On a functional inequality connected with quadratic functionals
The author finds functions \(f: X \to \mathbb R\), \(\phi: X \times X \to \mathbb R\) fulfilling the conditions \[ \begin{alignedat}{2} \frac{f(2x + 2y)}{4} - f(x) - f(y) &\geq \phi(x, y),&\quad &x,y \in X,\\ \phi(x, -y) &\geq -\phi(x,y),&\quad &x,y \in X,\\ \limsup_{n \to \infty} \frac{1}{4^{n}} \phi(2^{n}x, 2^{n}x) &< + \infty,&\quad &x \in X,\\ \liminf_{n \to \infty} \frac{1}{4^{n}} \phi(2^{n}x, 2^{n}y) &\geq \phi(x,y),&\quad &x,y \in X, \end{alignedat} \] where \((X,+)\) is an abelian group. The same is done for a real inner product space \(X\) with \( \dim X \geq 3 \) and the conditions \[ \begin{alignedat}{2} 2 \frac{f(2x + 2y)}{4} - f(x) - f(y) &\geq \phi(x, y),&\quad &x,y \in X,\\ \phi(x, -x) &\geq -\phi(x,x),&\quad &x \in X,\\ \phi(2x,2x) &\leq 4\phi(x,x),&\quad &x \in X,\\ \forall_{x \in X} \exists_{n_{0} \in \mathbb N} \forall_{n \geq n_{0}} f\biggl(\frac{x}{2^{n}}\biggr) &\geq 0, &&{}\\ \phi(x,y) &\geq 0, &\quad &x \perp y, x, y \in X . \end{alignedat} \]
scientific article; zbMATH DE number 5156431