Questions tagged [inequality]

Questions on proving, manipulating and applying inequalities.

19,201 questions
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Showing that two curves do not intersect

I want to show that $$x+1 \neq (x^3(x+2))^{1/4} + \sqrt{x+1-\sqrt{x^2+2x}}$$ for any real $x>0$. There are two approaches I've taken: showing they are equal and arriving at a contradiction (but ...
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Area of the region $\{(x,y):0\leq x \leq 1, 0 \leq y\leq 1, 3/4\leq x+y\leq 3/2\}$

Find the area of the region $\{(x,y):0\leq x \leq 1, 0 \leq y\leq 1, 3/4\leq x+y\leq 3/2\}$ (using definite integration). I cannot understand how to find this area. I have graphed the lines and found ...
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Inequality in a Polynomial

This is a problem I made. Let $x^3-ax^2+bx-c$ be a polynomial with real coefficients and three real roots, all greater than $1$. Prove, that $b+c \geq 3a-5$. Due to the discussion made (see the ...
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Calculate the maximum value of $\sqrt{\frac{3yz}{3yz + x}} + \sqrt{\frac{3zx}{3zx + 4y}} + \sqrt{\frac{xy}{xy + 3z}}$ where $x + 2y + 3z = 2$

$x$, $y$ and $z$ are positives such that $x + 2y + 3z = 2$. Calculate the maximum value of $$\sqrt{\frac{3yz}{3yz + x}} + \sqrt{\frac{3zx}{3zx + 4y}} + \sqrt{\frac{xy}{xy + 3z}}$$ This problem is ...
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Given $a_{k+1} \ge \frac{k a_{k}}{(a_{k}^{2} + k-1)}, \:\: k > 0$, prove $S_{n} = a_{1} + .. + a_{n} \ge n, \:\: n \ge 2$

Suppose a sequence of positive real numbers with $$a_{k+1} \ge \frac{k a_{k}}{(a_{k}^{2} + k-1)}, \:\: k > 0$$ prove that $$S_{n} = a_{1} + .. + a_{n} \ge n, \:\: n \ge 2$$ Solution: I will ...
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Prob. 5, Sec. 6.2, in Bartle & Sherbert's INTRO TO REAL ANALYSIS, 4th ed: How to show this function is strictly decreasing using derivative

Here is Prob. 5, Sec. 6.2, in the book Introduction To Real Analysis by Robert G. Bartle & Donald R. Sherbert, 4th edition: Let $a > b > 0$ and let $n \in \mathbb{N}$ satisfy $n \geq 2$. ...
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Series with exponential upper bound

Let $K > 0$. Is it then true that there is some constant $C$ independent of $K$ such that $$\sum_{n=0}^\infty e^{-2^n K} \leq C e^{-K/C}$$ Thanks for the help!
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Show that $\frac{j(j-1)}{2n}> \frac{j^2}{4nr}$

In Lemmas 8.5 and 8.6 in book Irrational Numbers by Ivan M. Niven it uses the following : $$\frac{j(j-1)}{2n}> \dfrac{j^2}{4nr}$$ $n \ge 2$, $2 \le j \le n$ and $r \ge 2$, that's it! How the ...
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Prove that $(x + y + z)^3 + 9xyz \ge 4(x + y + z)(xy + yz + zx)$ where $x, y, z \ge 0$. This has become the norm now... This problem is adapted from a recent competition. We have that $6(x^2y + xy^2 ... 1answer 61 views Can we use QM-AM inequality to solve this? There are two sequences (${a_1,a_2,a_3,....,a_n })$and$( {b_1,b_2,b_3,....,b_n})$such that $$\sum_{i=1}^n a_i = \sum_{i=1}^n b_i$$ Prove that: $$\sum_{i=1}^n \frac{a_i^2}{a_i+b_i} \ge \... 2answers 76 views Calculate the maximum value of \frac{x^2}{x^4 + yz} + \frac{y^2}{y^4 + zx} + \frac{z^2}{z^4 + xy} where x, y, z > 0 and x^2 + y^2 + z^2 = 3xyz. x, y and z are positives such that x^2 + y^2 + z^2 = 3xyz. Calculate the maximum value of$$\large \frac{x^2}{x^4 + yz} + \frac{y^2}{y^4 + zx} + \frac{z^2}{z^4 + xy}$$This is (obviously) ... 5answers 338 views Interesting inequality \frac{x^{m+1}+1}{x^m+1} \ge \sqrt[2m+1]{\frac{1+x^{2m+1}}{2}} Prove that$$\frac{x^{m+1}+1}{x^m+1} \ge \sqrt[k]{\frac{1+x^k}{2}}$$for all real x \ge 1 and for all positive integers m and k \le 2m+1. My work. If k \le 2m+1 then$$\sqrt[k]{\frac{1+x^k}{... 1answer 31 views A Euler summation like inequality. Let$f: \mathbb{R} \to \mathbb{C} $be a continuously differentiable function. Let$n$be an integer. Why do we have: $$\int_n^{n+1} f(t) dt = f(n)+O\left(\int_n^{n+1} |f’(t)| dt\right)?$$ 3answers 56 views $\frac{xy}{z^2(x + y)} + \frac{yz}{x^2(y + z)} + \frac{zx}{y^2(z + x)} \ge xy + yz + zx$given that where$x, y, z > 0$and$xyz = \frac{1}{2}$.$x$,$y$and$z$are positives such that$xyz = \dfrac{1}{2}$. Prove that $$\frac{xy}{z^2(x + y)} + \frac{yz}{x^2(y + z)} + \frac{zx}{y^2(z + x)} \ge xy + yz + zx$$ Before you complain, this problem ... 0answers 51 views is the last inequality in this page true?if yes how? [closed] enter image description herein the middle of the page its trying to integrate the expression using complex numbers integration and Res...this page has a inequality in the bottom(for the cas e goes to ... 2answers 58 views Prove the following inequality$\frac{1}{1+x}\leq\ln(1+x)\leq x$[closed] I need help proving the following inequality:$\frac{1}{1+x}\leq \ln(1+x)\leq x$if$x > 0 $Thanks 1answer 28 views Bound$\|(A+B)-(A^{1/4}(1+A^{-1/2}BA^{-1/2})^{1/2}A^{1/4})^2\|$in terms of commutator$\|AB-BA\|$For positive definite matrices$A$and$B$, can $$\|(A+B)-(A^{1/4}(1+A^{-1/2}BA^{-1/2})^{1/2}A^{1/4})^2\|$$ be bounded in terms of$\|AB-BA\|$? Note that if the matrices commute, then both norms ... 1answer 59 views Nice refinement of an inequality by Michael Rozenberg It's related to this If$a+b+c=abc$then$\sum\limits_{cyc}\frac{1}{7a+b}\leq\frac{\sqrt3}{8}$: In fact we have this refinement (wich I think much easier) : Let$a,b,c>0$then we have :$$\... 2answers 29 views Doubt in basics of inequalities While solving a problem I encountered with this$ 4m(4m-1) \leq 0 $Solving this one gets the solution$ 0 \leq m \leq 1/4 $Now, if the inequation is multiplied with -1 on both sides we get$ 4m(1-4m)...
I am trying to prove by induction that $\sqrt[n]{n}<2-\frac{1}{n}$ where $n\ge2$. It seemed simple at first, but I am stuck with $log(2n-1)$ in the RHS. I am in an elementary undergraduate Maths ...
Should $x=2$ be included in the solution?
Consider an inequation $\frac{(x-2)^6 (x-3)^3 (x-1)}{(x-2)^5 (x-4)^3}>0$. $(x-2)^5$ can be cancelled from both the numerator and denominator with the condition that $x\neq2$ leaving $(x-2)$ in the ...