Functional inequalities (1)


Problem 1. Find all functions f:\mathbb{R}\to\mathbb{R} such that

\displaystyle \frac{1}{2}f(xy)+\frac{1}{2}f(xz)-f(x)f(yz)\geq\frac{1}{4}\,\,\forall x,y,z\in\mathbb{R}.

Problem 2. Let f:(0;+\infty)\to (0;+\infty) be a function such that

f(2x)\geq x+f(f(x))\,\,\forall x\in (0;+\infty). Prove that f(x)\geq x\,\,\forall x\in (0;+\infty).

Problem 3. Let f:\mathbb{R}\to\mathbb{R} be a function such that

f(x+19)-19\leq f(x)\leq f(x+94)-94\,\,\forall x\in\mathbb{R}. Prove that f(x+1)=f(x)+1\,\,\forall x\in\mathbb{R}.

Problem 4. Find all functions f:[1;+\infty)\to [1;+\infty) such that

f(x)\leq 2x+2\,\,\text{and}\,\, xf(x+1)=f^2(x)-1\,\,\forall x\in [1;+\infty).

Problem 5. Find all functions f:\mathbb{N}\to \mathbb{N} such that

mf(n)+nf(m)=(m+n)f(m^2+n^2)\,\,\forall m,n\in \mathbb{N}.

Problem 6. Find all injective mappings f:\mathbb{N}^*\to\mathbb{N}^* such that for all positive integers n the following relation holds: f(f(n)) \leq \dfrac {n+f(n)}{2}.

Problem 7. Find all surjective mappings f:\mathbb{N}^*\to\mathbb{N}^* such that for all positive integers n the following relation holds: f(n) \geq n+(-1)^n.

Problem 8. Determine all functions f: \mathbb{N}^*\to \mathbb{N}^* such that for every positive integer n we have:

2n+2001\leq f(f(n))+f(n)\leq 2n+2002.

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