Inequalities (Table of contents)

Inequalities of Jensen and Karamata

Theorem 1 (Jensen)

If \(f: [a,b]\to\mathbb R\) is a convex function and \(\alpha_1, \dots, \alpha_n\) sequence of positive real numbers such that \(\alpha_1+\dots+ \alpha_n=1\), than for any sequence \(x_1, \dots, x_n\in[a,b]\) the following inequality holds: \[f(\alpha_1x_1+\cdots+ \alpha_nx_n)\leq \alpha_1f(x_1)+\cdots + \alpha_nf(x_n).\]

If \(f\) is concave and \((\alpha_k)_{k=1}^n\) and \((x_k)_{k=1}^n\) the sequences as above, then \[f(\alpha_1x_1+\cdots+ \alpha_nx_n) \geq \alpha_1f(x_1)+\cdots + \alpha_nf(x_n).\]

Problem 1 Let \(x_1\), \(\dots\), \(x_n\) be positive real numbers. Prove that \[\left(\frac{x_1+\cdots+ x_n}n\right)^{ x_1+\cdots +x_n }\leq x_1^{x_1}\cdots x_n^{x_n}.\]

Problem 2 (by Zuming Feng) If \(x\), \(y\), \(z\) are positive real numbers such that \( x+y+z=xyz\) prove that \[\frac1{1+xy}+\frac1{1+yz}+\frac1{1+zx}\leq \frac34.\]

Problem 3 Let \(x\), \(y\), \(z\) be positive real numbers such that \(x+y+z\geq 1\). Prove that \[\frac{x\sqrt x}{y+z}+\frac{y\sqrt y}{z+x}+\frac{z\sqrt z}{x+y}\geq\frac{ \sqrt 3}2.\]

Theorem 2 (Karamata)

Let \(f\) be a convex function and \(x_1, \dots, x_n\), \(y_1, y_2, \dots, y_n\) two non-increasing sequences of real numbers. If one of the following two conditions is satisfied:

  • (a) \((y)\prec (x)\);

  • (b) \(x_1\geq y_1\), \(x_1+x_2\geq y_1+y_2\), \(x_1+x_2+x_3\geq y_1+y_2+y_3\), \(\dots\), \(x_1+\cdots+ x_{n-1}\geq y_1+\cdots + y_{n-1}\), \(x_1+\cdots + x_n\geq y_1+\cdots + y_n\) and \(f\) is increasing;

then \begin{eqnarray} \sum_{i=1}^nf(x_i)\geq \sum_{i=1}^nf(y_i).\quad\quad\quad\quad\quad (1) \end{eqnarray}

If \(f\) is concave and \((\alpha_k)_{k=1}^n\) and \((x_k)_{k=1}^n\) the sequences as above, then \[f(\alpha_1x_1+\cdots+ \alpha_nx_n) \geq \alpha_1f(x_1)+\cdots + \alpha_nf(x_n).\]

Problem 4 If \(a_1\geq a_2 \geq \dots\geq a_n\) and \(b_1\geq b_2\geq \dots\geq b_n\) are two sequences of positive real numbers which satisfy the following conditions: \[a_1\geq b_2,\; a_1a_2\geq b_1b_2, \; a_1a_2a_3\geq b_1b_2b_3,\; \dots \geq a_1a_2\cdots a_n\geq b_1b_2\cdots b_n,\] prove that \[a_1+a_2+\cdots + a_n \geq b_1+b_2+\cdots +b_n.\]

Problem 5 If \(x_1, \dots, x_n \in [-\pi/6,\pi/6]\), prove that \[\cos(2x_1-x_2)+\cos(2x_2-x_3)+\cdots + \cos(2x_n-x_1)\leq \cos x_1+\cdots + \cos x_n.\]

Problem 6 Let \(a_1, \dots, a_n\) be positive real numbers. Prove that \[(1+a_1)(1+a_2)\cdots (1+a_n) \leq \left(1+\frac{a_1^2}{a_2}\right)\cdot \left(1+\frac{a_2^2}{a_3}\right)\cdot \cdots \cdot \left(1+\frac{a_n^2}{a_1}\right).\]