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ON SOME INEQUALITIES EQUIVALENT to the WRIGHT-CONVEXITY 451 and ( + )+ ( − ) − ( ) 2 ( ) = F X H F X H 2 F X

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ON SOME INEQUALITIES EQUIVALENT to the WRIGHT-CONVEXITY 451 and ( + )+ ( − ) − ( ) 2 ( ) = F X H F X H 2 F X Journal of Mathematical Inequalities Volume 9, Number 2 (2015), 449–461 doi:10.7153/jmi-09-38 ON SOME INEQUALITIES EQUIVALENT TO THE WRIGHT–CONVEXITY ANDRZEJ OLBRYS´ (Communicated by J. Pecari´ˇ c) Abstract. In the present paper we establish some conditions and inequalities equivalent to the Wright-convexity. 1. Introduction and terminology Let X be a real linear space, and let D ⊂ X be a convex set. A function f : D → R is called convex if f (λx +(1 − λ)y) λ f (x)+(1 − λ) f (y). (1) x,y∈D λ ∈[0,1] , ∈ λ = 1 If the above inequality holds for all x y D with 2 then f is said to be Jensen convex. In 1954 E.M. Wright [23] introduced a new convexity property. A function f : D → R is called Wright-convex if f (λx +(1 − λ)y)+ f ((1 − λ)x + λy) f (x)+ f (y). (2) x,y∈D λ ∈[0,1] One can easily see that convex functions are Wright-convex, and Wright-convex func- tions are Jensen-convex. On the other hand, if f : X → R is additive, that is, f (x + y)= f (x)+ f (y), x,y ∈ X; then f is also Wright-convex. The main result concerning Wright-convex functions is the much more surprising statement that any Wright-convex function can be decom- posed as the sum of such functions. The following theorem has been proved by Ng [14] for functions defined on convex subsets of Rn and was extended by Kominek [10] for functions defined on convex subsets of more general structures (see also [13]). THEOREM 1. Let X be a real linear space, and let D ⊂ X be an algebraically open and convex set. A function f : D → R is Wright-convex if and only if there exist a convex function F : D → R and an additive function a : X → R such that f (x)=F(x)+a(x), x ∈ D. (3) Mathematics subject classification (2010): 39B62, 26A51, 26B25. Keywords and phrases: Convexity, Jensen-convexity, Wright-convexity. Ð c ,Zagreb 449 Paper JMI-09-38 450 ANDRZEJ OLBRYS´ It follows immediately from the above representation that Wright-convex func- tions are either very regular or very irregular. It is known that quite week conditions e.g. locally boundedness from above or below at some point, Christensen measurabil- ity, continuity at least one point, imply the continuity of such functions. On the other hand, if they are discontinuous, they have a dense graph. A survey of other results concerning Wright-convex functions may be found in the papers [10], [13], [14], [16], [17], [24]. 2. Regularity properties Recall (see [22]) that a function f : D → R,whereD is an open subset of topo- logical space X , is said to be symmetric at a point x ∈ D,if lim[ f (x + h)+ f (x − h) − 2 f (x)] = 0. h→0 Of course, every continuous function is symmetric, but the converse is not true. Let us start with the following THEOREM 2. Let D ⊂ Rn be a convex and open set. If f : D → R is a Wright- convex function, then it is symmetric at every point x ∈ D. Proof. By Theorem 1 f has the form f (x)=a(x)+F(x), x ∈ D, where F : D → R is a convex function and a : Rn → R an additive function. Because convex functions defined on a subset of finite dimensional real linear space are contin- uous, then by additivity of a we get lim[ f (x + h)+ f (x − h) − 2 f (x)] = lim[F(x + h)+F(x − h) − 2F(x)] = 0. h→0 h→0 As we know from the classical theory of convex functions the convexity of a twice differentiable function can be inferred from the sign of its second derivative. Namely, the following theorem holds true (see [1], [12], [15], [19], [20]). THEOREM 3. Let I ⊂ R be an open interval, and let f : I → R be a convex func- tion. Then f is twice differentiable almost everywhere in I . Whenever the second derivative exists, f (x) 0. Moreover, if f is twice differentiable, then it is convex if and only if the function f is non-negative in I . Let us recall that the upper and the lower second symmetric derivative of f at x, are respectively defined by the formulas ( + )+ ( − ) − ( ) 2 ( ) = f x h f x h 2 f x , Ds f x : limsup 2 h→0 h ON SOME INEQUALITIES EQUIVALENT TO THE WRIGHT-CONVEXITY 451 and ( + )+ ( − ) − ( ) 2 ( ) = f x h f x h 2 f x . Ds f x : liminf h→0 h2 If the two extreme derivatives are equal and finite at a point x then f is said to have a 2 ( ) symmetric derivative at this point and the common value is denoted by Ds f x . It is easy to check that if f is twice differentiable at x,then 2 ( )= 2 ( )= ( ), Ds f x Ds f x f x 2 ( ) however, Ds f x can exist even at points of discontinuity. For example, for an arbitrary additive function a : R → R we have 2 ( )= , ∈ R, Ds a x 0 x and as we know (see [12]) such functions may be discontinuous at every point. For Wright-convex functions we have the following counterpart of Theorem 3 THEOREM 4. Let I ⊂ R be an open interval, and let f : I → R be a Wright- convex function. Then there exists a second symmetric derivative almost everywhere in 2 ( ) I . Whenever it exists, Ds f x 0. Proof. Let f : I → R be a Wright-convex function. By the representation (3) f has the form f = a + F ,wherea : R → R is an additive and F : I → R is a convex function. Observe that for all x ∈ I and h ∈ R such that x − h, x + h ∈ I we have f (x + h)+ f (x − h) − 2 f (x)=F(x + h)+F(x − h) − 2F(x) 0. By the above equality, convexity of F and on account of Theorem 3 we obtain 2 ( )= 2 ( )= ( ) , Ds f x Ds F x F x 0 almost everywhere in I . The proof of our theorem is complete. Since the second symmetric derivative of a Wright-convex function may not exist, we need to use the concept of the lower and upper second symmetric derivative to characterize this kind of convexity. Inspired by methods contained in [15, p. 24] we prove the following THEOREM 5. Let I ⊂ R be an open interval. A function f : I → R is Wright- convex if and only if the map x + y I × I (x,y) −→ f (x)+ f (y) − 2 f , (4) 2 2 ( ) ∈ is continuous, and Ds f x 0,x I. 452 ANDRZEJ OLBRYS´ Proof. If f is Wright-convex then f = a+F for some additive function a : R → R and a convex function F : I → R. Because f (x + h)+ f (x − h) − 2 f (x)=F(x + h)+F(x − h) − 2F(x) 2 ( ) ∈ then by convexity of F we get Ds f x 0, x I . The continuity of a map given by (4) follows from the continuity of F . 2 ( ) Conversely, suppose that the map given by (4) is continuous and Ds f x 0, x ∈ I . Let us consider the sequence of functions fn : I → R given by the formula 1 f (x) := f (x)+ x2, n ∈ N. n n Clearly, the mapping x + y I × I (x,y) −→ f (x)+ f (y) − 2 f , n n n 2 is continuous, for all n ∈ N, moreover, ( ) −→ ( ), , 2 ( ) > , ∈ , ∈ N. fn x n→∞ f x and Ds fn x 0 x I n We shall show that fn is a Wright-convex function for all n ∈ N. On account of The- orem 12 from [17] it is enough to show that fn is a convex function in the sense of Jensen, n ∈ N. Assume the contrary, that there exists a subinterval I0 =[a0,b0] ⊂ I such that f (a )+ f (b ) a + b J (a ,b ) := n 0 n 0 − f 0 0 < 0. fn 0 0 2 n 2 An easy calculation shows that a + b a + b a + 3b 3a + b J (a ,b )=J a , 0 0 + J 0 0 ,b + 2J 0 0 , 0 0 < 0, fn 0 0 fn 0 2 fn 2 0 fn 4 4 consequently, one of the intervals a + b a + b a + 3b 3a + b a , 0 0 , 0 0 ,b , 0 0 , 0 0 0 2 2 0 4 4 − =[ , ] − = b0 a0 can be chosen to replace I0 by a smaller interval I1 a1 b1 , with b1 a1 2 ( , ) < and Jfn a1 b1 0. Using induction, we can construct a sequence of intervals Ik =[ak,bk], k ∈ N such that b0 − a0 I + ⊂ I , b − a = , J (a ,b ) < 0, k ∈ N. k 1 k k k 2k fn k k Denote by x0 the unique element of the intersection { } = ∩∞ [ , ]. x0 k=1 ak bk Obviously, from the choice of the sequence {Ik}k∈N and by continuity of the function given by (4) we obtain 2 ( ) Ds fn x0 0 which is a contradiction. ON SOME INEQUALITIES EQUIVALENT TO THE WRIGHT-CONVEXITY 453 3. Characterizations Let start this section with the following THEOREM 6. A function f : D → R is Wright-convex if and only if for all x,y ∈ , λ ∈ [ , ], , ∈ N, ,..., , ,..., ∈ [ , ] k = n = λ D 0 1 k n s1 sk t1 tn 0 1 such that ∑ j=1 s j ∑i=1 ti the following inequality holds k n ∑ f (s jx +(1 − s j)y)+∑ f ((1 −ti)x +tiy) kf(y)+nf(x).
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