On Strongly Jensen m-Convex Functions - Hikari Ltd

Pure Mathematical Sciences, Vol. 6, 2017, no. 1, 87 - 94 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/pms.2017.61018

On Strongly Jensen m-Convex Functions1 Teodoro Lara Departamento de F´ısica y Matemáticas Universidad de Los Andes N. U. Rafael Rangel, Trujillo, Venezuela Roy Quintero Department of Mathematics University of Iowa, Iowa City, USA Edgar Rosales Departamento de F´ısica y Matemáticas Universidad de Los Andes N. U. Rafael Rangel, Trujillo, Venezuela Jos´ e L. S´ anchez Universidad Central de Venezuela Escuela de Matemáticas, Caracas, Venezuela c 2017 Teodoro Lara et al. This article is distributed under the Creative ComCopyright mons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract In this research we introduce the concept of a strongly Jensen m-convex function. We also present some interesting examples and prove general properties of this type of functions as well as its relationship with other classes of convexity. Finally, we demonstrate a discrete inequality of Jensen-type.

Mathematics Subject Classificatin: 26A51, 39B62 1This research has been partially supported by Central Bank of Venezuela.

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Keywords: m-convex, Jensen convex, Jensen m-convex, Strongly Jensen m-convex 1. Introduction The concept of a Jensen m-convex function was initially introduced in [11], more recently in [6] this concept is recasted, abundant properties and the algebra of this class of functions are set out. In this paper we go a little further and define, based on both, the original concept (of Jensen m-convex function) and the definition of strongly midconvex functions ([1, 4, 7, 8, 9] and many more) strongly Jensen m-convex functions. We begin by recalling both definitions. Definition 1. Let m ∈ (0, 1]. A function f : [0, +∞) → R which satisfies the inequality 1 x+y f (x) + f (y) , cm = 1 + f ≤ cm cm m being x, y ∈ [0, +∞) arbitrary will be called Jensen m-convex on the interval [0, +∞). The set of all Jensen m-convex functions on the interval [0, +∞) is denoted by Jm [+∞), and in the event that the domain is [0, b] the set is denoted by Jm [b]. Definition 2. A function f : [0, +∞) → R is called strongly midconvex with modulus c > 0 if x+y f (x) + f (y) c f ≤ − (x − y)2 , 2 2 4 for all x, y ∈ [0, +∞). We now introduce a new definition combining the two previous ones. Definition 3. Let m ∈ (0, 1]. A function f : [0, +∞) → R is said to be strongly Jensen m-convex on [0, +∞) with modulus c > 0 if x+y f (x) + f (y) c f ≤ − 2 (x − y)2 cm cm cm for all x, y ∈ [0, +∞). We shall denote the set of all strongly Jensen m-convex functions on [0, +∞) with modulus c > 0 by m c[+∞), and if the domain is the interval [0, b] then this set is referred to as SJm c[b]. We must point out that SJm c[+∞) ⊂ Jm [+∞) (SJm c[b] ⊂ Jm [b]) respectively. Building up examples of strongly Jensen m-convex functions from Jensen m-convex ones is an easy task; the following result shows one way of doing it. Proposition 4. Let g ∈ Jm [+∞) and c > 0 be a constant. Then the function f : [0, +∞) → R given by f (x) = g(x) + cx2 , x ∈ [0, +∞) is in SJm c[+∞).

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Proof. For any x, y ∈ [0, +∞), 2 x+y x+y x+y = g +c f cm cm cm g(x) + g(y) c 2 2 ≤ + 2 x + 2xy + y (since g ∈ Jm [+∞)) cm cm 2 f (x) − cx + f (y) − cy 2 c 2 2 = + 2 x + 2xy + y cm cm f (x) + f (y) c(2 − cm ) 2 c = − 2 (x − y)2 + (x + y 2 ), cm cm c2m and conclusion follows by taking into account that 2 − cm ≤ 0.

Example 1. In [2], the function g : [0, +∞) → R given by 1 4 (x − 5x3 + 9x2 − 5x) 12 is considered and, it is proved that g is 21 -convex (m = 12 ). Moreover, in [5], it is shown that g is a strongly 12 -convex with modulus 0 < c ≤ 31 . With this in mind, we are able to build up, according to the foregoing proposition, the function 1 f (x) = (x4 − 5x3 + 9x2 − 5x) + cx2 , c ∈ (0, 13 ] 12 which becomes, in this context, strongly Jensen 12 -convex with modulus c. g(x) =

2. Algebra In this section we shall show some algebra of the class of functions recently defined in both SJm c[+∞) and SJm c[b]. Proposition 5. If f ∈ SJm c1 [b] and g ∈ SJm c2 [b], then the function h : [0, b] → R given by h(x) = max{f (x), g(x)}, x ∈ [0, b] is in SJm c[b] with c = min{c1 , c2 }. Proof. By hypothesis, and for x, y ∈ [0, b], x+y f (x) + f (y) c1 h(x) + h(y) c (2.1) f ≤ − 2 (x − y)2 ≤ − 2 (x − y)2 cm cm cm cm cm and

x+y (2.2) g cm

≤

g(x) + g(y) c1 h(x) + h(y) c − 2 (x − y)2 ≤ − 2 (x − y)2 . cm cm cm cm

Now we combine (2.1) with (2.2) and the result follows.

Proposition 6. Let {fn }n≥1 be a sequence of functions in SJm c[b] such that fn (x) → f (x), x ∈ [0, b]. Then f ∈ SJm c[b].

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Proof. For any x, y ∈ [0, b] x+y fn (x) + fn (y) c x+y = lim fn ≤ lim − 2 (x − y)2 f n→∞ n→∞ cm cm cm cm or f (x) + f (y) c x+y ≤ − 2 (x − y)2 . f cm cm cm Proposition 7. Let f : [0, b] → R be a starshaped function; that is, f (tx) ≤ tf (x), x ∈ [0, b] and t ∈ [0, 1] arbitraries. If f ∈ SJm c[b], then f ∈ SJn c[b] for all n ∈ (0, m). Proof. For all x, y ∈ [0, b] c x c y cm x cm y m m f +f + x + y c cm x cm y 2 cn cn  cn cn  =f − 2 − f ≤ cn cm cm cm cn cn cm f (x) + f (y) c cm 2 ≤ − 2 (x − y)2 cn cm cm cn f (x) + f (y) c = − 2 (x − y)2 . cn cn Proposition 8. Let r > 0, k ≤ 0 and f : [0, b] → R be given. Define g : [0, b/r] → R by x 7→ f (rx) and h : [0, b] → R by x 7→ f (x) + k. If f ∈ SJm c[b], then g ∈ SJm r2 c[b/r] (in particular, if r ≥ 1, g ∈ SJm c[b/r]) and h ∈ SJm c[b]. Proof. For all x, y ∈ [0, b/r] x+y x + y f (rx) + f (ry) c =f r ≤ − 2 (rx − ry)2 g cm cm cm cm 2 g(x) + g(y) cr = − 2 (x − y)2 . cm cm On the other hand, if x, y ∈ [0, b] are arbitrary, x+y x+y f (x) + f (y) c h =f +k ≤ − 2 (x − y)2 + k cm cm cm cm f (x) + k + f (y) + k c k(cm − 2) = − 2 (x − y)2 + cm cm cm h(x) + h(y) c ≤ − 2 (x − y)2 . cm cm


Proposition 9. Let f ∈ SJm c[+∞) and α ≥ 0 be given. SJm αc[+∞). In particular, if α ≥ 1, αf ∈ SJm c[+∞).

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Then αf ∈

Proof. For all x, y ∈ [0, +∞) f (x) + f (y) c x + y ≤α − α 2 (x − y)2 (αf ) cm cm cm (αf )(x) + (αf )(y) αc − 2 (x − y)2 . ≤ cm cm Proposition 10. If f, g : [0, +∞) → R are functions in SJm c1 [+∞) and SJm c2 [+∞) respectively, then f +g ∈ SJm (c1 +c2 )[+∞). In particular, f +g ∈ SJm c1 [+∞) ∩ SJm c2 [+∞). Proof. For any x, y ∈ [0, +∞) x+y f (x) + f (y) c1 g(x) + g(y) c2 (f + g) ≤ − 2 (x − y)2 + − 2 (x − y)2 cm cm cm cm cm (f + g)(x) + (f + g)(y) c1 + c2 − (x − y)2 . = cm c2m Proposition 11. Let f : [0, b] → R and g : [0, b0 ] → R be functions such that Range(f ) ⊆ [0, b0 ] and g is nondecreasing. Suppose also that the functions f − id and f + id (id denotes the identity function) are similarly ordered on [0, b] ([6]). If f ∈ Jm [b] and g ∈ SJm c[b0 ], then g ◦ f ∈ SJm c[b]. Proof. By taking x, y ∈ [0, b] arbitrary x+y x + y g(f (x)) + g(f (y)) c (g ◦ f ) =g f ≤ − 2 (f (x) − f (y))2 . cm cm cm cm Since f − id and f + id are similarly ordered on [0, b], (f (x) − f (y))2 ≥ (x − y)2 . Therefore, x+y (g ◦ f )(x) + (g ◦ f )(y) c ≤ − 2 (x − y)2 . (g ◦ f ) cm cm cm Remark 12. The function f : [0, b] → R, defined by f (x) = x2 is strongly m-convex with modulus 1. Therefore, f ∈ SJm 1[b]. If g(x) = [f (x)]2 = x4 2 were in SJm c[b], then cc6 ≤ 0 which is not possible. This fact permits us to m conclude that the multiplication of functions in SJm c[b] is not necessarily in SJm c[b].

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3. More Properties In this section we show some results principally based on [1] and [7], at the same time a Jensen-type inequality is also obtained. We begin with a result similar to one given in [1]. The set of positive integers will be denoted by N. Lemma 13. Let c > 0 and f ∈ SJm c[+∞), then (3.1) k k k k k k f n x + m 1 − n y ≤ n f (x) + m 1 − n f (y) − cm n 1 − n (x − y)2 cm cm cm cm cm cm for all x, y ∈ [0, +∞), k, n ∈ N and k < 2n . Proof. We proceed by induction on n. If n = 1, then k = 0 or k = 1; in either case (3.1) holds. Let us suppose that (3.1) takes place for n ∈ N and k < 2n and show the inequality for n + 1. Let x, y ≥ 0, without loss of generality we may assume that x ≤ y and k < 2n . Then k k 1 k k 1 f n+1 x + m 1 − n+1 y = f x+m 1− n y + y cm cm cm cnm cm cm 2 1 k 1 k c k k ≤ f n x+m 1− n y + f (y) − 2 n x + m 1 − n y − y . cm cm cm cm cm cm cm Now by hypothesis, k k f n+1 x + m 1 − n+1 y cm cm 1 k k k k 1 2 ≤ f (x) + m 1 − n f (y) − cm n 1 − n (x − y) + f (y) n cm cm cm cm cm cm 2 c k k − 2 n x+m 1− n y−y cm cm cm k m k 1 k k = n+1 f (x) + 1− n + f (y) − cm n+1 1 − n (x − y)2 cm cm cm cm cm cm 2 c k − 2 n (x − my) + my − y . cm cm k But cmm 1 − ckn + c1m = m 1 − cn+1 and ckn (x − my) + my − y ≤ ckn (x − y), m m m m we get, by taking into account the assumption x ≤ y, k k f n+1 x + m 1 − n+1 y cm cm k k k k ≤ n+1 f (x) + m 1 − n+1 f (y) − cm n+1 1 − n+1 (x − y)2 , cm cm cm cm and proof is complete.


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Definition 14. Let t be a fixed number in (0, 1), c > 0 a constant and m as before. A function f : [0, +∞) → R is called strongly t − m−convex with modulus c if f (tx + m(1 − t)y) ≤ tf (x) + m(1 − t)f (y) − cmt(1 − t)(x − y)2 ; x, y ∈ [0, +∞). n o 1 Theorem 15. Let c > 0 as in definition 3.1, m ∈ k : k ∈ N and f : [0, +∞) → R be a continuous function. Then for any t ∈ 0, c2m , f is strongly t − m−convex with modulus c if and only if f ∈ SJm c[+∞). Proof. If f is strongly t − m−convex with modulus c for all t ∈ 0, c2m , then 1 2 f ∈ SJm c[+∞) since cm ∈ 0, cm . In the other direction, if f ∈ SJm c[+∞), n o (3.1) holds and because cm is a positive integer, the set ckn : k, n ∈ N, k < 2n m i h 2 is a dense set in 0, cm . Thus by continuity of f the result is true for any 2 t ∈ 0, cm . Theorem 16. Let f ∈ SJm c[+∞) and starshaped, c > 0, n ∈ N, n ≥ 2 and x1 , x2 , . . . , xn ∈ [0, ∞). Then " # X n n−1 n−1 n−k X X 1 1 1 f (c x ) k m f xk ≤ cn−1−k (ak − xk+1 )2 , f (cn−1 −c m m x1 ) + n−k n k=1 n cn−1 c m m k=1 k=1 Pk where ak = j=1 xj , for k = 1, 2, . . . , n. Proof. Proof runs by induction; we only need to check that X n n−1 n−1 X X 1 f (cn−k m xk ) n−1 n−1−k xk ≤ n−1 f (cm x1 ) + f −c cm (ak − xk+1 )2 , n−k c c m m k=1 k=1 k=1 since for being f starshaped, X n n 1 1 X f xk ≤ f xk . n k=1 n k=1 The case n = 2 holds because function f is strongly Jensen m-convex with modulus c. We assume inequality true for n and show for n + 1. X Pn n+1 k=1 cm xk + cm xn+1 f xk = f cm k=1 Pn n 2 f c X k=1 cm xk + f (cm xn+1 ) ≤ − 2 cm xk − cm xn+1 cm cm 1 X n 2 f cm xn+1 1 = f cm x k + − c an − xn+1 cm cm k=1

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and by the inductive hypothesis, n−1 n−k X f c (c x ) 1 1 m k+1 m ≤ f cn−1 m (cm x1 ) + cm cn−1 cn−k m m k=1 n−1 X 2 2 f (cm xn+1 ) − c an − xn+1 −c cn−1−k cm ak − cm xk+1 + m cm k=1 (n+1)−k n n X X f cm xk+1 1 n n−k −c = n f cm x 1 + cm (ak − xk+1 )2 . (n+1)−k cm cm k=1 k=1 References [1] A. Az´ ocar, J. Giménez, K. Nikodem and J. L. Sánchez, On strongly midconvex functions, Opuscula Mathematica, 31 (2011), no. 1, 15–26. https://doi.org/10.7494/opmath.2011.31.1.15 [2] S. S. Dragomir, On Some New inequalities of Hermite-Hadamard Type for m-Convex Functions, Tamkang J. of Math., 33 (2002), no. 1, 45–55. [3] S. S. Dragomir and Gh. Toader, Some Inequalities for m-Convex Functions, Studia Univ. Babes-Bolyai, Math., 38 (1993), no. 1, 21–28. [4] M.V. Jovanoviˇc, A note on strongly convex and strongly quasiconvex functions, Math. Notes, 60 (1996), no. 5, 778–779. (in Russian). https://doi.org/10.4213/mzm1892 [5] T. Lara, N. Merentes, R. Quintero and E. Rosales, On strongly m-convex functions, Mathematica Aeterna, 5 (2015), no. 3, 521–535. [6] T. Lara, R. Quintero, E. Rosales and J. L. Sáncez, On a generalization of the class of Jensen convex functions, Aequat. Math., 90 (2016), 569–580. https://doi.org/10.1007/s00010-016-0406-2 [7] N. Merentes and K. Nikodem, Remarks on strongly convex functions, Aequat. Math., 80 (2010), 193–199. https://doi.org/10.1007/s00010-010-0043-0 [8] K. Nikodem and Zs. P´ ales, Characterizations of inner product spaces by strongly convex functions, Banach J. Math. Anal., 5 (2011), no. 1, 83–87. https://doi.org/10.15352/bjma/1313362982 [9] E.S. Polovinkin, Strongly convex analysis, Sb. Mathematics, 187 (1996), no. 2, 259–286. https://doi.org/10.1070/sm1996v187n02abeh000111 [10] A. W. Roberts and D. E. Varberg, Convex Functions, Academic Press, New York, 1973. [11] G. Toader, The hierarchy of convexity and some classic inequalities, J. Math. Inequalities, 3 (2009), 305–313. https://doi.org/10.7153/jmi-03-30

Received: October 21, 2016; Published: August 21, 2017