# 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;

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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