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Inequality results about some function

Source: Romania National Olympiad 2016, grade x, p.2

August 25, 2019
functioninequalitiesinduction

Problem Statement

Let be a function f:R⟶R f:\mathbb{R}\longrightarrow\mathbb{R} satisfying the conditions: {f(x+y)≤f(x)+f(y)f(tx+(1−t)y)≤t(f(x))+(1−t)f(y), \left\{\begin{matrix} f(x+y) &\le & f(x)+f(y) \\ f(tx+(1-t)y) &\le & t(f(x)) +(1-t)f(y) \end{matrix}\right. , for all real numbers x,y,t x,y,t with t∈[0,1]. t\in [0,1] .
Prove that: a) f(b)+f(c)≤f(a)+f(d), f(b)+f(c)\le f(a)+f(d) , for any real numbers a,b,c,d a,b,c,d such that a≤b≤c≤d a\le b\le c\le d and d−c=b−a. d-c=b-a. b) for any natural number n≥3 n\ge 3 and any n n real numbers x1,x2,…,xn, x_1,x_2,\ldots ,x_n, the following inequality holds. f(∑1≤i≤nxi)+(n−2)∑1≤i≤nf(xi)≥∑1≤i<j≤nf(xi+xj) f\left( \sum_{1\le i\le n} x_i \right) +(n-2)\sum_{1\le i\le n} f\left( x_i \right)\ge \sum_{1\le i<j\le n} f\left( x_i+x_j \right)
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