Inequalities (Table of contents)

Introduction to Inequalities

Our study starts with the following fundamental result:

Theorem 1. If \(x\) is a real number, then \(x^2\geq 0\). The equality holds if and only if \(x=0\).

No proofs will be omitted in this text. Except for this one. We will assume that the reader has mature enough to not multiply inequality by a negative number.

We continue our exposition with the first consequence of the previous fact:

Theorem 2. If \(a,b\in \mathbb R\) then: \begin{eqnarray*} a^2+b^2\geq 2ab. \quad\quad\quad\quad\quad (1) \end{eqnarray*} The equality holds if and only if \(a=b\).

Problem 1. Prove the inequality \(a^2+b^2+c^2\geq ab+bc+ca\), if \(a,b,c\) are real numbers.

Problem 2. Find all real numbers \(a, b, c\), and \(d\) such that \[a^2+b^2+c^2+d^2=a(b+c+d).\]

Problem 3. If \(a,b,c\) are positive real numbers that satisfy \(a^2+b^2+c^2=1\), find the minimal value of \[S=\frac{a^2b^2}{c^2}+\frac{b^2c^2}{a^2}+\frac{c^2a^2}{b^2}.\]

Problem 4. If \(x\) and \(y\) are two positive numbers less than \(1\), prove that \[\frac1{1-x^2}+\frac1{1-y^2}\geq \frac2{1-xy}.\]

Problem 5. If \(a\) and \(b\) are positive real numbers, prove that \(a^3+b^3\geq a^2b+ab^2\).

Problem 6. If \(a,b,c\) are positive real numbers that satisfy \[\frac{a^2}{1+a^2}+\frac{b^2}{1+b^2}+\frac{c^2}{1+c^2}=1,\] prove that \(\left|abc\right|\leq \frac1{2\sqrt 2}\).

Problem 7. (Nesbit’s inequality) If \(a\), \(b\), \(c\) are positive real numbers prove that \[\frac a{b+c}+\frac b{c+a}+\frac c{a+b}\geq \frac32.\]

Theorem 3. If \(a\), \(b\), \(c\) are non-negative real numbers then \begin{eqnarray}\frac{a^3+b^3+c^3}3\geq abc.\end{eqnarray} The equality holds if and only if \(a=b=c\).

Problem 8. If \(a\) and \(b\) are positive real numbers, prove that \(2a^3+b^3\geq 3a^2b\).

Problem 9. If \(a\), \(b\), \(c\) are positive real numbers prove that \[a^3+b^3+c^3\geq a^2b+b^2c+c^2a.\]

Problem 10. If \(a,b,c,d > 0\), prove that \[\frac a{b+c}+\frac b{c+d}+ \frac c{d+a}+ \frac d{a+b}\geq 2.\]

Problem 11. Prove that \[\frac{a^3}{a^2+ab+b^2}+ \frac{b^3}{b^2+bc+c^2}+ \frac{c^3}{c^2+ca+a^2} \geq \frac{a+b+c}3,\] for \(a,b,c > 0\).

Problem 12. If \(a_1,a_2, \dots, a_n,b_1,b_2,\dots, b_n\) are two sequences of positive real numbers prove that \[\frac{a_1b_1}{a_1+b_1}+\frac{a_2b_2}{a_2+b_2}+\cdots+\frac{a_nb_n}{a_n+b_n}\leq \frac{(a_1+\cdots+a_n)(b_1+\cdots+b_n)}{a_1+\cdots+a_n+b_1+\cdots+b_n}.\]

Problem 13. If \(a\), \(b\),\(c\), \(d\), \(e\), \(f\) are positive real numbers prove that \[\sqrt{ab}+\sqrt{cd}+\sqrt{ef}\leq\sqrt{(a+c+e)(b+d+f)}.\]

Problem 14. If \(a\), \(b\), and \(c\) are positive real numbers, prove that \[\frac{5a^3-ab^2}{a+b}+\frac{5b^3-bc^2}{b+c}+\frac{5c^3-ca^2}{c+a}\geq 2(a^2+b^2+c^2).\]

Problem 15. Let \(n\geq 3\) be an integer and let \(x_1, x_2, \dots, x_n\) be non-negative real numbers such that \(x_1=0\), \(x_n=1\). Prove that there exists \(j\in\{1,2,\dots, n-1\}\) for which: \[|x_{j+1}+x_{j-1}-2x_j|\geq \frac4{n^2}.\]