Adam Najdecki, Józef Tabor CONDITIONALLY APPROXIMATELY CONVEX FUNCTIONS

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1 DEMONSTRATIO MATHEMATICA Vol. 49 No 06 Adam Najdecki, Józef Tabor CONDITIONALLY APPROXIMATELY CONVEX FUNCTIONS Communicated by A. Fryszkowski Abstract. Let X be a real normed space, V be a subset of X and α: r0, 8q Ñ r0, 8q be a nondecreasing function. We say that a function f : V Ñ r8, 8q is conditionally α- convex if for each convex combination ř n tivi of elements from V such that ř n tivi P V, the following inequality holds true ÿ n n f t iv i t ifpv iq ` α` ipt0,...,nu ti}vi ÿ t iv i}. We present some necessary and some sufficient conditions for f to be conditionally α-convex.. Introduction Theory of convex functions occurred to be very useful in different domains of mathematics, in particular in economics, optimal control and mathematical programming. Many interesting applications of convex functions are included in [6]. Therefore, there appeared many generalizations of this notion. An increasing role of computers stimulated investigations of approximate convexity. It started with the problem possed by D. H. Hyers and S. M. Ulam in []. They introduced the notion of δ-convex function. Definition.. Let X be a real vector space, V be a convex subset of X, and let δ 0 be given. A function f : V Ñ R is called δ-convex if fptx ` p tqyq tfpxq ` p tqfpyq ` δ for x, y P V, t P r0, s. A different generalizations of δ-convex function has been investigated in numerous papers (see e.g. [], [], [4], [8]). An interesting generalization of the notion of convex function has been considered in [7]. 00 Mathematics Subject Classification: 6A5, 9B6. Key words and phrases: convex function, approximately convex function, conditionally convex function. DOI: 0.55/dema c Copyright by Faculty of Mathematics and Information Science, Warsaw University of Technology Download Date /9/8 5: PM

2 A. Najdecki, J. Tabor Definition.. Let X be a real vector space and V be a subset of X. A ř function f : V Ñ r8, 8s is called convex if for all convex combinations n j t jx j of elements x j of V for which not both 8 and 8 are contained řn in tfpx j qu n j, we have f j t jx j ř n j t jfpx j q whenever ř n j t jx j is in V. We think that the term convex function used in Definition. could be misleading. We suggest to replace it by the term conditionally convex function. In [5] we joined the ideas of Definitions. and.. For the sake of simplicity we considered there only functions with values in r8, 8q. Definition.. Let X be a real vector space, V be a subset of X and let δ 0. A function f : V Ñ r8, 8q will be called conditionally δ-convex if for each convex combination ř n t ix i of elements x i of V, the following implication holds true ÿ n t i x i P V ñ f t i x i t i fpx i q ` δ. In case δ 0 we say that f is conditionally convex. We have given in [5] a characterization of conditionally δ-convex function. Though Definition. is a resonable generalization of Definitions. and. it is not subtle enough. In the inequality ÿ n f t i x i t i fpx i q ` δ the error term δ does not depend on the values of x i and t i for i 0,,..., n. Therefore, we are going to partially generalize Definition. in such a way that to remove the above mentioned drawback. In the sequel we will assume that X is a real normed space, V is a subset of X and α: r0, 8q Ñ r0, 8q is a nondecreasing function. Definition.4. We say that a function f : V Ñ r8, 8q is conditionally α-convex if for every convex combination ř n t iv i of elements v 0,..., v n P V the following implication holds true ÿ n P V ñ f t i fpv i q ` α v i. The estimation depending on the differences of x ř n t iv i and v i seems to be natural. Obviously it should be also depending on values of t i, i 0,,..., n. One can immediately notice that taking in Definition.4 α δ we get Definition. (obviously for normed space). Download Date /9/8 5: PM

3 Conditionally approximately convex functions As we know family of convex functions on a given convex set is invariant with respect to addition, multiplication by a nonnegative scalar and the operation imum. The question arises if the similar properties holds for conditionally α-convex function. One can easily notice that imum of two conditionally α-convex functions is conditionally α-convex, the sum of two conditionally α, α, respectively, convex functions is conditionally α ` α - convex and if f is conditionally α-convex and c 0 then cf is conditionally cα-convex. Further analogue to convex functions will be presented later on. Let f : V Ñ R be an arbitrary function. Following [9] we define the function convf : convv Ñ r8, 8q by the formula pconvfqpxq : infty P R : px, yq P convpepifqu epif stands here for the epigraph of f, i.e. epif tpx, yq P V ˆ R : y fpxqu. for x P convv. convf is the greatest convex function defined on convv and majorized by f. One can easily notice (see also [9]) that! pconvfqpxq inf t i fpv i q : x, t i P r0, s, v i P V for i 0,,..., n; ) t i, n P N, for x P convv. To simplify the above formula, we introduce some additional denotation. For v P convv we put C v :!`pt0,..., t n q; pv 0,..., v n q : t 0 v 0 `... ` t n v n v, t i P r0, s, v i P V for i 0,,..., n; ) t i, n P N.. The results We are going to present now some necessary and some sufficient conditions for a function to be conditionally α-convex. Theorem.. Assume that f : V Ñ R is conditionally α-convex. Then there exists a convex function f : convv Ñ r8, 8q such that fpvq fpvq fpvq () ` sup v i for v P V. α Download Date /9/8 5: PM

4 4 A. Najdecki, J. Tabor Proof. We are going to prove that f convf satisfies (). The first inequality in () is obvious. Suppose for the proof by a contradiction that the second one does not hold, i.e. that there exists a v P V such that fpvq ą pconvfqpvq ` sup α v i Then there exist `p t 0,..., t m q; p v 0,..., v m q P C v such that t i v i ă fpvq sup α v i.. Applying now the above inequality and conditional α-convexity of f we obtain ÿ m fpvq f a contradition. ă fpvq ` α t i v i sup t i fp v i q ` α α ipt0,...,mu t i vi ipt0,...,mu t i v i fpvq, t i vi v i t i v i Theorem.. Let f : V Ñ R be a given function. Suppose that there exists a convex function f : convv Ñ r8, 8q such that () fpvq fpvq fpvq ` for v P V. Then f is conditionally α-convex. inf α v i Proof. Consider an arbitrary convex combination v : ř m t i v i of elements from V and assume that v P V. Then by () and by convexity of f we have f t i v i fpvq fpvq ` inf α f ÿ m t i v i ` inf α v i v i Download Date /9/8 5: PM

5 Conditionally approximately convex functions 5 t i fp vi q ` t i fp v i q ` t i fp v i q ` α inf α inf α ipt0,...,nu t i vi t i v i. v i v i Theorems. and. yield immediately two natural questions: Is condition () sufficient, is condition () necessary for a function f : V Ñ R to be conditionally α-convex? It happens that the answer to the both above questions is negative. Example.. Let V t0,, u, α id and let f : V Ñ R be defined as follows fp0q fpq 0, fp q 4. We are going to show that f is conditionally α-convex. We have to consider all convex combinations of the form x : µ 0 ` ν ` η P V. If x 0 or x then the condition of Definition.4 is obviously satisfied. Consider now the case when x. Then we have µ 0 ` ν ` η, µ ` ν ` η, µ, ν, η P r0, s. The general solution of the above system has the form $ & µ P r0, s, ν µ, % η µ. Whence we obtain ˆ ˆ µfp0q ` νf ` ηfpq ` µ 0, ν, η p µq 4 ` µ 4 f ˆ. It shows that in the case x, Definition.4 is satisfied, too. Suppose that there exists a convex function f : convv Ñ R satisfying (). Then fp0q fp0q 0, fpq fpq 0 and consequently fp q 0. Hence Download Date /9/8 5: PM

6 6 A. Najdecki, J. Tabor ˆ ˆ ˆ f ` inf µ 0 µpr0, s, p µq, µ ˆ 0 ` inf µpr0, s µ a contradiction. 0 ă 4 fˆ Example.4. Let V t0,,, u, α id and let f : V Ñ R be defined as follows: fp0q f` fpq 0, f` 6. Take f : convv Ñ R, f 0. We are going to prove that condition () is satisfied. It is obviously satisfied for v P t0,, u. We have 4 0 ` 0 ` 4 ` and 0 ` ` 4 0, 0, 4, 6. Hence 0` sup ppµ,µ,µ,µ 4 q;p0,,,qqpc α` pµ 0, µ, µ, µ 4 6 fp q., It means that condition () for v is also satisfied. Now we show that f is not conditionally α-convex. We have 0 0 ` ` ` 6 and f` 6 ą 6 ` `00,, Hence f is not conditionally α-convex.,. In case of δ-convex function (and consequently in conditionally δ-convex one), we cannot obtain regularity conditions of a function. It is caused by the fact that then we assume only that the convexity error is bounded above by δ, but it does not depend on the arguments of a function. In case of conditionally α-convex function situation is different. Proposition.5. We assume that lim rñ0` αprq 0. Let f : V Ñ R be conditionally α-convex. Then f is continuous in intv. Proof. Consider an arbitrary x 0 P intv and arbitrary r ą 0 such that B 0 px 0, rq Ă V, where B 0 px 0, rq denotes an open ball centered at x 0 with the radius r. The function f B0 is obviously conditionally α-convex. Therefore, by Theorem., there exists a convex function f : B 0 Ñ r8, 8q such that fpxq fpxq fpxq ` sup αp t i}v i x}q ipt0,...,nu ppt 0,...,t nq;pv 0,...,v nqqpc x v 0,...,v npb 0 Download Date /9/8 5: PM 6

7 Conditionally approximately convex functions 7 for x P B 0. Then we have fpxq fpxq fpxq ` αprq for x P B 0. Whence we obtain fpxq fpx 0 q fpxq fpx 0 q ` αprq for x P B 0. Now, letting r Ñ 0 and making use of continuity of f and the condition lim rñ0` αprq 0, we obtain that f is continuous at x 0. Acknowledgements. The research has been partially supported by the Centre for Innovation and Transfer of Natural Science and Engineering Knowledge of University of Rzeszów. References [] A. Daniilidis, P. Georgiev, Approximate convexity and submonotonicity, J. Math. Anal. Appl. 9 (004), 9 0. [] A. Házy, On approximate t-convexity, Math. Inequal. Appl. 8() (005), [] D. H. Hyers, S. M. Ulam, Approximately convex functions, Proc. Amer. Math. Soc. (95), [4] J. Makó, Zs. Páles, Strengthening of strong and approximate convexity, Acta Math. Hungar. (-) (0), [5] A. Najdecki, Jacek Tabor, Józef Tabor, On conditionally δ-convex functions, Acta Math. Hungar. 8(-) (00), 8. [6] C. Niculescu, L. E. Persson, Convex Functions and Their Applications, CMS Books in Mathematics, Springer, 006. [7] H. J. M. Peters, P. P. Wakker, Convex functions on non-convex domains, Econom. Lett. (987), [8] S. Rolewicz, On αp q-paraconvex and strongly αp q-paraconvex functions, Control Cybernet. 9 (000), [9] R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ, 970. A. Najdecki FACULTY OF MATHEMATICS AND NATURAL SCIENCES UNIVERSITY OF RZESZÓW Rejtana 6A RZESZÓW, POLAND najdecki@ur.edu.pl J. Tabor FACULTY OF MATHEMATICS AND NATURAL SCIENCES UNIVERSITY OF RZESZÓW Rejtana 6A RZESZÓW, POLAND tabor@ur.edu.pl Received May 6, 04; revised version November 9, 04. Download Date /9/8 5: PM