} + \frac{x}{4}$$ I know that is necessary to put the function $>$ than $0$, but then? $$\arctan\frac{x+1}{x-3} + \frac{x}{4}>0$$ It's a sum, so I can't set up a "false ...
1
vote
0answers
37 views

Lower bounds for holomorphic functions on annuli with explicit bounds on their power series

Let $f$ be a holomorphic function on $\mathbf{C}$ and consider its restriction to the annulus $X=B(0,1) - B(0,3/4)$ in the complex plane. (Here $B(0,r)$ is the open disc with radius $r$ around $0$.) ...
0
votes
2answers
67 views

Trouble understanding how an equality is obtained

(This is from a proof by contradiction, so that's why the equality does not actually hold. Edited for brevity; I don't think I've omitted anything pertinent to my questions.) [...] The ...
4
votes
2answers
122 views

How to prove $|f(x) - f(y)| < |x - y|$ if $f(x) = x + 1/x$ where $x > 1$

I have attempted as follows: $|f(x) - f(y)| = |x + 1/x - y - 1/y|$ $\leq |x - y| + |1/x - 1/y|$ Struck here. Any help.
8
votes
2answers
284 views

$ \sum\limits_{i=1}^{p-1} \Bigl( \Bigl\lfloor{\frac{2i^{2}}{p}\Bigr\rfloor}-2\Bigl\lfloor{\frac{i^{2}}{p}\Bigr\rfloor}\Bigr)= \frac{p-1}{2}$

I was working out some problems. This is giving me trouble. If $p$ is a prime number of the form $4n+1$ then how do i show that: $$ \sum\limits_{i=1}^{p-1} \Biggl( ...
2
votes
2answers
202 views

$f\left(\sum X_i\right) \leq \sum f(X_i)$, where $X_i\gt 0$; for what functions is this true?

In a previous post, the following inequality has been proven $${\left( {\sum\limits_{i = 1}^n {{W_i}} } \right)^a} \le \sum\limits_{i = 1}^n {{W_i}^a}$$ where $W_i\gt 0$, $0\lt a\lt 1$. I guess it is ...
2
votes
1answer
108 views

Inequality based on the minimum of a function

Given $1<q\leq2$ and $0\leq p\leq1$, let us consider the following function: $$\phi\left(\alpha\right)=p\times\left|1-\alpha\right|^{q}+\left(1-p\right)\times\left|1+\alpha\right|^{q}$$ The ...