20 In praise of inequalities
Let \(\langle a, b \rangle \) be an inner product on a real vector space \(V\) (with the norm \(|a|^2 := \langle a, a \rangle \)). Then
holds for all vectors \(a, b \in V\), with equality if and only if \(a\) and \(b\) are linearly dependent.
The following (folklore) proof is probably the shortest. Consider the quadratic function
in the variable \(x\). We may assume \(a \neq 0\). If \(b = \lambda a\), then clearly
If, on the other hand, \(a\) and \(b\) are linearly independent, then \(|x a + b|^2 {\gt} 0\) for all \(x\), and thus the discriminant \(\langle a, b \rangle ^2 - |a|^2 |b|^2\) is less than 0.
Let \(a_1, \dots a_n\) be positive real numbers, then
with equality in both cases if and only if all \(a_i\)’s are equal.
TODO
Let \(a_1, \dots a_n\) be positive real numbers, then
with equality in both cases if and only if all \(a_i\)’s are equal.
TODO
Let \(a_1, \dots a_n\) be positive real numbers, then
with equality in both cases if and only if all \(a_i\)’s are equal.
TODO
Suppose all roots fo the polynomial \(x^n + a_{n - 1}x^{n - 1} + \dots +a_0\) are real. Then the roots of are contained in the interval with the endpoints
TODO
Let \(f(x)\) be a real polynomial of defree \(n \ge 2\) with only real roots, such that \(f(x){\gt} 0\) for \( -1 {\lt} x {\lt} 1\) amd \(f(-1) = f(1) = 0\). Then
and equality holds in both cases only for \(n=2\).
TODO
Suppose \(G\) is a graph on \(n\) vertices without triangles. Then \(G\) has at most \(\frac{n^2}{4}\) edges, and equality holds only when \(n\) is even and \(G\) is the complete bipartite graph \(K_{n/2, n/2}\).
TODO
Suppose \(G\) is a graph on \(n\) vertices without triangles. Then \(G\) has at most \(\frac{n^2}{4}\) edges, and equality holds only when \(n\) is even and \(G\) is the complete bipartite graph \(K_{n/2, n/2}\).
TODO