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/lp/springer_journal/functions-generating-m-m-varvec-psi-schur-convex-sums-05CK5Q66Yp
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Abstract
Aequat. Math. 93 (2019), 79–90 c The Author(s) 2018 0001-9054/19/010079-12 published online May 30, 2018 Aequationes Mathematicae https://doi.org/10.1007/s00010-018-0569-0 Functions generating (m,M,Ψ)-Schur-convex sums Silvestru Sever Dragomir and Kazimierz Nikodem Dedicated to Professor Karol Baron on his 70th birthday. Abstract. The notion of (m, M, Ψ)-Schur-convexity is introduced and functions generating (m, M, Ψ)-Schur-convex sums are investigated. An extension of the Hardy–Littlewood–P´ olya majorization theorem is obtained. A counterpart of the result of Ng stating that a function generates (m, M, Ψ)-Schur-convex sums if and only if it is (m, M, ψ)-Wright-convex is proved and a characterization of (m, M, ψ)-Wright-convex functions is given. Mathematics Subject Classification. Primary 26A51; Secondary 39B62. Keywords. Strongly convex functions, (m, M, ψ)-convex (Jensen-convex, Wright-convex) functions, (m, M, Ψ)-Schur-convexity. 1. Introduction Let (X, ·) be a real normed space. Assume that D is a convex subset of X and c is a positive constant. A function f : D → R is called: – strongly convex with modulus c if f (tx +(1 − t)y) ≤ tf (x)+(1 − t)f (y) − ct(1 − t)x − y (1) for all x, y ∈ D and t ∈ [0, 1]; – strongly Wright-convex with modulus c if f (tx +(1 − t)y)+ f ((1 − t)x + ty) ≤ f (x)+ f (y) − 2ct(1 − t)x − y (2) for all x, y ∈ D and t ∈ [0, 1]; – strongly Jensen-convex with modulus c if (1) is assumed only for t = , that is x + y f (x)+ f (y) c f ≤ − x − y ,x,y ∈ D. (3) 2 2 4 80 S. S. Dragomir and K. Nikodem AEM The usual concepts of convexity, Wright-convexity and Jensen-convexity correspond to the case c = 0, respectively. The notion of strongly convex functions was introduced by Polyak [22] and they play an important role in optimization theory and mathematical economics. Many properties and appli- cations of them can be found in the literature (see, for instance, [10, 15, 19, 22– 24, 27]). Let us also mention the paper [18] by the second author which is a survey article devoted to strongly convex functions and related classes of functions. In [1] the first author introduced the following concepts of (m, ψ)-lower convex, (M, ψ)-upper convex and (m, M, ψ)-convex functions (see also [2–4]): Assume that D is a convex subset of a real linear space X, ψ : D → R is a convex function and m, M ∈ R. A function f : D → R is called (m, ψ)-lower convex ((M, ψ)-upper convex ) if the function f − mψ (the function Mψ − f ) is convex. We say that f : D → R is (m, M, ψ)-convex if it is (m, ψ)-lower convex and (M, ψ)-upper convex. Denote the above classes of functions by: L(D, m, ψ)= {f : D → R | f − mψ is convex}, U (D, M, ψ)= {f : D → R | Mψ − f is convex}, B(D, m, M, ψ)= L(D, m, ψ) ∩U (D, M, ψ). Let us observe that if f ∈B(D, m, M, ψ) then f − mψ and Mψ − f are convex and then (M − m)ψ is also convex, implying that M ≥ m whenever ψ is not trivial (i.e. is not the zero function). If m> 0 and (X, ·) is an inner product space (that is, the norm · in X is induced by an inner product: x = x, x ) the notions of (m, · )- lower convexity and strong convexity with modulus m coincide. Namely, in this case the following characterization was proved in [19]: A function f is 2 n strongly convex with modulus c if and only if f − c· is convex (for X = R this result can be also found in [8, Prop. 1.1.2]). However, if (X, ·) is not an inner product space, then the two notions are different. There are functions f ∈L(D, m, · ) which are not strongly convex with modulus m, as well as there are functions strongly convex with modulus m which do not belong to L(D, m, · ) (see the examples given in [6]). If M> 0and f ∈U (D, M, ψ), then f is a difference of two convex functions. Such functions are called d.c. convex or δ-convex and play an important role in convex analysis (cf. e.g. [26] and the reference therein). Functions from the class U (D, M, · ) with M> 0 were also investigated in [13] under the name approximately concave functions. In [5] Dragomir and Ionescu introduced the concept of g-convex dominated functions, where g is a given convex function. Namely, a function f is called g-convex dominated, if the functions g + f and g − f are convex. Note that this concept can be obtained as a particular case of (m, M, ψ)-convexity by choosing m = −1, M =1 and ψ = g. Observe also (cf. [1]), that in the case Vol. 93 (2019) Functions generating (m,M, Ψ)-Schur-convex sums 81 where I ⊂ R is an open interval and f, ψ : I → R are twice differentiable, f ∈B(I, m, M, ψ) if and only if mψ (t) ≤ f (t) ≤ Mψ (t), for all t ∈ I. In particular, if I ⊂ (0, ∞), f : I → R is twice differentiable and ψ(t)= − ln t, then f ∈B(I, m, M, − ln) if and only if m ≤ t f (t) ≤ M, for all t ∈ I, (4) which is a convenient condition to verify in applications. Let I ⊂ R be an interval and x =(x ,...,x ), y =(y ,...,y ) ∈ I , where 1 n 1 n n ≥ 2. Following I. Schur (cf. e.g. [12, 25]) we say that x is majorized by y,and write x y, if there exists a doubly stochastic n × n matrix P (i.e. a matrix containing nonnegative elements with all rows and columns summing up to 1) such that x = y · P . A function F : I → R is said to be Schur-convex if F (x) ≤ F (y) whenever x y, x, y ∈ I . It is known, by the classical works of Schur [25], Hardy et al. [7] and Kara- mata [9] that if a function f : I → R is convex then it generates Schur-convex sums, that is the function F : I → R defined by F (x)= F (x ,...,x )= f (x )+ ··· + f (x ) 1 n 1 n is Schur-convex. It is also known that the convexity of f is a sufficient but not necessary condition under which F is Schur-convex. A full characterization of functions generating Schur-convex sums was given by Ng [16]. Namely, he proved that a function f : I → R generates Schur-convex sums if and only if it is Wright-convex (cf. also [17]). Recently Nikodem et al. [20] obtained similar results in connection with strong convexity in inner product spaces. Let us also mention the paper by Olbry´s[21] in which delta Schur-convex mappings are investigated. The aim of this paper is to present some generalizations and counterparts of the above mentioned results related to (m, ψ)-lower convexity, (M, ψ)-upper convexity and (m, M, ψ)-convexity. We introduce the notion of (m, M, Ψ)- Schur-convex functions and give a sufficient and necessary condition for a function f to generate (m, M, Ψ)-Schur-convex sums. As a corollary we obtain a counterpart of the classical Hardy–Littlewood–P´ olya majorization theorem. Finally we introduce the concept of (m, M, ψ)-Wright-convex functions, prove a representation theorem for them and present an Ng-type characterization of functions generating (m, M, Ψ)-Schur-convex sums. It is worth underlining, that our results concern a few different classes of functions related to convexity and are formulated in vector spaces, that is in a much more general setting than the original ones. 82 S. S. Dragomir and K. Nikodem AEM 2. Main results Let X be a real vector space. Similarly as in the classical case we define majorization in the product space X . Namely, given two n–tuples x =(x ,..., x ), y =(y ,...,y ) ∈ X we say that x is majorized by y, written x y,if n 1 n (x ,...,x )=(y ,...,y ) · P 1 n 1 n for some doubly stochastic n × n matrix P . In what follows we will assume that D is a convex subset of a real vector space X, ψ : D → R is a convex function and m, M ∈ R. For any n ≥ 2 define Ψ : D → R by Ψ (x ,...,x )= ψ(x )+ ··· + ψ(x ),x ,...,x ∈ D. (5) n 1 n 1 n 1 n We say that a function F : D → R is (m, M, Ψ )-Schur-convex if for all x, y ∈ D x y =⇒ F (x) ≤ F (y) − m Ψ (y) − Ψ (x) (6) n n and x y =⇒ F (x) ≥ F (y) − M Ψ (y) − Ψ (x) . (7) n n If only condition (6) [condition (7)] is satisfied, we say that F is (m, Ψ )-lower ((M, Ψ )-upper) Schur-convex. Note that if x y then Ψ (x) ≤ Ψ (y). It follows from the fact that the n n function ψ is convex and so it generates Schur-convex sums Ψ . Given a function f : D → R andaninteger n ≥ 2 we define the function F : D → R by F (x ,...,x )= f (x )+ ··· + f (x ),x ,...,x ∈ D. (8) n 1 n 1 n 1 n Now, let D be a convex subset of a real vector space X, and let m, M ∈ R. Assume that ψ : D → R is a convex function and Ψ : D → R is defined by (5). We will prove now that (m, M, ψ)-convex functions generate (m, M, Ψ )- Schur-convex sums. Theorem 1. (i) If f ∈L(D, m, ψ), then the function F defined by (8) is (m, Ψ )-lower Schur-convex; (ii) If f ∈U (D, M, ψ), then the function F defined by (8) is (M, Ψ )-upper n n Schur-convex; (iii) If f ∈B(D, m, M, ψ), then the function F defined by (8) is (m, M, Ψ )- n n Schur-convex. Proof. To prove (i) fix x =(x ,...,x )and y =(y ,...,y )in D with x y. 1 n 1 n There exists a doubly stochastic n × n matrix P =[t ] such that x = y · P . ij Then x = t y,j =1,...,n. j ij i i=1 Vol. 93 (2019) Functions generating (m,M, Ψ)-Schur-convex sums 83 Since f ∈L(D, m, ψ), the function g = f − mψ is convex and hence n n n n g(x )+ ··· + g(x )= g t y ≤ t g(y ) 1 n ij i ij i j=1 i=1 j=1 i=1 n n n n = t g(y )= g(y ) t = g(y )+ ··· + g(y ). ij i i ij 1 n i=1 j=1 i=1 j=1 Consequently, F (x)= f (x )+ ··· + f (x ) n 1 n = g(x )+ ··· + g(x )+ m ψ(x )+ ··· + ψ(x 1 n 1 n ≤ g(y )+ ··· + g(y )+ m ψ(x )+ ··· + ψ(x ) 1 n 1 n = f (y )+ ··· + f (y ) − m ψ(y )+ ··· + ψ(y ) 1 n 1 n + m ψ(x )+ ··· + ψ(x ) 1 n = F (y) − m Ψ (y) − Ψ (x) . n n n This shows that F satisfies (6), i.e. it is (m, Ψ )-lower Schur-convex. n n The proof of part (ii) is similar. Since f ∈U (D, M, ψ), the function h = Mψ − f is convex. Hence, for x and y as previously, we have F (x)= f (x )+ ··· + f (x ) n 1 n =+M ψ(x )+ ··· + ψ(x ) − h(x ) − ··· − h(x ) 1 n 1 n ≥ M ψ(x )+ ··· + ψ(x ) − h(y ) − ··· − h(y ) 1 n 1 n = M ψ(x )+ ··· + ψ(x ) − M ψ(y )+ ··· + ψ(y ) 1 n 1 n + f (y )+ ··· + f (y ) 1 n = F (y) − M Ψ (y) − Ψ (x) . n n n Part (iii) follows from (i) and (ii). As an immediate consequence of the above theorem, we obtain the following counterpart of the classical Hardy–Littlewood–P´ olya majorization theorem [7]. Corollary 2. Let I ⊂ R be an interval and n ≥ 2. Assume that x =(x ,...,x ), 1 n y =(y ,...,y ) ∈ I satisfy: 1 n (a) x ≤ ··· ≤ x ,y ≤ ··· ≤ y ; 1 n 1 n (b) y + ··· + y ≤ x + ··· + x , k =1,...,n − 1; 1 k 1 k (c) y + ··· + y = x + ··· + x . 1 n 1 n Assume also that f, ψ : I → R and ψ is convex. (i) If f ∈L(D, m, ψ), then f (x )+ ··· + f (x ) ≤ f (y )+ ··· + f (y ) − m Ψ (y) − Ψ (x) ; 1 n 1 n n n 84 S. S. Dragomir and K. Nikodem AEM (ii) If f ∈U (D, M, ψ), then f (x )+ ··· + f (x ) ≥ f (y )+ ··· + f (y ) − M Ψ (y) − Ψ (x) ; 1 n 1 n n n (iii) If f ∈B(D, m, M, ψ), then f (y )+ ··· + f (y ) − M Ψ (y) − Ψ (x) ≤ f (x )+ ··· + f (x ) 1 n n n 1 n ≤ f (y )+ ··· + f (y ) − m Ψ (y) − Ψ (x) . 1 n n n Proof. Note that assumptions (a)–(c) imply x y (see e.g. [12]) and apply Theorem 1. Remark 3. Specifying the functions ψ and f in Corollary 2 above, one can get various analytic inequalities. For example, if I ⊂ (0, ∞)and f ∈B(I, m, M, −ln), then for all (x ,...,x ), (y ,...,y ) ∈ I satisfying conditions (a)–(c), we get 1 n 1 n n n n n x x i i m ln ≤ f (y ) − f (x ) ≤ M ln , i i y y i i i=1 i=1 i=1 i=1 or, equivalently, n m n n M x exp [ f (y )] x i i i i=1 ≤ ≤ . (9) y exp [ f (x )] y i i i i=1 i=1 i=1 1 p If we take, for instance, I =[k, K] ⊂ (0, ∞)and f (t)= t , with p(p−1) 2 p p p p> 0, p = 1, then t f (t)= t ∈ [k ,K ] , which means [cf. (4)] that p p f ∈B(I, k ,K , − ln). Therefore, by (9), we then have p p n n p(p−1)k n p p(p−1)K x exp ( y ) x i i i=1 i ≤ ≤ . n p y exp ( x ) y i i i=1 i=1 i=1 One can give other examples by choosing f (t)= t with q< 0,f (t)= t ln t, etc. We say that a function f : D → R is (m, ψ)-lower Jensen-convex ((M, ψ)- upper Jensen-convex ) if the function f − mψ (the function Mψ − f ) is Jensen- convex, i.e. satisfies (3) with c = 0. We say that f : D → R is (m, M, ψ)-Jensen- convex if it is (m, ψ)-lower Jensen-convex and (M, ψ)-upper Jensen-convex. In the next theorem we show that functions generating (m, M, Ψ )-Schur- convex sums must be (m, M, ψ)-Jensen–convex. Theorem 4. Let f : D → R. (i) If for some n ≥ 2 the function F defined by (8) is (m, Ψ )-lower Schur- n n convex, then f is (m, ψ)-lower Jensen-convex; (ii) If for some n ≥ 2 the function F defined by (8) is (M, Ψ )-upper Schur- n n convex, then f is (M, ψ)-upper Jensen-convex; (iii) If for some n ≥ 2 the function F defined by (8) is (m, M, Ψ )-Schur- n n convex, then f is (m, M, ψ)- Jensen-convex. Vol. 93 (2019) Functions generating (m,M, Ψ)-Schur-convex sums 85 Proof. To prove (i) take y ,y ∈ D and put x = x = (y + y ). Consider 1 2 1 2 1 2 the points y =(y ,y ,y ,...,y ),x =(x ,x ,y ,...,y ) 1 2 2 2 1 2 2 2 (if n = 2, then we take y =(y ,y ), x =(x ,x )). One can check easily that 1 2 1 2 x y. Therefore, by (6), F (x) ≤ F (y) − m Ψ (y) − Ψ (x) , n n n n that is y + y y + y 1 2 1 2 2f ≤ f (y )+ f (y ) − m ψ(y )+ ψ(y ) − 2ψ . 1 2 1 2 2 2 Hence, for g = f − mψ we have y + y y + y y + y 1 2 1 2 1 2 2g =2f − 2mψ 2 2 2 ≤ f (y )+ f (y ) − m (ψ(y )+ ψ(y ) = g(y )+ g(y ), 1 2 1 2 1 2 which means that f is (m, ψ)-lower Jensen-convex. The proof of part (ii) is similar. Part (iii) follows from (i) and (ii). We say that a function f : D → R is (m, ψ)-lower Wright-convex ((M, ψ)- upper Wright-convex ) if the function f − mψ (the function Mψ − f ) is Wright- convex, i.e. satisfies (2) with c = 0. We say that f : D → R is (m, M, ψ)- Wright-convex if it is (m, ψ)-lower Wright-convex and (M, ψ)-upper Wright- convex. As was shown above in Theorems 1 and 2, if a function f : D → R is (m, M, ψ)-convex, then for every n ≥ 2 the corresponding function F de- fined by (8)is(m, M, Ψ )-Schur-convex and if for some n ≥ 2 the function F is (m, M, Ψ )-Schur-convex, then f is (m, M, ψ)-Jensen-convex. The next n n theorem characterizes all the functions f for which F are (m, M, Ψ )- Schur– n n convex. It is a counterpart of the result of Ng [16] on functions generating Schur–convex sums. Recall also that a subset D of a vector space X is said to be algebraically open if for every x ∈ D and for every y ∈ X there exists ε> 0 such that {ty +(1 − t)x | t ∈ (−ε, ε)}⊂ D. Theorem 5. Let f : D → R,where D is an algebraically open convex subset of a vector space X.Then: (i) If f is (m, ψ)-lower Wright-convex, then for every n ≥ 2 the function F defined by (8) is (m, Ψ )-lower Schur-convex. Conversely, if for some n n n ≥ 2 the function F is (m, Ψ )-lower Schur-convex, then f is (m, ψ)- n n lower Wright-convex; (ii) If f is (M, ψ)-upper Wright-convex, then for every n ≥ 2 the function F defined by (8) is (M, Ψ )-upper Schur-convex. Conversely, if for some n ≥ 2 the function F is (M, Ψ )-upper Schur-convex, then f is (M, ψ)- n n upper Wright-convex; 86 S. S. Dragomir and K. Nikodem AEM (iii) If f is (m, M, ψ)- Wright-convex, then for every n ≥ 2 the function F defined by (8) is (m, M, Ψ )- Schur-convex. Conversely, if for some n n n ≥ 2 the function F is (m, M, Ψ )- Schur-convex, then f is (m, M, ψ)- n n Wright-convex. Proof. To prove (i) assume that f is (m, ψ)-lower Wright-convex and fix an n ≥ 2. Since the function g = f − mψ is Wright-convex, it is of the form g = g + a, where g is convex and a is additive (cf. [11]; here the assumption 1 1 that D is algebraically open is needed). Therefore it generates Schur-convex sums. Thus, for x =(x ,...,x ) y =(y ,...,y ), we have 1 n 1 n g(x )+ ··· + g(x ) ≤ g(y )+ ··· + g(y ). 1 n 1 n Hence f (x )+ ··· + f (x ) − m ψ(x )+ ··· + ψ(x ) 1 n 1 n ≤ g(y )+ ··· + g(y ) − m ψ(y )+ ··· + ψ(y ) , 1 n 1 n which means that F (x) ≤ F (y) − m Ψ (y) − Ψ (x) , n n n n that is F is (m, Ψ )-lower Schur-convex. Now, assume that for some n ≥ 2 n n the function F is (m, Ψ )-lower Schur-convex. Take y ,y ∈ D and t ∈ (0, 1). n n 1 2 Put x = ty +(1 − t)y ,x =(1 − t)y + ty 1 1 2 2 1 2 and, if n> 2, take additionally x = y = z ∈ D for i =3,...,n. Then i i x =(x ,...,x ) y =(y ,...,y ). Therefore, by (6), 1 n 1 n F (x) ≤ F (y) − m Ψ (y) − Ψ (x) , n n n n that is f (ty +(1 − t)y )+ f ((1 − t)y + ty ) ≤ f (y )+ f (y ) − m ψ(y ) 1 2 1 2 1 2 1 +ψ(y ) − ψ(x ) − ψ(x ) . 2 1 2 Hence, for g = f − mψ we get g(ty +(1 − t)y )+ g((1 − t)y + ty ) 1 2 1 2 = f (ty +(1 − t)y )+ f ((1 − t)y + ty ) − mψ(ty +(1 − t)y ) 1 2 1 2 1 2 −mψ((1 − t)y + ty ) 1 2 ≤ f (y )+ f (y ) − mψ(y ) − mψ(y )= g(y )+ g(y ). 1 2 1 2 1 2 Thus g is Wright-convex, which means that f is (m, ψ)-lower Wright-convex. The proof of part (ii) is similar. Part (iii) follows from (i) and (ii). Remark 6. In the special case where (X, ·) is an inner product space, ψ = · and m = c> 0, parts (i) of the above Theorems 1, 4, 5 reduce to the results obtained in [20] for strong Schur-convexity. For m =0 and X = R they coincide with the Ng theorem [16]. Vol. 93 (2019) Functions generating (m,M, Ψ)-Schur-convex sums 87 Finally, we give a representation theorem for (m, M, ψ)-Wright-convex func- tions. It is known (and easy to check) that every convex function is Wright- convex, and every Wright-convex function is Jensen-convex, but not the con- verse (some examples can be found in [18]). In [16] Ng proved that a function f defined on a convex subset of R is Wright-convex if and only if it can be represented in the form f = f + a, where f is a convex function and a is an 1 1 additive function (see also [18]). Kominek [11] extended that result to functions defined on algebraically open subset of a vector space. An analogous result for strongly Wright-convex functions was obtained in [14]. In the next theorem we give a similar representation for (m, M, ψ)-Wright-convex functions. In the proof we will use the following fact: Lemma 7. Assume that f, g : D → R are convex functions, a : X → R is additive and a(x)= f (x) − g(x) for all x ∈ D.Then a is an affine function on D. Proof. Fix x, y ∈ D and consider the function ϕ :[0, 1] → R defined by ϕ(s)= a(sx +(1 − s)y)= f (sx +(1 − s)y) − g(sx +(1 − s)y),s ∈ [0, 1]. As a difference of convex functions on [0, 1], ϕ is continuous on (0, 1). Fix any t ∈ (0, 1) and take a sequence (q ) of rational numbers in (0, 1) tending to t. By the additivity of a we have a(q x +(1 − q )y)= q a(x)+(1 − q )a(y), n n n n whence ϕ(q )= q a(x)+(1 − q )a(y). n n n Going to the limit we get ϕ(t)= ta(x)+(1 − t)a(y). Hence a(tx +(1 − t)y)= ta(x)+(1 − t)a(y), which proves that a is affine on D. Theorem 8. Let f : D → R,where D is an algebraically open convex subset of a vector space X.Then: (i) f is (m, ψ)-lower Wright-convex if and only if f = g + a ,where g ∈ 1 1 1 L(D, m, ψ) and a : X → R is additive; (ii) f is (M, ψ)-upper Wright-convex if and only if f = g + a ,where g ∈ 2 2 2 U (D, M, ψ) and a : X → R is additive; (iii) f is (m, M, ψ)- Wright-convex if and only if f = g + a,where g ∈ B(D, m, M, ψ) and a : X → R is additive. 88 S. S. Dragomir and K. Nikodem AEM Proof. To prove (i) assume first that f is (m, ψ)-lower Wright-convex, that is h = f −mψ is Wright-convex. By the Ng representation theorem [16] (extended by Kominek [11] to functions defined on algebraically open domains), there exist a convex function h : D → R and an additive function a : X → R such 1 1 that h = h + a on D. Then g = h + mψ belongs to L(D, m, ψ)and 1 1 1 1 f = h + mψ = h + a + mψ = g + a , 1 1 1 1 which was to be proved. Conversely, if f = g + a with some g ∈L(D, m, ψ) 1 1 1 and a additive, then f − mψ = g − mψ + a is Wright-convex as a sum of 1 1 1 a convex function and an additive function. This shows that f is (m, ψ)-lower Wright-convex. The proof of part (ii) is analogous. Part (iii). If f = g + a, where g ∈B(D, m, M, ψ)and a : X → R is additive, then, by (i) and (ii) f is (m, ψ)-lower Wright-convex and (M, ψ)-upper Wright- convex. Consequently, it is (m, M, ψ)-Wright-convex. The proof in the opposite direction is more delicate. If f is (m, M, ψ)- Wright-convex, then f − mψ and Mψ − f are Wright-convex. Then f − mψ = h + a and Mψ − f = h + a 1 1 2 2 with some convex functions h ,h and additive functions a ,a . Hence 1 2 1 2 a + a =(M − m)ψ − (h + h ) 1 2 1 2 which, by Lemma 5, implies that A = a + a is affine. Denote a = a and 1 2 1 g = f − a. Then g − mψ = f − a − mψ = h , which implies that g ∈L(D, m, ψ) because h is convex. Also Mψ − g = Mψ − f + a = h + a + a = h + A, 2 2 2 which implies that g ∈U (D, m, ψ) because h + A is convex. Thus g ∈ B(D, m, ψ)and f = g + a, which finishes the proof. Open Access. This article is distributed under the terms of the Creative Commons At- tribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. References [1] Dragomir, S.S.: On the reverse of Jessen’s inequality for isotonic linear functionals. J. Inequal. Pure Appl. Math. 2(3), 1–13 (2001). (Art. 36) [2] Dragomir, S.S.: Some inequalities for (m, M )-convex mappings and applications for Csisz´ ar Φ-divergence in information theory. Math. J. 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Box 14428 Melbourne VIC 8001 Australia e-mail: sever.dragomir@vu.edu.au and DST-NRF Centre of Excellence in the Mathematical and Statistical Sciences, School of Computer Science and Applied Mathematics University of the Witwatersrand (Wits) Private Bag 3 Johannesburg 2050 South Africa Kazimierz Nikodem Department of Mathematics University of Bielsko-Biala ul. Willowa 2 43-309 Bielsko-Biala Poland e-mail: knikodem@ath.bielsko.pl Received: January 20, 2018 Revised: April 17, 2018
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aequationes mathematicae – Springer Journals
Published: May 30, 2018