Some Properties of Convex Functions

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1 Filomat 31: ), Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: Some Properties of Conve Functions Miloljub Albijanić a a Faculty of Economics, Finance and Administration, Belgrade, Serbia Abstract. We treat two problems on conve functions of one real variable. The first one is concerned with properties of tangent lines to the graph of a conve function and essentially is related to the questions on the first derivative if it eists). The second problem is related to Schwarz s derivative, in fact its upper limit modification. It gives an interesting characterization of conve functions. Let us recall the definition of a conve functions. Definition. A function f : a, b) R, defined on an interval a, b), is conve if for all distinct points 1, 2 a, b) and every real number λ 0, 1) the following inequality holds: f λ λ) 2 ) λ f 1 ) + 1 λ) f 2 ). If the reverse inequality holds for a function f : a, b) R f λ λ) 2 ) λ f 1 ) + 1 λ) f 2 ). then the function f is said to be concave. If we write < instead of, then we say that the function is strictly conve or instead we write >, the function is strictly concave). Figure 1: Conveity 2010 Mathematics Subject Classification. Primary 26A51; Secondary 26A24 Keywords. Conveity, asymptote, Schwarz derivative Received: 21 Septempber 2016; Revised: 24 December 2016; Accepted: 03 January 2017 Communicated by Eberhard Malkowsky address: malbijanic@fefa.edu.rs Miloljub Albijanić)

2 M. Albijanić / Filomat 31: ), An equivalent definition of conveity on an interval a, b) is given by the following condition: f ) f 1 ) 1 f 2) f ) 2 1 < < 2, 1, 2 a, b) ). Conveity and tangent line. A differentiable function f : a, b) R is strictly) conve if and only if for each point P 0 = 0, f 0 )) on the graph of f, the graph of f lies above the tangent L to the graph at P 0 the graph of f is strictly above the tangent L, ecept at the point P 0 ). An analogous statement holds for strictly) concave differentiable functions. Proof. Necessity). Let 0 a, b). The equation of L is given by y = L) = f 0 ) + f 0 ) 0 ). Using Lagrange s theorem we obtain Figure 2: Conve function f ) L) = f ) f 0 ) f 0 ) 0 ) = f ξ) f 0 ) ) 0 ), for some ξ 0, ). Since f is conve, f is increasing on a, b) and the difference f ξ) f 0 ) has the same sign as 0. Hence f ) L) 0, for a, b), as desired. If the function f is strictly conve, then f ) L) > 0 for a, b) and 0. The sufficiency.) By the assumption we have, for all, 0 a, b): then f ) L) = f ) f 0 ) f 0 ) 0 ) 0, f ) f 0 ) f 0 ) 0 for < 0, f ) f 0 ) f 0 ) 0 for > 0. Therefore, for all 1, 0, 2 a, b), such that 1 < 0 < 2 we have f 0 ) f 1 ) 0 1 f 2) f 0 ) 2 0, which is the second definition of conveity. The inspiration for this paper comes from the Bourbaki treatise [2]. Solutions to some of the problems from [2] are presented in [1]. We direct the reader to the classical monograph on inequalities [4] for a thorough eposition of classical inequalities. Conveity plays a crucial role in such considerations, there is etensive literature on these topics. It is present in classical tets on Analysis, like Hardy [3] and in modern ones, like Zorich [9]. An eample of this line of research can be found in [6].

3 M. Albijanić / Filomat 31: ), Let us mention a more specialized monograph D. Mitrinović [8] on inequalities. Serbian mathematicians made their contributions to that field, for eample M. Petrovic s inequality and Karamata s inequality, which will be stated below. M. Petrovic s inequality. If f : [0, + ) R is a conve function and 1, 2,..., n is a sequence of positive numbers. Then f 1 ) + + f n ) f n ) + n 1) f 0). Majorization. Let a = a 1, a 2,..., a n ) and b = b 1, b 2,..., b n ) be two finite sequences of real numbers. We say that the sequence a majorizes the sequence b and write a b, if a 1 a 2 a n, b 1 b 2 b n, a 1 + a a k b 1 + b b k, k = 1, 2,..., n 1, a 1 + a a n = b 1 + b b n. Karamata s inequality. Let a = a 1, a 2,..., a n ) and b = b 1, b 2,..., b n ) be two sequences of real numbers in an interval α, β). If a b and if f : α, β) R is conve, then f a 1 ) + + f a n ) f b 1 ) + + f b n ). Proposition A. Let f be differentiable and conve on a, b), a 0. 1) Then f ) f ) decreases strictly decreases if f is strictly conve) on a, b). 2) If f admits a finite right limit at a, then lim a+0 a) f ) = 0. 3) Function f ) on a, b) either increases or decreases or there eists c a, b) so that and increases on c, b). 4) Let us assume that b = +. If β = lim f ) f ) ) f ) decreases on a, c) f ) is finite, then the limit α = lim also eists and is finite. The straight line y = α + β is an asymptote of the function f and it lies below the graph of f. Proof. 1) If f has a second derivative then it is easy to prove that h) = f ) f ) decreases, because h ) = f ) f ) f ) = f ) 0. Figure 3: Let us now present a proof assuming only eistence of the first derivative. We choose 1 < 2. h 2 ) h 1 ) = f 2 ) 2 f 2 ) f 1 ) + 1 f 1 ) = = f 2 ) f 1 ) 2 f 2 ) 1 f 1 ) ) = = f 2 ) f 1 ) 2 f 2 ) + 1 f 2 ) 1 f 2 ) + 1 f 1 ) = f 2 ) f 1 ) f 2 ) 2 1 ) 1 f 2 ) f 1 ) ) = f ξ) 2 1 ) f 2 ) 2 1 ) 1 f 2 ) f 1 ) ) Lagrange s theorem) = 2 1 ) f ξ) f 2 ) ) 1 f 2 ) f 1 ) ),

4 M. Albijanić / Filomat 31: ), where ξ is between 1 and 2. Since f is conve, the first derivative f is increasing and therefore we obtain f ξ) f 2 ) 0, ξ < 2, f 2 ) f 1 ) 0, 1 < 2 and the result follows immediately. 2) Let us etend the function f at point a by setting f a) = f a + 0) = c. Let us assume that, for some 0 > a we have f 0 ) = f a). Then, by conveity, f ) f a) for all a < < 0. In fact, there are only two possibilities: either f ) = c for all a < < 0 or f ) < c for all a < < 0. Since the first possibility is trivial we can assume that f ) c for near a. Then, by the Lagrange s theorem, there is ξ = ξ) a, ) such that a) f ) = f ) f ) f a) a f ) f a)) = f ) f ξ) f ) f a) ). Case a. If for some sufficiently small a, f ) < 0, then f ξ) < f ) < 0 so f ) f ξ) 1, while f ) f a) c c = 0 when a + 0. We conclude a) f ) 0. Case b. f ) 0, for every a, b). Then f is increasing on a, b). Let us fi d a, b). For < d we have 0 f ) f d). Because tends to a we can assume that < d. We can conclude at once a) f ) a f ) 0, which suffices. 3) Let h) = f ). Then, h ) = f ) f ) 2 = f ) f ) 2 = ) 2. We already proved, in 1), that ) is decreasing function on a, b). Let us consider three cases for ): Figure 4: In the first case ) < 0 for all a, b), so h ) > 0, which means that h) is increasing. The second case is that takes both positive and negative values on a, b). This easily implies that h at first decreases and then increases; similarly we deal with the case when is strictly positive on a, b). 4) f ) is monotonically increasing because of the conveity of f for > a). Let us fi c > a and let, for > a, ϕ) be the unique real number such that the points Mc, f c)), L, f )) and N0, ϕ)) are collinear. We distinguish two cases: a) f ) is not bounded from above on a, + ); b) f ) is bounded from above on a, + ).

5 M. Albijanić / Filomat 31: ), Figure 5: Figure 6: The first case a). When +, then ϕ) see Figure 5). Indeed, assuming the contrary, we would have ϕ) s for all sufficiently large. That gives: the graph of f lies below the line q determined by points M and S0, s), for large. Let y = γ + δ be the equation of the line q. Then, for large : f ) γ + δ lim sup f ) γ. and therefore Net, there is an 0 such that f ) γ + 1 for all 0. Using Lagrange s theorem we have but the last inequality easily implies lim sup f ) f 0 ) = 0 ) f ξ) γ + 1) 0 ) 0, f ) lim ϕ) =. b) The equation of the tangent line at point L, f )) is y = f ) + f )t ), t > a, γ + 1 which gives a contradiction and we proved where t is the independent variable. Let us denote the value of this function at t = 0 by ψ = ψ), i.e. ψ) = f ) f ), see Figure 6. Let us note that, under our assumptions, f ) is bounded from above. Hence, since f is increasing and bounded from above, it has a finite limit α as +. Net, by assumption, f ) = f ) + β + o1) which gives, passing to the limit: Note that we proved: f ) lim = lim f ) + β + o1) = α. f ) lim = lim f ).

6 M. Albijanić / Filomat 31: ), The straight line y = α + β is an asymptote of the function f. The asymptote is under the graph of f because f is conve. Proposition B. Let f : I R be upper semi-continuous on an open interval I R. Then: f is conve on I if lim sup h 0 f + h) + f h) 2 f ) h 2 0, 1) for every I. Proof. The following should be proved: if f is not conve, then the condition 1) fails. Let us assume that f is not conve on I. That means that there are a < c < b in I such that f c) > c) where ) is represents a straight line which contains a, f a)) and b, f b)). Let Φ) = f ) + ε 2. For sufficiently small ε > 0 we also have Φc) > Gc) where G) represents a straight line which contains points a, Φa)) and b, Φb)) that is, the function Φ is not conve on [a, b]. Figure 7: Figure 8: Let Ψ) = Φ) G), [a, b]. The function Ψ attains its maimum on [a, b] at some point ξ a, b) because Φ is upper semi-continuous on the compact interval [a, b]. Therefore, there is δ > 0 such that so Ψξ h) + Φξ + h) 2 That means, Ψξ h) + Ψξ + h) 2Ψξ), Ψξ), for all h < δ. Φξ h) Gξ h) ) + Φξ + h) Gξ + h) ) 2 Φξ) Gξ) ), and since G) is linear we obtain Gξ h) + Gξ + h) = 2Gξ).

7 M. Albijanić / Filomat 31: ), Now we complete the proof by establishing the following inequalities: Φξ h) + Φξ + h) 2Φξ), f ξ h) + εξ h) 2 + f ξ + h) + ɛξ + h) 2 2 f ξ) + 2εξ 2, f ξ h) + f ξ + h) 2 f ξ) 2εh 2, f ξ h) + f ξ + h) 2 f ξ) 2ε and finally h 2 f ξ h) + f ξ + h) 2 f ξ) lim sup 2ε < 0. h 0 h 2 Conclusion The proofs presented have a strong geometrical flavor and we stress the role played by Lagrange s theorem. Although these results are quite classical and appeared in the literature, we believe the novelty of presentation will be of interest. These and related results have generalizations in many directions. References [1] M. Albijanić, Abstraction and Application in Mathematical Analysis in Serbian), Zavod za udžbenike, Beograd, [2] N. Bourbaki, Fonctions d une variable réelle: Théorie élémentaire, Springer, [3] G. H. Hardy, A Course of Pure Mathematics, Cambridge University Press, [4] G. H. Hardy, J. E. Littlewood and G. Plya Inequalities, Cambridge Mathematical Library, [5] P. Loh, Conveity, ploh/docs/math/mop2013/conveity-soln.pdf, [6] M. Mateljević, M. Svetlik, M. Albijanić, N. Savić, Generalizations of the Lagrange mean value theorem and applications, FILOMAT 27:4 2013), [7] N. Merentes, K. Nikodem, Remarks on strongly conve functions, Aequationes Mathematicae 801) 2010), [8] D. S. Mitrinović, Analytic Inequalities, Springer, [9] V. A. Zorich Mathematical Analysis I, Springer, 2015.