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Some remarks on subquadratic functions. (English) Zbl 1127.39048
A function \(\varphi:x\to\mathbb{R}\), where \(X\) is a real linear space, satisfying the inequality \[ \varphi(x+y)+\varphi(x-y)\leq 2\varphi(x)+2 \varphi(y) \] is called subquadratic. The authors prove several theorems on subquadratic functions including the facts that if \(\varphi(kx)=k^2 \varphi(x)\) for \(x\in X\), then \(\varphi\) is quadratic and that if \(f:\mathbb{R} \to\mathbb{R}\) is non-decreasing, concave, and subadditive, and \(\varphi\) is subquadratic, then \(f\circ\varphi\) is subquadratic. They raise the question of characterizing continuous non-positive subquadratic functions.

MSC:
39B22 Functional equations for real functions
26A51 Convexity of real functions in one variable, generalizations
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