An increasing function with infinitely changing convexity (Q2786835)

The aim of the paper is to construct an example of the strictly increasing function changing its convexity infinitely many time in a neighborhood of a point.NEWLINENEWLINESuch example has the form NEWLINE\[NEWLINE u(t) = \int\limits_{0}^{t} (t - s) e^{-\alpha/s} [\sin 1/s + C \sin^2 1/s] ds, \;\;\; t\in [0, 1], \; \alpha, C > 0. \eqno{(1)} NEWLINE\]NEWLINENEWLINENEWLINEThis example is generalized for the case of two variables, i.e., the authors construct a function strictly increasing along each ray from the origin and changing its curvature infinitely many times: NEWLINE\[NEWLINE v: {\mathbb D} \rightarrow {\mathbb R}, NEWLINE\]NEWLINE \(v(x,y) = u(x^2 + y^2)\), where \(u\) is taken from (1).NEWLINENEWLINEBoth examples are useful in understanding the reversed Hopf lemma.