Aequat. Math. c The Author(s) 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 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 Functions generating (m,M, Ψ)-Schur-convex sums 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. 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 Functions generating (m,M, Ψ)-Schur-convex sums 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 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. Functions generating (m,M, Ψ)-Schur-convex sums 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; 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]. Functions generating (m,M, Ψ)-Schur-convex sums 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. 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. 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[19] Nikodem, K., P´ ales, Zs: Characterizations of inner product spaces by strongly convex functions. Banach J. Math. Anal. 5(1), 83–87 (2011) [20] Nikodem, K., Rajba, T., Waso ¸ wicz, Sz: Functions generating strongly Schur-convex sums. In: Bandle, C., et al. (eds.) Inequalities and Applications 2010, International Series of Numerical Mathematics 161, pp. 175–182. Springer, New York (2012) [21] Olbry´ s, A.: On delta Schur-convex mappings. Publ. Math. Debrecen 86, 313–323 (2015) [22] Polyak, B.T.: Existence theorems and convergence of minimizing sequences in extremum problems with restrictions. Sov. Math. Dokl. 7, 72–75 (1966) [23] Rajba, T., Waso ¸ wicz, Sz: Probabilistic characterization of strong convexity. Opusc. Math. 31(1), 97–103 (2011) [24] Roberts, A.W., Varberg, D.E.: Convex Functions. Academic Press, New York (1973) [25] Schur, I.: Uber eine Klasse von Mittelbildungen mit Anwendungen auf die Determinan- tentheorie. Sitzungsber. Berl. Math. Ges. 22, 9–20 (1923) [26] Vesely, ´ L., Zaj´ ıˇ cek, L.: Delta-convex mappings between Banach spaces and applications. Dissert. Math. 289, PWN, Warszawa (1989) [27] Vial, J.P.: Strong convexity of sets and functions. J. Math. Econ. 9, 187–205 (1982) S. S. Dragomir and K. Nikodem AEM Silvestru Sever Dragomir Mathematics, College of Engineering and Science Victoria University P.O. 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
aequationes mathematicae – Springer Journals
Published: May 30, 2018
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