Then we discuss monotonicity, concavity, and convexity of fractional derivative of some functions and derive corresponding criteria. An elementary account of this work and partial bibliography has been given elsewhere 9, 10. Convexity, inequalities, and norms convex functions you are probably familiar with the notion of concavity of functions. Taking into account the above definition of concavity and convexity, a function is concave in an interval when the value of the second derivative of a point in that interval is greater than zero. In the arrowdebreu model of general economic equilibrium, agents have convex budget sets and convex preferences. Concavity, convexity and points of inflection if the graph of flies above all of its tangents on an interval i, then it is called concave upward convex downward on i. Confusion related to convexity and concavity of a function. We also introduce the notions of 1qlog convexity and 1qlog concavity, and the notion of higher order log concavity with respect to 1qlogconvexor 1qlog concavity. To visualize the idea of concavity using the first derivative, consider the tangent line at a point. Concavity, convexity and points of inflexion submitted by. Testing for convexity of a single variable function. Convex, concave, strictly convex, and strongly convex functions.
Inflection points an inflection point is defined as the point in which the function changes from being convex to concave. Thus to check if demand functions, obtained from di. Duration and convexity are two tools used to manage the risk exposure of fixedincome investments. Finding regions of concavity and convexity high school math. In other words, we need to determine the curvature of the function. As you move along an interval, if the slope of the line is increasing, then is increasing and so the function is concave up. Concavity convexity cw maxima and minima convex set.
Understanding duration and convexity of fixed income. You should be able to find a proof for that inequality here. It seems that m2nx are log concave of any order with respect to 1qlog convexity. Convexity, concavity and optimization without constraints the sign of hessian is a possible criterium for convexity equivalent condition for a c2 function f. It is also possible to characterize concavity or convexity of functions in terms. Brenti 4, 6, 7 has successfully applied total positivity techniques and results to study the log concavity problems. This implies that cccp can be applied to almost any optimization problem. R, we say that is convex or concave up if 00x 0 for all x2r.
One characteristic of the inflection points is that they are the points where the derivative function has maximums and minimums. A differentiable function is called strongly convex with parameter m 0 if the following inequality holds for all points x, y in its domain. The convexity and the concavity derived from newtons. Convex analysis is the standard tool for analyzing textbook economics. Convex sets separation concave and convex functions quasiconcave functions convex sets in rn 2 another ways of writing down the convex combination of two points is. The role of the properties of concavity convexity and quasiconcavity quasiconvexity of the relevant objective functions in determining the nature of their unconstrained or constrained critical points is considered in lecture note 3, section 3. If the graph of flies below all of its tangents on i, it is called concave downward convex upward on i. The rst function is said to be concave up and the second to be concave down.
First, every concave function must also be continuous except possible at the boundary points. Convexity, concavity and inflexion points of a function. In this lesson i will explain how to calculate the concavity and convexity of a function in a given interval without the need for a function graph but first, so as not to confuse terms, lets define what is a concave function and what is a convex function. A function is convex if its slope is non decreasing or 2. Prove that a strictly concave function of a strictly concave function maybe strictly convex. Some results on convexity and concavity of multivariate copulas ali dolati, akbar dehgan nezhad department of mathematics, yazd university, 89195741, yazd, iran. The monotonicity of the solutions of a class of nonlinear fractional differential equations is studied first, and the existing results were extended. Pdf convexity, concavity, type and cotype of lorentz spaces. Recall that the slope of the tangent line is precisely the derivative. A strongly convex function is also strictly convex, but not vice versa. Indices, convexity and concavity of calderonlozanovskii spaces article pdf available in mathematica scandinavica 921 march 2003 with 37 reads how we measure reads. The property you are referring to simply states that the directional derivative of a 1dimensional concave function is nonincreasing. We can characterize the concavity or convexity of a differentiable function using the. Increasing and decreasing functions, min and max, concavity studying properties of the function using derivatives typeset by foiltex 1.
This paper is devoted to show that there existtwoindices s and t with s t,suchthat a0,a1. Concave and convex functions1 washington university. Convex set convex function strictly convex function. Denition denition 11 concave function let c rn be a convex set, and let f. Theorem 1 shows that any function, subject to weak conditions, can be expressed as the sum of a convex and concave part this decomposition is not unique. As we can see in the graph above, interest rates or yields are the affecting. As in case of equities, the investors in fixed income securities is concerned about the volatility of bond prices in relation to yields. Logconcavity and logconvexity in passage time densities. Pdf different types of convexity and concavity for. This paper provides some results on di erent types of convexity and concavity in the class of multivariate copulas.
More generally, a function is said to be concave up on an interval if the graph of the. Monotonicity, concavity, and convexity of fractional. Convexity of sets in linear spaces, and concavity and convexity of functions, lie at the root of beautiful theoretical results that are at the same time extremely useful in the analysis and solution of optimization problems, including problems of either single objective or multiple objectives. Concavity convexity cw free download as powerpoint presentation. For example, the function in iv is convex on the part where it is solid and concave on the part where it is dotted. Does knowing the file type of an encrypted file make it easier to decrypt. Conversely, the preceding argument shows that if the rst inequality in 1 holds then fis concave take any a concavity or convexity of functions in terms of. Concavity, convexity and points of inflection study page.
To study the concavity and convexity, perform the following steps. The twin notions of concavity and convexity are used widely in economic theory, and are also central to optimization theory. Inflection points an inflection point is defined as the point in which the function changes from being convex to concave or vice versa. It should however be noted that, due to their extrinsic nature, convexity concavity measures are not isometry invariant and hence, at least on their own, not directly suited. The concept of strong convexity extends and parametrizes the notion of strict convexity. Another central theme that will make its first brief appearance in this paper is the holographic properties of traversing flows on manifolds with boundary. Singular point a point on the curve at which the curve exhibits on extra ordinary behavior is called a singular point. Concavityconvexity of lagrangian suppose lagrangian lx is concave convex in x. The convexity and the concavity derived from newtons inequality xuntuan su andweiwei zhang abstract.
Hence it follows, that if the second derivative f exists in an inflexion point x 0, then f x 0 0. That is, whether or not a function is concave depends on the numbers which the function assigns to its level curves, not just to their shape. The author used a change detection task and simple polygons to systematically manipulate convexity concavity. Logconcavity and logconvexity in passage time densities of. A conjecture on the 1qlog convexity of the borosmoll polynomials is presented. Study the intervals of concavity and convexity of the following function.
Concavity convexity relates to unconstrained maximum minimum problems, while quasi concavity quasi convexity relates to constrained maximum minimum problems. Concavity and convexity, inflection points of a function. The study of the concavity and convexity is done using the inflection points. Different types of convexity and concavity for copulas. Convex sets and concave functions carmen astornefigari. Convexity, concavity, type and cotype of lorentz spaces article pdf available in indagationes mathematicae 93. Increasing and decreasing functions, min and max, concavity. Convexity is a measure of the curvature in the relationship between bond prices and bond yields. Indeed, many log concave sequences arising in combinatorics are actually pf sequences. One problem with concavity and convexity which well encounter again when we look at homogeneity is that they are cardinal properties. Convexityandconcavityandquasi convexity and quasi concavity of functions play an essential role in economics. We can determine the concavity convexity of a function by determining whether the hessian is negative or positive semidefinite, as follows. In the first instance the graph of the function fx is convex downward concave upward and the function itself is called convex.
By newtons inequality, a sequence a in i0 of nonnegative real numbers is unimodal if its generating function. Efficient computation of shortest pathconcavity for 3d meshes. A function is convex in an interval when the value of the second derivative of a point in that interval is less than zero. Lesson 2convexity and concavity of functions of one and. Concave downward, downward, is an interval, or youre gonna be concave downward over an interval when your slope is decreasing. Convexity is a riskmanagement tool, used to measure and manage a portfolios exposure to market risk. Duration measures the bonds sensitivity to interest rate changes. The profit function is the convex conjugate of the cost function. Log concavity and log convexity in passage time densities of diffusion and birthdeath processes volume 8 issue 2 julian keilson. Critics of this account often rely on assumptions about preferences over wide ranges of. These are concavity and convexity and quasiconcavity quasiconvexity. Convexity concavity encodes certain characteristics of shapes and can thus be used to derive shape descriptors for 3d retrieval or matching. Scribd is the worlds largest social reading and publishing site. A di erentiable function is concave i it lies on or below the tangent line or plane, for n1 at any point.
Feb 12, 2017 here i discuss about working methods to solve the sums. To extend the notions of concavity and convexity to functions of many variables we first define the notion of a convex set. Detection of convexity and concavity in context marco bertamini university of liverpool sensitivity to shape changes was measured, in particular detection of convexity and concavity changes. Checking the convexity or the concavity of a function. Several examples are provided to illustrate the applications of our results. Suppose that convexity and concavity interchange at. This paper is devoted to the study of the log convexity of combinatorial sequences and is organized as follows. If you havent already, label the local maximaminima, absolute maximumminimum, in ection points, and where the graph is concave up or concave down. Concavity and convexity for the analysis of a function we also need to determine where the function is concave or convex.
Convex, concave, strictly convex, and strongly convex functions first and second order characterizations of convex functions optimality conditions for convex problems 1 theory of convex functions 1. So g, so concave upward means that your first derivative increasing, increasing, which means, which means that your second derivative is greater than zero. Pdf bell numbers, logconcavity, and logconvexity hui. Here are some of the topics that we will touch upon. The ultimate aim here is to reconstruct perhaps, only partially. While they are both increasing, their concavity distinguishes them. The twist is that while concavity requires that v0d2fxvbe negative for all v, quasi concavity requires only that v0d2fxvbe negative for vsuch that rfx v 0.
The assumption of convexity has two important implications. A function of a single variable is concave if every line segment joining two points on its graph does not lie above the graph at any point. Increasing and decreasing functions characterizing functions behaviour typeset by foiltex 2. The link proves it for convex functions, for concave functions simply reverse the sign of the inequality. We say that is concave or concave down if 00x 0 for all x2r. Certain key passage time densities and sojourn time densities for such processes have a simple property of log concavity or log convexity and associated unimodality. Concavity, convexity, quasiconcavity and quasiconvexity. To take the first derivative of this equation, use the power rule. Given the graph of a function, the hypograph of f, written hypf, is the set of points that lies on or below the graph of f, while the epigraph of f, written epif, is the set of points that lies on or above the graph of f. Show that the sum of strictly concave function is strictly concave. Convexity and concavity article about convexity and. To find the concavity, we need to look at the first and second derivatives at the given point. Understanding concavity wolfram demonstrations project. However, note that a function that fails to be globally convex concave can be convex concave on parts of their domains.
Convexity 5 convexity just as dollar duration describes dollar price sensitivity, dollar convexity describes curvature in dollar performance. In conclusion, its important to know if the control set has this nice property called convexity, because this naturally lead to efficient and reliable numerical solution, that can be. Roughly speaking concavity of a function means that the graph is above chord. Learn more about concavity, convexity image processing toolbox. How to find concavity or convexity part in an image. At equilibrium prices, the budget hyperplane supports the best attainable indifference curve. Notice that when we approach an inflection point the function increases more every timeor it decreases less, but once having exceeded the inflection point, the function begins increasing less or decreasing more. Operations that preserve convexity practical methods for establishing convexity of a function 1. The purpose of this article is to present necessary and sufficient conditions on convexity and concavity, lower and upper estimates and type and cotype of weighted lorentz spaces. Let fx be continuous on interval j and let there exists second derivative at each interior point of this interval. If fx 0, and the concavity changes, there is a point of inflection always check whether concavity changes around the point of inflection now lets look at some examples of concavity of a curve. It is also possible to characterize concavity or convexity of functions in terms of the convexity of particular sets.
Convexity, concavity and optimization without constraints what about suf. Free practice questions for high school math finding regions of concavity and convexity. Some results on convexity and concavity of multivariate. We say that the convexityconcavity is strict if the graph of fx over the interval i contains no straight. Convexity and concavity a property of the graph of the function y fx a curve that consists in the fact that each arc of the curve lies neither higher nor lower than its chord. Determine for which values of a the following is concave, convex or neither. If a function changes a convexity to a concavity or vice versa at passage through some point, then this point is called an inflexion point an inflexion point. Note that it is possible for f to be neither convex nor concave. Consider maximizing the function u x,y with respect to x and y without any con straints on the values chosen for x or y. Generalized concavity in fuzzy optimization and decision.