Questions on proving and manipulating inequalities.

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5
votes
2answers
137 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
436 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
3k 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
177 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
3answers
148 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
588 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
230 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
88 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
4answers
152 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
107 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
1answer
99 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
202 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
673 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 $x$.
2
votes
1answer
133 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 ...
2
votes
4answers
156 views

Three Variables-Inequality with $a+b+c=abc$

$a$,$b$,$c$ are positive numbers such that $~a+b+c=abc$ Find the maximum value of $~\dfrac{1}{\sqrt{1+a^{2}}}+\dfrac{1}{\sqrt{1+b^{2}}}+\dfrac{1}{\sqrt{1+c^{2}}}$
2
votes
1answer
126 views

Is this induction procedure correct? ($2^n<n!$)

I am rather new to mathematical induction. Specially inequalities, as seen here How to use mathematical induction with inequalities?. Thanks to that question, I've been able to solve some of the form ...
2
votes
1answer
87 views

How can I give a bound on the $L^2$ norm of this function?

I came across this question in an old qualifying exam, but I am stumped on how to approach it: For $f\in L^p((1,\infty), m)$ ($m$ is the Lebesgue measure), $2<p<4$, let $$(Vf)(x) = ...
2
votes
3answers
128 views

Prove an inequality with a $\sin$ function

$$\forall{x\in(0,\frac{\pi}{2})}\ \sin(x) > \frac{2}{\pi}x $$ I suppose that solving $ \sin x = \frac{2}{\pi}x $ is the top difficulty of this exercise, but I don't know how to think out such ...
2
votes
3answers
751 views

Cauchy-Schwarz Inequality

In Luenberger book Cauchy-Schwarz Inequality is defined like this: For all $x,y$ in an inner product space $|(x|y)| \le \|x\|\|y\|$. Equality holds if and only if $x = \lambda y$ or $y = \theta$. ...
1
vote
2answers
157 views

Does this inequality hold true, in general?

Let $$N = \prod_{i=1}^{\omega(N)}{{p_i}^{\alpha_i}}$$ be the prime factorization of the positive integer $N$. Does the following inequality hold true in general? ...
1
vote
3answers
61 views

What is the maximum value of $\frac{2x}{x + 1} + \frac{x}{x - 1}$, if $x \in \mathbb{R}$ and $x > 1$?

What is the maximum value of $$f(x) = \frac{2x}{x + 1} + \frac{x}{x - 1},$$ if $x \in \mathbb{R}$ and $x > 1$? A 2-D plot of of $f$ for $x \in (\infty, \infty)$ is here. Lastly, note that ...
1
vote
1answer
389 views

Prove: $a_n \leq b_n \implies \limsup a_n \leq \limsup b_n$

Is my proof correct? Prove: $a_n \leq b_n \implies \limsup a_n \leq \limsup b_n$ Proof: Let $a_n$ and $b_n$ be sequences such that $a_n \leq b_n \forall_n$. Suppose $\limsup a_n \nleq \limsup b_n$. ...
1
vote
1answer
235 views

$ b_{n + 1} = \frac {b_n^2 + 2b_n}{b_n^2 + 2b_n+2}$ and $ b_1 = 1$, show that $ \left|\frac{2}{n}-\frac{2\ln{n}}{n^2}-b_n\right|\leq\frac{1}{n^2}$ [closed]

In the recursion $ b_{n + 1} = \frac {b_n^2 + 2b_n}{b_n^2 + 2b_n+2}$, with $ b_1 = 1,$ how can one prove that $ \left|\frac{2}{n}-\frac{2\ln{n}}{n^2}-b_n\right|\leq\frac{1}{n^2}$?
0
votes
1answer
142 views

How to prove that $\frac{x^2}{yz+2}+\frac{y^2}{zx+2}+\frac{z^2}{xy+2}\geq \frac{x+y+z}{3}$ holds for any $(x,y,z)\in[1,2]^3$

Prove that for $x,y,z\in [1,2]$ the following inequality holds: $$\frac{x^2}{yz+2}+\frac{y^2}{zx+2}+\frac{z^2}{xy+2}\geq \frac{x+y+z}{3}.$$ I tried to apply the Cauchy-Schwarz inequality or the power ...
0
votes
7answers
87 views

Induction and convergence of an inequality

Problem statement: Prove that $\frac{1*3*5*...*(2n-1)}{2*4*6*...(2n)}\leq \frac{1}{\sqrt{2n+1}}$ and that there exists a limit when $n \to \infty $. , $n\in \mathbb{N}$ My progress LHS is ...
8
votes
3answers
391 views

Basic Algebra Proof on Integers - Weak Inequalities Work but Strict Inequalities Don't?

Let $a, b, \& \, m$ be integers. Prove that if $2a + 3b \geq 12m + 1$, then $a \geq 3m + 1$ or $b \geq 2m + 1$. My Attempt: I don't conceive apace how to contrive, from the one inequality in ...
8
votes
4answers
3k views

Proof of Bernoulli's inequality

The question reads $$U_n = (1+x)^n - 1 - nx$$ Show that $U_2 \geq 0$ Hence or otherwise show that $(1+x)^n \geq 1 + nx$ for all $x \gt -1$. Obviously the $U_2 \geq 0$ is very easy, I can do ...
7
votes
1answer
2k views

A simpler proof of Jensen's inequality

Jensen's inequality states that if $(X,\mu)$ is a measure space with $\mu(X) = 1$, $\phi$ is convex, and $f:X \rightarrow \mathbb R$ is integrable, then $$\phi\left(\int fd\mu\right) \leq \int \phi ...
6
votes
2answers
342 views

Inequality involving the regularized gamma function

Prove that $$Q(x,\ln 2) := \frac{\int_{\ln 2}^{\infty} t^{x-1} e^{-t} dt}{\int_{0}^{\infty} t^{x-1} e^{-t} dt} \geqslant 1 - 2^{-x}$$ for all $x\geqslant 1$. ($Q$ is the regularized gamma function.) ...
6
votes
5answers
746 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
3answers
829 views

Bernoulli inequality

In an ordered field show that $x \geq 0 \implies (1+x)^{n} \geq 1+nx+ \frac{1}{2}n(n-1)x^2$ for every positive integer $n$. I know that $(1+x)^{n} \geq 1+nx$ (Bernoulli's inequality). To get the ...
5
votes
2answers
186 views

Inequality involving absolute values and square roots

I could use some help with proving this inequality: $$\left|\,x_1\,\right|+\left|\,x_2\,\right|+...+\left|\,x_p\,\right|\leq\sqrt{p}\sqrt{x^2_1+x^2_2+...+x^2_p}$$ for all natural numbers p. Aside ...
5
votes
4answers
297 views

How find this maximum $S_{\Delta ABC}$

in $\Delta ABC$,and $\angle ABC=60$,such that $PA=10,PB=6,PC=7$, find the maximum $S_{\Delta ABC}$. My try:let $AB=c,BC=a,AC=b$, then $$b^2=a^2+c^2-2ac\cos{\angle ABC}=a^2+c^2-2ac$$ then ...
5
votes
2answers
422 views

Proof of an inequality about $\frac{1}{z} + \sum_{n=1}^{\infty}\frac{2z}{z^2 - n^2}$

I've encountered an inequality pertaining to the following expression: $\frac{1}{z} + \sum_{n=1}^{\infty}\frac{2z}{z^2 - n^2}$, where $z$ is a complex number. After writing $z$ as $x + iy$ we have ...
4
votes
2answers
456 views

Inequality for the combined resistance of two resistors connected in parallel

Exact question: In electric circuit theory, the combined resistance $R$ of two resistors $R_1>0$ and $R_2>0$ connected in parallel obeys $$\frac{1}{R}=\frac{1}{R_1}+\frac{1}{R_2}$$ Show that ...
4
votes
0answers
106 views

Rising Sun Inequality (Dunford-Schwartz maximal inequality)

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be an absolutely integrable function, and let $f^*:\mathbb{R} \rightarrow \mathbb{R}$ be the one-sided signed Hardy-Littlewood maximal function $$f^*(x) := ...
4
votes
1answer
52 views

Simple upper bound for $\binom{n}{k}$

I remember seeing an upper bound for the binomial $\binom{n}{k}$ with an exponential function, something like $\binom{n}{k}\leq \left(ne/k\right)^k$. What exactly is it, and are there other similar ...
4
votes
0answers
294 views

Inequalities involving the probability density function and variance

I am wondering whether anyone knows of any any inequalities involving the probability density function of an unknown distribution (as opposed to the cumulative distribution function) and its known ...
3
votes
2answers
97 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 ...
3
votes
1answer
133 views

A determinant inequality

Let $A,B$ be two $m\times n$ real matrices. Then $$|AA'|\cdot |BB'|\geq |AB'|^2.$$ For square matrices, it is the equality. How to prove this inequality then?
3
votes
1answer
82 views

$2(1+abc)+\sqrt2(1+a^2)(1+b^2)(1+c^2)\ge(1+a)(1+b)(1+c)$ for real numbers $a,b,c$

$a,b,c$ reel sayılar için ; $$2(1+abc)+\sqrt2(1+a^2)(1+b^2)(1+c^2)\ge(1+a)(1+b)(1+c)$$ Olduğunu gösteriniz. Translation:1 For real numbers $a,b,c$, show that: ...
3
votes
2answers
95 views

Prove through induction that $3^n > n^3$ for $n \geq 4$

I'm new to induction and have not done induction with inequalities before, so I get stuck at proving after the 3rd step. The question is: Use induction to show that $3^n > n^3$ for $n \geq ...
3
votes
3answers
297 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
88 views

An inequality with radicals

If $s_{1}\ge t_{1}\ge t_{2}\ge s_{2}\ge0$, does one always have $(s_{1}-t_{1}+s_{2}+t_{2})^{1/2}\ge\sqrt{s_{1}}-\sqrt{t_{1}}+\sqrt{t_{2}}-\sqrt{s_{2}}$? Thanks a lot!
2
votes
1answer
114 views

Use induction to prove the following: $1! + 2! + … + n! \le (n + 1)!$

Use induction to prove the following: $1! + 2! + .... + n! < (n + 1)!$ Base case: $n = 1$ $1! < 2!$ true Inductive step: Assume that $1! + 2! + .... + k! \le (k + 1)!$ is true let $n = k ...
2
votes
1answer
58 views

Interior $H^2$ regularity - applying “Cauchy's inequality with $\epsilon$”

This is from PDE Evans, 2nd edition, pages 327, 328, and 330. I have a question regarding one piece of the proof. The theorem concerned is THEOREM 1 (Interior $H^2$ regularity) which is stated on ...
2
votes
2answers
288 views

Solving $x\; \leq \; \sqrt{20\; -\; x}$

This is how I tried to solve it: By squaring both sides: $x^{2}\; \leq \; 20\; -\; x$ $x^{2}\; +\; x\; -\; 20\; \leq \; 0$ Thus $-5\; \leq \; x\; \leq \; 4$ However, it seems that values less ...
2
votes
3answers
182 views

Proving that $n!≤((n+1)/2)^n$ by induction

I'm new to inequalities in mathematical induction and don't know how to proceed further. So far I was able to do this: $V(1): 1≤1 \text{ true}$ $V(n): n!≤((n+1)/2)^n$ $V(n+1): ...
2
votes
3answers
220 views

I don't understand this proof of the AM-GM inequality?

The proof uses this lemma which I understand: $\mathbf {Lemma}$: Suppose $x$ and $y$ are positive real numbers such that $x>y$. If we decrease $x$ and increase $y$ by some positive quantity $E$ ...
2
votes
1answer
605 views

Poincare Inequality

In page 290 of this book, Evans prove the Poincare inequality (Theorem 1) arguing by contradiction. Is there a direct proof of this theorem (Theorem 1) without arguing by contradiction?