alattiEgyenes

alattiEgyenes is a term used in some Hungarian mathematical contexts to describe a straight line that lies on or below a given curve over a specified interval and is tangent to the curve at at least one point. The word combines alatti, meaning beneath or under, with egyenes, meaning straight line, highlighting the directional relation of the line to the curve.

Definition and interpretation. Let f be a real-valued function defined on an interval I. An alattiEgyenes at a point x0 in I is a line L(x) = a x + b that satisfies L(x0) = f(x0) and L(x) ≤ f(x) for all x in I. If f is differentiable at x0, a natural candidate is the tangent line L(x) = f(x0) + f′(x0)(x − x0). For convex functions on I, the tangent line lies below the graph, making it an alattiEgyenes on that interval. In general, alattiEgyenes can be used to obtain lower bounds for f and to study supporting lines and sublevel structures in convex analysis and optimization.

Examples. For the parabola f(x) = x^2 on I = R, the tangent line at x0 is L(x) = 2x0 x − x0^2, which satisfies L(x) ≤ f(x) for all x in R; thus it is an alattiEgyenes at every x0. More complex functions yield a family of such lines, each tangent from below at a chosen point.

Usage and relation. The concept relates to supporting lines and lower bounds in optimization and convex geometry. It is not a universally standardized term, but appears in discussions emphasizing lines that lie beneath a curve and touch it from below. See also: tangent line, supporting line, convex analysis.