On locally convex curves

V. S. Klimov

P.G. Demidov Yaroslavl State University, 14 Sovetskaya str., Yaroslavl 150003, Russia

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 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 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.

Keywords: regular curve, corner function, degree, straight line, differential equation, polyline.

UDC: 513.7

Received: 27.02.2017

DOI: 10.18255/1818-1015-2017-5-567-577

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