Journal of Mathematical Analysis and Applications

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1 J. Math. Anal. Appl Contents lists available at ScienceDirect Journal of Mathematical Analysis and Applications Takagi functions and approximate midconvexity Jacek Tabor a,,józeftabor b a Wydział Matematyki i Informatyki, Gołȩbia, Kraków, Poland b Institute of Mathematics, University of Rzeszów, Rejtana 6A, Rzeszów 35-30, Poland article info abstract Article history: Received November 008 Available online March 009 Submitted by A. Dontchev Keywords: Approximate convexity Jensen convexity Paraconvexity Takagi function Let V be a convex subset of a normed space and let ε 0, p > 0 be given constants. Afunction f : V R is called ε, p-midconvex if x + y f x + f y f + ε x y p for all x, y V. We consider the case p [, ] and investigate the relations between continuous ε, p- midconvex functions and Takagi-like functions given by ω p x := dist kp k x; Z for x R. It occurs that functions ω p are optimal, p-midconvex functions. This gives us sharp estimations for continuous ε, p-midconvex functions. We also compute the maximum of the function ω p for a certain set of parameter values. 009 Elsevier Inc. All rights reserved.. Introduction In the year 903 T. Takagi introduced the Takagi function [8] T x := dist k k x; Z for x R to give an easy example of a continuous nowhere differentiable function. This function was used in various parts of mathematics, see [,8,0,9]. In this paper we investigate the relations between approximate midconvexity and Takagi-like functions of the form ω p x := d kp k x for x R, where dx = distx; Z for more information concerning functions of this type see [7]. Notice that ω = T. Investigation of approximate convexity has a long history. One should probably begin by mentioning D. Hyers and S. Ulam [6] who in the year 95 defined and investigated ε-convex functions. Since then many papers on this subject have appeared. Two trends in this papers can be pointed out. One is focused on investigation of the regularity properties * Corresponding author. addresses: tabor@ii.uj.edu.pl J. Tabor, tabor@univ.rzeszow.pl J. Tabor. 00-7X/$ see front matter 009 Elsevier Inc. All rights reserved. doi:0.06/j.jmaa

2 730 J. Tabor, J. Tabor / J. Math. Anal. Appl mainly differentiability of approximately convex function for more references see [,6]. The other concerns, roughly speaking, estimations of the bounds which satisfy approximately convex functions, see for example []. Our considerations lie in the second current and are motivated by the fact that Takagi-like functions appear naturally in the investigation of approximate convexity [ 5,,7]. To explain the problems briefly we will need the following definition [3 5]. Definition.. Let V be a convex subset of a normed space and let ε 0, p > 0 be given constants. We say that a function f : V R is ε, p-midconvex if x + y f x + f y f + ε x y p for x, y V. The function f is called ε, p-convex if f tx+ ty tfx + t f y + ε x y p for x, y V, t [0, ]. A. Házy and Zs. Páles considered in [] the case p = and proved that if f : V R is a continuous ε, -midconvex function then the following inequality holds: f tx+ ty tfx + t f y + εω t x y for t [0, ], x, y V. It yields a natural question if in the class of continuous ε, -midconvex functions the above estimation is optimal. Z. Boros [] proved that ω is, -midconvex, which by the observation of Zs. Páles [] gives the positive answer to the above question. In this paper we are going to study a similar problem for p [, ]. Thecasep > is not interesting as it leads to convexity, see [,5], with the case p [0, ] we plan to deal in the next paper. The following theorem plays an essential role in motivating our investigations. Its proof is contained in [7], however since the paper has not appeared, for the readers convenience we provide a short sketch of the proof. Theorem.. See [7, Theorem 3]. Let h : V R be a continuous ε, p-midconvex function. Then h tx+ ty thx + thy + εω p t x y p for x, y V,t [0, ]. Idea of the proof. One can easily verify that the general case can be reduced to the case when V =[0, ], ε =, x =, y = 0. By the continuity of h it is enough to prove that k h k N h + N k h0 + N N i=0 d i k N ip for k { 0,..., N}, N N. The proof goes by induction over N. ForN = 0 is obvious. Suppose that is valid for some N. We are going to show that it holds for N +. So let q = k/ N+, where k {0,..., N+ }.Ifk = l for a certain l {0,..., N },thend N q = dl = 0, and the estimation follows directly from the inductive hypothesis. Let k = l + foracertainl {0,..., N }. Applying and the inductive hypothesis we obtain Since di hq h l + h l+ N N l N h + + / Np l N N d i l h0 + i=0 N i d = qh + qh0 + ip l l+ N +di N i=0 N ip + l + d i l+ N N + / Np. l + h + N l + N N h0 + i=0 d i l+ N ip = d i l+ for i = 0,...,N and d N l+ =, the last formula simplifies to. N+ N+ + p / Np Problem. Modified Problem of Zs. Páles. Can the bound ω p in Theorem. be improved? More precisely, does there exist afunctionv p ω p, v p ω p such that the assertion of Theorem. holds with ω p replaced by v p? We will prove that for p [, ] the answer to the above problem is negative. To do this we will need the following observation.

3 J. Tabor, J. Tabor / J. Math. Anal. Appl Proposition.. Let p > 0 and v p :[0, ] R be a function such that ht v p t for t [0, ] 3 for every, p-midconvex continuous function h :[0, ] Rsuch that h0 = h = 0. Then for every continuous ε, p-midconvex function f : V R the following inequality holds f tx+ ty tfx + t f y + εv p t x y p for x, y V, t [0, ]. Moreover, if v p itself is continuous and, p-midconvex, then it gives the minimal estimation in the above inequality. Proof. Clearly it is sufficient to consider the case ε > 0. To prove it is sufficient to fix arbitrary x, y V, x y and apply 3 for the function h f :[0, ] t [ f tx+ ty tfx t f y ] / ε x y p R. Now let us consider the minimality part. Suppose that u p :[0, ] R is a function such that f tx+ ty tfx + t f y + εu p t x y p for x, y V, t [0, ], for all convex sets V and continuous ε, p-midconvex functions f : V R. Asv p is continuous, p-midconvex function, we can put it in the place of f in the above inequality and obtain with x = and y = 0 v p t u p t for t [0, ], which obviously yields the minimality of v p. In the next part of the paper we show that for p [, ] the answer to Problem. is negative. We first prove it for p =, and then express ω p by a series depending on ω. This enables us to generalize the result of Z. Boros [] onto the case p [, ] by using a completely different method. Let us just add that in the case p 0, the function ω p is not, p-midconvex numerical verifications support the conjecture that in this case the optimal estimation is given by the function dp k x/ k obtained by A. Házy and Zs. Páles []. It is obvious that the functions ω p are bounded. Hence it follows from Theorem. that there exists a minimal constant Mp 0 such that every continuous ε, p-midconvex function is Mpε, p-convex. Thus we obtain naturally the following problem. Problem.. Find the formula or at least a good estimation from above for the constant Mp. From the optimality of the function ω p which we show in Theorem.3 we easily conclude that in fact Mp = max x [0,] ω px. Thus we arrive at the problem of evaluating the maximum of the function ω p. Directly from the definition of ω p we obtain the following estimation Mp = p kp p. Clearly, this estimation is very rough. Let us recall here that the maximum of the classical Takagi function is /3 and is attained on a Cantor set [7] see also [9]. We generalize this result and give an analytic formula for the maximum of ω p on a certain sequence of parameter values p n [, ] such that p =, p n. The problem of determining such a formula for M is in general open.. Optimality of the Takagi-like functions By BR, R we denote the space of bounded functions with the supremum norm. In our investigation we will need the following reformulation of the G. de Rham s Theorem [3]: Theorem.. Let h BR, R,a, b R, a <.LetT h : BR, R BR, R be an operator defined as follows: Then T h f x := hx + afbx for f BR, R, x R.

4 73 J. Tabor, J. Tabor / J. Math. Anal. Appl i T h is a contraction which has a unique fixed point f h ; ii f h is given by the formula f h x = n=0 an hb n x, x R; iii if h is continuous, then so is f h ; iv if a 0 and g BR, R is such that T h g g, then f h g. We are now ready to proceed with our investigation. Let I be a subinterval of R and let f : I R be a given function. By J f we denote the Jensen difference of f, that is x + y f x + f y J f x, y := f for x, y I. We are going to prove that J ω x, y d x y for x, y R. Webeginwithsomepreliminaryresults. Lemma.. Let f : R R be a -periodic function such that Then f x = x for x [ /, /]. J f x, y x y for x, y R. 5 Proof. Since Im f [ /, 0], weget J f x, y [ /, /] for x, y R. Thus 5 holds if x y. Assume now that x y <. Since f is -periodic and J f is symmetric with respect to x and y it is sufficient to consider the case y [ /, /], 0 x y <. Three subcases may occur. x [ /, /]. Then x + y J f x, y = + x + y x y =. x+y x [/, 3/], [0, /]. Then x + y J f x, y = + [ x + y ] = x y x + 3 x [/, 3/], x+y [/, ]. Then x + y J f x, y = + [ x + y ] = x y + y Lemma.. We have ω x = x x for x [0, ]. x y. x y. Proof. Let ϕ denote a -periodic function such that ϕx = x x for x [0, ]. One can easily check that ϕx = dx + ϕx for x R. So by the de Rham s Theorem to prove that ω = ϕ it is enough to observe that also ω x = dx + ω x for x R. Now we are ready to proceed to the simple theorem which we will need to prove the main result of the paper.

5 J. Tabor, J. Tabor / J. Math. Anal. Appl Theorem.. We have J ω x, y d x y for x, y R. Proof. We first show that J ω x, y x y for x, y R. 6 Notice that by Lemma. ω x = f x / +, where f is the function defined in Lemma.. By Lemma. we obtain 6. Consider an arbitrary x, y R. Wecanfindak Z such that x y k [, ]. Then J ω x, y = J ω x k, x + k [ x k y + k ] x k y + k x y = d = d. Now we are going to obtain a similar estimation of J ω p for p [,. In this purpose we will use Theorem. and the representation of ω p x as an infinite linear combination of ω x, ω x,... Proposition.. For every p > 0 we have ω p x = ω x + p / + ω i x / ip for x R. 7 i= Proof. Since ω x = dk x/ k, the right-hand side of 7 can be written in the form n=0 a nd n x, where a n = / n + p / + [ / n p + / n p + +/ np] = / n + / n + / np = / np. Consequently ω x + p / + ω i x / ip = d n x / np = ω p x. i= n=0 Lemma.3. We have x p + x x p for x [/, ] and p [, ]. Proof. Let gx = x p x p + x. Obviously, g is continuous on [/, ], twice differentiable on /, and g x = pp [ x p x p ] > 0 for x /,. Hence g is convex. Since g/ = 0 = g we obtain that gx 0forx [/, ]. Now we are ready to present the desired result. Theorem.3. For every p [, ] we have x y J ω p x, y d p for x, y R. Proof. By Proposition. and Theorem. for all x, y R we have J ω p x, y = J ω x, y + p / + J ω i x, i y / ip x y d + i= d i x y / ip. i=

6 73 J. Tabor, J. Tabor / J. Math. Anal. Appl Let ψ p r := d r + d i r / ip for r R. i= Thus to prove the assertion it is enough to show that ψ p r d p r for r R. 8 One can easily check that ψ p r = ψ p r/ p + d r d r/ for r R. 9 We define an operator S p : BR, R BR, R by the formula S p φr := φr/ p + d r d r/ for φ BR, R, r R. It follows from 9 that ψ p is a fixed point of S p. Thus by Theorem. to prove 8 it is sufficient to show that S p d p r d p r for r R. 0 The function d is -periodic and symmetric with respect to /, and consequently the same properties has S p d p. Therefore we need to verify 0 only for r [0, /]. Let r [0, /]. Then S p d p r = d p r/ p + d r d r/ = p / p + / r p = d p r. Let us now consider the case r [/, /]. Thenr [/, ]. Applying Lemma.3 we get S p d p r = d p r/ p + d r d r/ = [ r ] p / p + r [ r ] / = r p + r r p = d p r. As a direct corollary from Theorem.3 we obtain the following result taking p = we obtain the result of Boros []: Corollary.. For every p [, ] we have J ω p x, y x y p for x, y R, i.e. ω p is, p-midconvex function. 3. Maximum of the function ω p By Proposition. we observe that Corollary. jointly with Theorem. implies that ω p [0,] gives a minimal bound for continuous, p-midconvex functions. Consequently then Mp = max x [0,] ω px for p [, ]. In this section we will investigate properties of the function Mp for p [, ]. It follows directly from the definition of ω p that the family ω p p [,] is continuous with respect to p in the supremum norm. Hence the function M is continuous. Obviously it is also a decreasing function. Making use of the fact that the maximum of the classical Takagi function is /3 [7,9] and ω = T we obtain that M = /3. From Lemma. we obtain that M =. We are going to give a simple analytic formula of Mp for a certain increasing sequence p n [, ]. By x n we understand the unique positive solution to xn k =. One can easily notice that x =, x n / asn. The above equation can be rewritten for n in an equivalent form x n+ n = x n, x n /,. We denote p n := log x n. 3 Clearly p =, p n. Now we are ready to present the main result of this section.

7 J. Tabor, J. Tabor / J. Math. Anal. Appl Theorem 3.. For p, we put Cp := Then we have p p/p. M = /3 = lim p + Cp, M = = lim Cp p and Mp n = Cp n for n N, n. Before proceeding to the proof we present a few preliminary results. For K {, 6, 8,...} and n N where by N we understand the set of nonnegative integers we define A n := { x [0, ]: K l x [/ /K, / + /K ]+Z for l N, l n }, A := { x [0, ]: K l x [/ /K, / + /K ]+Z for l N }. We will need the following observation. Observation 3.. Let K {, 6, 8,...} be fixed. The set A is nonempty and A = { + ɛ K + + ɛ } n K n + : ɛ i {, }. Sketch of the proof. One first shows by induction that for n N we have A n = { + ɛ K + + ɛ } n K n : ɛ i {, } + [ /K n+, /K n+]. Then the assertion is an immediate consequence of the above formula. As we see A is a Cantor-like set. As a direct consequence of the above observation we get Proposition 3.. Let K {, 6, 8,...} be fixed and let f : R [0, ] be a -periodic function such that x [0, ], f x = x [/ /K, / + /K ]. Let a n n N 0, be a given sequence and let F x := n=0 a n f K n x.then M = max F x = a n, x R and for x [0, ] we have n=0 f x = M x A. Lemma 3.. For every n N we define the function g n : R R by the formula g n x := kp n d k x. Then g n is symmetric with respect to / and -periodic. Moreover, g n is increasing on the interval [0, / / n ] and equal to on [/ / n, /]. Proof. We first prove that g n is increasing on [0, /] and constant on [/ / n+, /]. Sinceg n is symmetric with respect to / and g n / =, this will consequently prove the assertion of the lemma. Because g n is a piecewise linear function, to prove this it is obviously enough to verify that g n x 0forx [0, /] at points where the derivative exists, and that g n x = 0forx [/ /n+, /]. Clearly for x [0, /] we have

8 736 J. Tabor, J. Tabor / J. Math. Anal. Appl g n x = k kp d k x = + n kp n d k x. Since d x =, we obtain that g n x kp n = xn k = 0, for x [0, /]. One can easily notice that d l x = forx [/ / n+, /], which implies that in this case g n x = kp n = 0. Proof of Theorem 3.. Consider the function g n from Lemma 3.. Clearly, ω pn x = pnn+ k g n n+ k x. Now we apply Proposition 3. to the function g n, K = n+ and sequence a n = max ω p n x = x R pnn+ = k p n n+. Thus to verify the assertion of the theorem it is enough to check if p n p n /p n = p n n+, or equivalently if p n = p n n+. pn+ k, and obtain that Since by 3 x n = p n, this is exactly Eq.. Remark 3.. Looking at Theorem 3. one may expect that the equality holds for all p,. However, it is not the case. For example for p =. using strict numerical verification one can show that Mp = max x [0,] ω p x ω p , while C. is less than.. This can be also seen from the comparison of the graphs of the functions Mp and Cp see Fig.. Fig..

9 J. Tabor, J. Tabor / J. Math. Anal. Appl Fig.. For a more precise information let us look at Fig. which shows the difference Mp Cp in the log-scale. The vertical lines at Fig. are drawn in the points p = uptop and. Fig. supports the conjecture that Mp = Cp if and only if p = p n for a certain n. References [] P. Allaart, K. Kawamura, Extreme values of some continuous nowhere differentiable functions, Math. Proc. Cambridge Philos. Soc [] Z. Boros, An inequality for the Takagi function, Math. Inequal. Appl [3] A. Házy, On approximate t-convexity, Math. Inequal. Appl [] A. Házy, Zs. Páles, On approximately midconvex functions, Bull. London Math. Soc [5] A. Házy, Zs. Páles, On approximately t-convex functions, Publ. Math. Debrecen [6] D.H. Hyers, S. Ulam, Approximately convex functions, Proc. Amer. Math. Soc [7] J.-P. Kahane, Sur lexemple, donne par M. de Rham, dune fonction continue sans derivee, Enseign. Math [8] Hans-Heinrich Kairies, Functional equations for peculiar functions, Aequationes Math [9] M. Krüppel, On the extrema and the improper derivatives of Takagi s continuous nowhere differentiable function, Rostock. Math. Kolloq [0] R. Mauldin, S. Williams, On the Hausdorff dimension of some graphs, Trans. Amer. Math. Soc [] H. Ngai, J.-P. Penot, Approximately convex functions and approximately monotonic operators, Nonlinear Anal [] Zs. Páles, 7. Problem in report of meeting, in: The Forty-First International Symposium on Functional Equations, Aequationes Math [3] G. de Rham, Sur une exemple defonction continue sans dériv ee, Enseign. Math [] S. Rolewicz, On paraconvex multifunctions, Oper. Res. Verf. Methods Oper. Res [5] S. Rolewicz, On γ -paraconvex multifunctions, Math. Jpn [6] S. Rolewicz, On differentiability of strongly α -paraconvex functions in non-separable Asplund spaces, Studia Math [7] Jacek Tabor, Józef Tabor, Generalized approximate midconvexity, Control Cybernet., in press. [8] T. Takagi, A simple example of continuous function without derivative, Proc. Phys. Math. Soc. Jpn [9] S. Tasaki, I. Antoniou, Z. Suchanecki, Deterministic diffusion, De Rham equation and fractal eigenvectors, Phys. Lett. A