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 \[ (1+a_1)(1+a_2)\cdots (1+a_n) \leq \left(1+\frac{a_1^2}{a_2}\right)\cdot \left(1+\frac{a_2^2}{a_3}\right)\cdot \cdots \cdot \left(1+\frac{a_n^2}{a_1}\right).\]


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