When M-convexity of a Function Implies its N-convexity for Some Means M and N

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1 International Mathematical Forum,, 007, no. 47, When M-convexity of a Function Implies its N-convexity for Some Means M and N Janusz Matkowski Institute of Mathematics, University of Zielona Góra PL Zielona Góra, Poland and Institute of Mathematics, Silesian University PL Katowice, Poland J.Matkowski@im.uz.zgora.pl Devoted to Professor Boris Paneah on the occasion of his 70th birthday Abstract. Let M be an arbitrary strict mean in an interval J and M p [ϕ] be a weighted quasi-arithmetic mean of the weight p (0; 1) and a generator ϕ : J R. We prove that, for all intervals I J and for all continuous functions f : I J, the condition f is M p [ϕ] -affine implies f is M-convex is satisfied iff M is a quasi-arithmetic mean. Some variants of this result are proved and an open problem is posed. Mathematics Subject Classification: Primary 6B5, 6E60; Secondary 39B Keywords: mean, quasi-arithmetic mean, M convex function (convex function with respect to a mean M), M-affine function 1. Introduction Let J R be an interval. A function M : J R is called a mean in J if min(x, y) M(x, y) max(x, y), x,y J. If, for all x, y J, x y, these inequalities are strict, M is called strict; and symmetric, if M(x, y) =M(y, x). Every mean M in J is reflexive, i.e. M(x, x) =x, x J; consequently, M(I )=I for every subinterval I J and M is a mean in I. Obviously, every reflexive function M : J R which is increasing in each variable is a mean.

2 3 J. Matkowski Let I J be an interval. A function f : I J is said to be M-convex, M-concave, M-affine if, respectively, (cf. J. Aczél [1], G. Aumann [3], and J. Matkowski & J. Rätz [8]), f (M(x, y)) M(f(x),f(y)), x,y I; f (M(x, y)) M(f(x),f(y)), x,y I; f (M(x, y)) = M(f(x),f(y)), x,y I. If M = A where A(x, y) := x+y, then A-convexity coincides with the Jensenconvexity. Let us mention that the convexity of a function with respect to a non arithmetic mean appears in some characterization of L p -norm [7] and the Gamma function [5]. Let N and M be some means in an interval J. In the present paper we show that: if N is a weighted quasi-arithmetic mean and for any subinterval I J and any continuous function f : I J, f is N-affine implies that f is M-convex, then M must be quasi-arithmetic too. If moreover both means are symmetric, then M = N. We also show that assuming either some special condition on the generator of quasi-arithmetic mean N or the continuity of M, one can substantially weaken the assumptions in this result. A simple example shows that the quasi-arithmeticity of the means is meaningful. An open problem is posed.. Auxiliary results Recall that for every continuous and strictly monotonic function ϕ : J R and p [0, 1], the function M p [ϕ] : J J, M p [ϕ] (x, y) :=ϕ 1 (pϕ(x)+(1 p)ϕ(y)), x,y J, is a mean. M p [ϕ] is called a weighted quasi-arithmetic mean; the function ϕ is referred to as its generator, and the numbers p and 1 p its weights. Forp = 1 this mean is denoted by M [ϕ] and is called quasi-arithmetic. Remark 1. The mean M p [ϕ] is strict iff p (0, 1). Moreover we have M [ϕ] 0 (x, y) = y and M [ϕ] 1 (x, y) =x for all x, y I, i.e. M [ϕ] 0 and M [ϕ] 1 are the projective means. Remark. Suppose that ϕ, φ : J R are continuous and strictly monotonic, and p, q (0, 1). Then M p [ϕ] = M q [φ] if, and only if, q = p and there are a, b R, a 0, such that ϕ(x) =aφ(x)+b, x J,

3 When M-convexity implies N-convexity 33 (cf. [], Corollary 5, p. 46 where the case p = q = 1 is considered). This remark allows to assume, without any loss of generality, that the generator ϕ of the mean M p [ϕ] is increasing and, if it is convenient, that 0 ϕ(j). Remark 3. Let M be a mean in an interval J. Considering M-convex (Maffine) functions we can assume, without any loss of generality, that 0 int J (or 0 J). To show this take an arbitrary x 0 J, put J x 0 := {x x 0 : x J}, and define N :(J x 0 ) R by N(u, v) :=M(u + x 0,v+ x 0 ) x 0, u,v J x 0. It is easy to verify that N is a mean in J x 0, and if M = M [ϕ] p then N = M [φ] p where φ :(J x 0 ) R is given by φ(u) :=ϕ(u + x 0 ). Moreover, if f : I J is M-convex, then g :(I x 0 ) J x 0 defined by g(u) :=f(u + x 0 ) x 0, u J x 0, is N-convex. Indeed, for all u, v J x 0 we have g(n(u, v)) = f (N(u, v)+x 0 ) x 0 = f(m(u + x 0,v+ x 0 )) x 0 M(f(u + x 0 ),f(v + x 0 )) x 0 = M([f(u + x 0 ) x 0 ]+x 0, [f(v + x 0 )) x 0 ]+x 0 ) x 0 = M(g(u)+x 0,g(v)+x 0 ) x 0 = N(g(u),g(v)). If f : I J is M-affine, then a similar reasoning shows that g is N-affine. Let J R be an interval and p (0, 1). In the sequel we say that a function g : J R is p-convex (resp., p-concave, p-affine) ifg is convex (resp. concave, affine) with respect to the weighted arithmetic mean A p (x, y) := px+(1 p)y. In particular, 1-convexity ( 1-concavity, 1 -affinity) coincides with Jensen convexity (Jensen cocavity, Jensen affinity, respectively). Remark 4. It follows from the Daróczy-Páles identity ( x + y = p (1 p)x + p x + y ) ( +(1 p) (1 p) x + y ) + py that every p-convex ( p-concave, p-affine) function is Jensen convex (resp., Jensen concave, Jensen affine). Lemma 1. Let J R be an interval, ϕ : J R continuous and strictly increasing and p (0, 1). Then 1. f : I J is M p [ϕ] -convex (M p [ϕ] ϕ f ϕ 1 is p-convex (p-concave, p-affine) in ϕ(i); -concave, M p [ϕ] -affine) iff the function. if f : I J is M p [ϕ] -affine and continuous at least at one point, then there are a, b R such that ϕ f ϕ 1 (u) =au + b for all u ϕ(i).

4 34 J. Matkowski Proof. Suppose that f is M p [ϕ] -convex in convex in I. Then, for all x, y I. f ( ϕ 1 (pϕ(x)+(1 p)ϕ(y)) ) ϕ ( 1 ϕ 1 (pϕ(f(x)) + (1 p)ϕ(f(y))) ). For arbitrary u, v ϕ(i), taking here x := ϕ 1 (u), y:= ϕ 1 (v) and making use of the increasing monotonicity of ϕ, we hence get ϕ f ϕ 1 (pu +(1 p)v) pϕ f ϕ 1 (u)+(1 p)ϕ f ϕ 1 (u), which proves that ϕ f ϕ 1 is p-convex. In the same way we can show the remaining assertions of part 1. Now the second part of the lemma is a consequence of the Daróczy-Páles identity lemma and the classical theory of Jensen convex (or affine) functions (cf. M. Kuczma p. [6]). 3. Some results Clearly, every M-affine function is M-convex. We begin with the following Theorem 1. Let ϕ : J R be continuous and strictly monotonic in an open interval J R such that ϕ(j) =R, and p (0, 1) a fixed number. Suppose that M : J J is a mean. If, for all continuous functions f : J J, f is M p [ϕ] -affine = f is M-convex, then M = M q [ϕ] for some q [0, 1]. If, moreover, M is a strict mean, then q (0, 1). Furthermore, if p = 1 and M is strict and symmetric, then M = M [ϕ]. Proof. By Lemma 1, taking into account that ϕ(j) =R, we infer that, for all a, b R, the function f : J J given by f(x) =ϕ 1 (aϕ(x)+b), x J, is M p [ϕ] -affine. From the assumed implication we infer that this function f is M-convex, that is ϕ 1 (aϕ(m(x, y)) + b) M ( ϕ 1 (aϕ(x)+b),ϕ 1 (aϕ(y)+b) ), x,y J, for all a, b R. By Remark, without any loss of generality, we can assume that ϕ is strictly increasing. Therefore, repleacing x by ϕ 1 (u) and y by ϕ 1 (v), we hence get aϕ(m(ϕ 1 (u),ϕ 1 (v))) + b ϕ ( M ( ϕ 1 (au + b),ϕ 1 (av + b) )), u,v R, for all a, b R, that is (1) am (u, v)+b M (au + b, av + b), a,b,u,v R, where M : R R R is defined by M (u, v) :=ϕ(m(ϕ 1 (u),ϕ 1 (v))). For b = 0 we hence get am (u, v) M (au, av), a,u,v R,

5 which, of course, implies that () that is M is homogeneous. Taking a = 1 in (1) we get When M-convexity implies N-convexity 35 am (u, v) =M (au, av), a,u,v R, M (u, v)+b M (u + b, v + b), b,u,v R. Repleacing here u by u b and v by v b we obtain whence Consequently, (3) M (u b, v b) M (u, v) b, b, u, v R, M (u + b, v + b) M (u, v)+b, b, u, v R. M (u + b, v + b) =M (u, v)+b, b, u, v R. Applying in turn (3) and () we obtain, for all u.v R, where M (u, v) =M ((u v)+v, 0+v) =M (u v, 0) + v =(u v)m (1, 0) + v = qu +(1 q)v, Hence, by the definition of M, q := M (1, 0). M(x, y) =ϕ 1 (qϕ(x)+(1 q)ϕ(y)), x,y J. Since M is a mean, we have q [0, 1]. This completes the proof. Denote by Q the set of rational numbers. The continuity of the mean M in Theorem 1 (as well as in Corollary 1) allows to weaken the basic assumption significally. Theorem. Let ϕ : J R be continuous and strictly monotonic in an open interval J R, ϕ(j) = R, and p (0, 1) a fixed number. Suppose that M : J J is a continuous mean. log b If there are a, b, c, d R\{0}, 0 < a < 1 < b, / Q such that the log a functions and the function ϕ 1 (aϕ), ϕ 1 (bϕ), ϕ 1 (cϕ + d) are M-convex ϕ 1 ( ϕ) is M-affine, then M = M [ϕ] q for some q [0; 1]. If moreover M is strict, then q (0; 1).

6 36 J. Matkowski Proof. We can assume that ϕ is strictly increasing. From the M-convexity of the function ϕ 1 (aϕ) we have ϕ 1 (aϕ(m(x, y))) M(ϕ 1 (aϕ(x)),ϕ 1 (aϕ(y))), x,y J. Repleacing x by ϕ 1 (u) and y by ϕ 1 (v), we hence get aϕ(m(ϕ 1 (u),ϕ 1 (v))) ϕ(m ( ϕ 1 (au),ϕ 1 (av) ) ), u,v R, whence, by induction, a n ϕ(m(ϕ 1 (u),ϕ 1 (v))) ϕ(m ( ϕ 1 (a n u),ϕ 1 (a n v) ) ), u,v R, n N. From the M-convexity of the function ϕ 1 (bϕ), in the same way, we get b m ϕ(m(ϕ 1 (u),ϕ 1 (v))) ϕ(m ( ϕ 1 (b m u),ϕ 1 (b m v) ) ), u,v R, n N. Making use both these inequalities we obtain a n b m ϕ(m(ϕ 1 (u),ϕ 1 (v))) ϕ(m ( ϕ 1 (a n b m u),ϕ 1 (a n b m v) ) ) for all u, v R, n N. Since, by assumption, 0 <a<1 <b, log g / Q, the set log a {a n b m : n, m N} is dense in (0, ). The last inequality and the continuity of M imply that tϕ(m(ϕ 1 (u),ϕ 1 (v))) ϕ(m ( ϕ 1 (tu),ϕ 1 (tv) ) ), u,v R, t>0. Replacing u and v by u/t and v/t, respectively, we obtain the reversed inequality, whence tϕ(m(ϕ 1 (u),ϕ 1 (v))) = ϕ(m ( ϕ 1 (tu),ϕ 1 (tv) ) ), u,v R, t>0. This proves that the mean M : R R defined by (4) M (u, v) :=ϕ(m(ϕ 1 (u),ϕ 1 (v))), u,v R, is positively homogeneous. Since, by assumption, ϕ 1 ( ϕ) ism-affine, we have ϕ 1 ( ϕ(m(x, y))) = M(ϕ 1 ( ϕ(x)),ϕ 1 ( ϕ(y))), x,y J, which can be written in the form ϕ(m(ϕ 1 (u),ϕ 1 (v))) = ϕ(m ( ϕ 1 ( u),ϕ 1 ( v) ) ), u,v R, whence, by the definition of M, (5) M ( u, v) := M (u, v), u,v R. This relation and the positively homogeneity of M imply that M is homogeneous, i.e. (6) M (tu, tv) :=tm (u, v), t,u,v R. By assumption, there are real nonzero c, d such that the function ϕ 1 (cϕ+d) is M-convex. Consequently, ϕ 1 (cϕ(m(x, y)) + d) M ( ϕ 1 (cϕ(x)+d),ϕ 1 (cϕ(y)+d) ), x,y J,

7 When M-convexity implies N-convexity 37 which can be written in the form cϕ(m(ϕ 1 (u),ϕ 1 (v))) + d ϕ(m ( ϕ 1 (cu + d),ϕ 1 (cv + d) ) ), u,v R, whence cm (u, v)+d M (cu + d, cv + d), u,v R. Hence, applying (6), the homogeneity of M, we get M (cu, cv)+d M (cu + d, cv + d), u,v R, and, consequently, (7) M (u, v)+d M (u + d, v + d), u,v R. Replacing here u by u d and v by v d we get M (u d, v d) M (u, v) d, u, v R, whence, by replacing u by u and v by v, M ( u d, v d) M ( u, v) d, u, v R. Making use of (5), we obtain M (u + d, v + d) M (u, v) d, u, v R, that is (8) M (u + d, v + d) M (u, v)+d, u, v R, The inequalities (7) and (8) imply that M (u + d, v + d) =M (u, v)+d, u, v R. Hence, by the homogeneity of M, M (tu + td, tv + td) =M (tu, tv)+td, t, u, v R, whence M (u + td, v + td) =M (u, v)+td, t, u, v R. Since t R is arbitrary, we conclude that (9) M (u + w, v + w) =M (u, v)+w, u, v, w R. Now, similarly as in the proof of Theorem 1, applying (6) and (9) we obtain M (u, v) =M ((u v)+v, 0+v) =M (u v, 0) + v =(u v)m (1, 0) + v) =qu +(1 q)v for all u, v R, and the result is a consequence of (4). In the case ϕ(j) Rthe following result holds true:

8 38 J. Matkowski Theorem 3. Let ϕ : J R be continuous and strictly monotonic in an open interval J R and let p (0, 1) be a fixed number. Suppose that M : J J is a mean. If for all compact intervals I J and for all continuous functions f : I J, f is M p [ϕ] -affine = f is M-convex, then M = M q [ϕ] for some q [0; 1]. If moreover M is a strict mean, then q (0; 1). Furthermore, if M is symmetric, then M = M [ϕ]. Proof. By Remark we may assume that 0 int ϕ(j). By Remark 3 we can also assume that 0 int J. Take a compact subinterval I J such that 0 int(i) and a continuous function f : I J. By Lemma 1, if f is M p [ϕ] -affine, then there are a, b R such that (4) Conversely, for all a, b R such that f(x) =ϕ 1 (aϕ(x)+b), x I. (aϕ(i)+b) ϕ(j), the function f given by this formula is M p [ϕ] -affine. Since I J is compact, there are α = α(i) > 1 and β = β(i) > 0 such that this inclusion holds true for all a, b R such that 0 a<αand b <β.suppose that condition 1 is satisfied. Then, for all a, b R, such that 0 a<αand b <β,we have ϕ 1 (aϕ(m(x, y)) + b) M ( ϕ 1 (aϕ(x)+b),ϕ 1 (aϕ(y)+b) ), x,y I, or equivalently, for all a, b R, such that 0 a<αand b <β, aϕ(m(ϕ 1 (u),ϕ 1 (v))) + b ϕ ( M ( ϕ 1 (au + b),ϕ 1 (av + b) )), (5) Thus, for all a, b R, such that 0 a<αand b <β, am 1 (u, v)+b M 1 (au + b, av + b), where M 1 : ϕ(j) ϕ(j) ϕ(j) is defined by M 1 (u, v) :=ϕ(m(ϕ 1 (u),ϕ 1 (v))), Taking b = 0 we hence get am 1 (u, v) M 1 (au, av), for all a R such that 0 a<α,which implies that (6) am 1 (u, v) =M 1 (au, av), u,v ϕ(i), u,v ϕ(j). u,v ϕ(i), u,v ϕ(i), for all a R such that 0 a<α. For an arbitrary (u, v) R R take t>0 such that u t, v t M (u, v) :=tm 1 ( u t, v t ). u,v ϕ(i). ϕ(i) and put

9 When M-convexity implies N-convexity 39 To show that M (u, v) does not depend on the choice of t, take an s>0 such that u, v ϕ(i). Without any loss of generality we can assume that s>t. s s Since 0 < s < 1, from (6) we have t tm 1 ( u t, v t )=s t s M 1( u t, v t )=sm 1( t u s t, t v s t )=sm 1( u s, v s ). This proves that the function M : R R R is correctly defined. Clearly, M is positively homogeneous, that is M (tu, tv) =tm (u, v), u,v R, t 0. Taking a = 1 in (1) we get, for all b R, b <β, M 1 (u, v)+b M 1 (u + b, v + b), u,v ϕ(i). Repleacing here u by u b and v by v b we obtain M 1 (u b, v b) M 1 (u, v) b, u, v ϕ(i), for all b R, b <β.therefore M 1 (u + b, v + b) =M 1 (u, v)+b, b <β; u, v ϕ(i). Take arbitrary u, v R, b R and choose a t>0 such that u, v ϕ(i) and u t b t <β.then, by the definition of M and its homogeneity, we have ( u M (u + b, v + b) =tm 1 t + b t, v t + b ) [ ( u = t M 1 t t, v ) + b ] t t ( u = tm 1 t, v ) + b = M (u, v)+b. t Now the same resoning as in the proof of Theorem 1 shows that there is a q [0, 1] such that M (u, v) =qu +(1 q)v, u, v R. Since the function M is uniquelly determined and the definition does not depend on the the choice of the compact interval I, we infer that ϕ(m(ϕ 1 (u),ϕ 1 (v))) = M (u, v), u,v ϕ(j). Consequently, M(x, y) =ϕ 1 (qϕ(u)+(1 q)ϕ(y)), x,y J. Since remaining statements are obvious, the proof is complete.

10 330 J. Matkowski 4. A remark and open problem Remark 5. In the basic supposition of the above results that N-affinity of some f implies its M-convexity, we assume that N is a quasi-arithmetic mean. This assumption is essential because there are different non-quasiarithemetic means with the same classes of affine functions. For instance, the logarithmic mean L :(0, ) (0, ), { x y L(x, y) =, x y log x log y x, x = y and the mean M :(0, ) (0, ), M(x, y) := x + xy + y, 3 x + y are symmetric and have the same classes continuous L-affine functions and M-affine functions coincide (cf. [9], [10]). We end up this paper with the following open Problem 1. Let M and N be strict, symmetric and continuous means in an interval J. Suppose that for all intervals I J and for all continuous functions f : I J, the implication f is N-convex = f is M-convex holds true. Is then M = N? Acknowledgement 1. The research was supported by the Silesian University Mathematical Department (Iterative Functional Equations and Real Analysis program). 5. References [1] J. Aczél, A generalization of the notion of convex functions, Norske Vid. Selsk. Forhdl. 19, Nr 4 (1947), [] J. Aczél, J. Dhombres, Functional equations in several variables, Encyclopedia of Mathematics and its applications 31, Cambridge University Press, Cambridge, New York, New Rochelle, Melbourne, Sydney, [3] G. Aumann, Konvexe Funktionen und Induktion bei Ungleichungen zwischen Mittelverten S.-B. math.-naturw. Abt. Bayer. Akad. Wiss. München (1933), [4] Z. Daróczy, Zs. Páles, Convexity with given infinite weight sequences, Stochastica II(1987), 5-1. [5] D. Gronau, J. Matkowski, Geometrical convexity and generalization of the Bohr-Mollerup theorem on the Gamma function, Math. Pannonica 4/(1993), [6] M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities, Cauchy s Equation and Jensen s inequality, Prace Nauk. Uniw. Ślask. 489, Polish Scientific Publishers, Warszawa - Kraków - Katowice, 1985.

11 When M-convexity implies N-convexity 331 [7] J. Matkowski, L p -like paranorms, in Selected Topics in Functional Equations and Iteration Theory, Proceedings of the Austrian-Polish Seminar, Universität Graz, October 4-6, 1991, edited by D. Gronau and L. Reich, Grazer Math. Ber. 316(199), [8] J. Matkowski, J. Rätz, Convex functions with respect to an arbitrary mean, Internat. Ser. Num. Math. 13(1997), [9] J. Matkowski, Affine functions with respect to the logarithmic mean, Colloq. Math. 95(003), [10] J. Matkowski, Convex and affine functions with respect to a mean and a characterizationn of the quasi-arithmetic means, Real Anal. Exchange 9 (1), 003/004, Received: November 9, 006