On Locally Convex Curves

Abstract

We introduce the definition of locally convex curves and establish some properties of such curves. In the section 1, we consider the curve \(K\) allowing the parametric representation \(x = u(t),\, y = v(t), \, (a \leqslant t \leqslant b)\), where \(u(t)\), \(v(t)\) are continuously differentiable on \([a,b]\) functions such that \(|u'(t)| + |v'(t)| > 0 \,\forall t \in [a,b]\). A continuous on \([a,b]\) function \(\theta(t)\) is called it the angle function of the curve \(K\) if the following conditions hold: \(u'(t) = \sqrt{(u'(t))^2 + (v'(t))^2}\, \cos \theta(t), \quad v'(t) = \sqrt{(u'(t))^2 + (v'(t))^2}\, \sin \theta(t)\). The curve \(K\) is called it locally convex if its angle function \(\theta(t)\) is strictly monotonous on \([a,b]\). For a closed curve \(K\) the number \(deg K= \cfrac{\theta(b)- \theta(a)}{2 \pi}\) is whole. This number is equal to the number of rotations that the speed vector \((u'(t),v'(t))\) performs around the origin. The main result of the first section is the statement: if the curve \(K\) is locally convex, then for any straight line \(G\) the number \(N(K;G)\) of intersections of \(K\) and \(G\) is finite and the estimate \(N(K;G) \leqslant 2 |deg K|\) holds. We discuss versions of this estimate for closed and non-closed curves. In the sections 2 and 3, we consider curves arising in the investigation of a linear homogeneous differential equation of the form \(L(x) \equiv x^{(n)} + p_1(t) x^{(n-1)} + \cdots p_n(t) x = 0 \) with locally summable coefficients \(p_i(t)\, (i = 1, \cdots,n)\). We demonstrate how conditions of disconjugacy of the differential operator \(L\) that were established in works of G.A. Bessmertnyh and A.Yu.Levin, can be applied.