On boundedly-convex functions on pseudo-topological vector spaces (Q1975944)

It was proved by \textit{J. Focke} [Math. Operationsforsch. Stat. Optimization 8, 505-507 (1977; Zbl 0386.46038)] that the Fréchet derivative of continuous boundedly convex function satisfies the Lipschitz condition. Recall that \(f: B\to\mathbb{R}\) is boundedly convex if there is \(M>0\) such that for any elements \(x_1\), \(x_2\) in the Banach space \(B\) and any \(\lambda_1,\lambda_2\geq 0\), \(\lambda_1+ \lambda_2= 1\), \[ 0\leq \lambda_1 f(x_1)+ \lambda_2 f(x_2)- f(\lambda_1x_1+ \lambda_2x_2)\leq\textstyle{{1\over 2}} M\lambda_1\lambda_2\cdot\|x_1-x_2\|^2. \] The author generalizes this result to the case of functions defined on a pseudo-topological (or convergence) vector space. About the basic notions of the latter theory see the author's papers [Math. Nachr. 75, 153-183 (1976; Zbl 0364.46054) and (with \textit{O. G. Smolyanov}) Vestnik Moskov Univ., Ser. I 27, No. 2, 3-9 (1972; Zbl 0244.46057)].