On Strongly Jensen m-convex Functions 1

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1 Pure Matheatical Sciences, Vol. 6, 017, no. 1, HIKARI Ltd, On Strongly Jensen -Convex Functions 1 Teodoro Lara Departaento de Física y Mateáticas Universidad de Los Andes N. U. Rafael Rangel, Trujillo, Venezuela Roy Quintero Departent of Matheatics University of Iowa, Iowa City, USA Edgar Rosales Departaento de Física y Mateáticas Universidad de Los Andes N. U. Rafael Rangel, Trujillo, Venezuela José L. Sánchez Universidad Central de Venezuela Escuela de Mateáticas, Caracas, Venezuela Copyright c 017 Teodoro Lara et al. This article is distributed under the Creative Coons Attribution License, which perits unrestricted use, distribution, and reproduction in any ediu, provided the original work is properly cited. Abstract In this research we introduce the concept of a strongly Jensen -convex function. We also present soe interesting exaples and prove general properties of this type of functions as well as its relationship with other classes of convexity. Finally, we deonstrate a discrete inequality of Jensen-type. Matheatics Subject Classificatin: 6A51, 39B6 1 This research has been partially supported by Central Bank of Venezuela.

2 88 Teodoro Lara et al. Keywords: -convex, Jensen convex, Jensen -convex, Strongly Jensen -convex 1. Introduction The concept of a Jensen -convex function was initially introduced in [11], ore 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 -convex function and the definition of strongly idconvex functions [1, 4, 7, 8, 9] and any ore strongly Jensen -convex functions. We begin by recalling both definitions. Definition 1. Let 0, 1]. A function f : [0, + R which satisfies the inequality x + y fx + fy f, = being x, y [0, + arbitrary will be called Jensen -convex on the interval [0, +. The set of all Jensen -convex functions on the interval [0, + is denoted by J [+, and in the event that the doain is [0, b] the set is denoted by J [b]. Definition. A function f : [0, + R is called strongly idconvex with odulus c > 0 if x + y fx + fy f 4 x y, for all x, y [0, +. We now introduce a new definition cobining the two previous ones. Definition 3. Let 0, 1]. A function f : [0, + R is said to be strongly Jensen -convex on [0, + with odulus c > 0 if x + y fx + fy f x y c for all x, y [0, +. We shall denote the set of all strongly Jensen -convex functions on [0, + with odulus c > 0 by c[+, and if the doain is the interval [0, b] then this set is referred to as SJ c[b]. We ust point out that SJ c[+ J [+ SJ c[b] J [b] respectively. Building up exaples of strongly Jensen -convex functions fro Jensen -convex ones is an easy task; the following result shows one way of doing it. Proposition 4. Let g J [+ and c > 0 be a constant. Then the function f : [0, + R given by fx = gx + cx, x [0, + is in SJ c[+.

3 On strongly Jensen -convex functions 89 Proof. For any x, y [0, +, x + y x + y x + y f = g + c gx + gy + c x + xy + y since g J c [+ = fx x + fy y + c x + xy + y c = fx + fy c x y + c x + y, c and conclusion follows by taking into account that 0. Exaple 1. In [], the function g : [0, + R given by gx = 1 1 x4 5x 3 + 9x 5x is considered and, it is proved that g is 1-convex = 1. Moreover, in [5], it is shown that g is a strongly 1-convex with odulus 0 < c 1. With this 3 in ind, we are able to build up, according to the foregoing proposition, the function fx = 1 1 x4 5x 3 + 9x 5x + cx, c 0, 1 3 ] which becoes, in this context, strongly Jensen 1 -convex with odulus c.. Algebra In this section we shall show soe algebra of the class of functions recently defined in both SJ c[+ and SJ c[b]. Proposition 5. If f SJ c 1 [b] and g SJ c [b], then the function h : [0, b] R given by hx = ax{fx, gx}, x [0, b] is in SJ c[b] with c = in{c 1, c }. Proof. By hypothesis, and for x, y [0, b], x + y fx + fy.1 f 1 x y c and x + y. g gx + gy 1 x y c hx + hy hx + hy Now we cobine.1 with. and the result follows. x y c x y. c Proposition 6. Let {f n } n 1 be a sequence of functions in SJ c[b] such that f n x fx, x [0, b]. Then f SJ c[b].

4 90 Teodoro Lara et al. Proof. For any x, y [0, b] x + y x + y f = li f n n or x + y f f n x + f n y li x y n c fx + fy x y. c Proposition 7. Let f : [0, b] R be a starshaped function; that is, ftx tfx, x [0, b] and t [0, 1] arbitraries. If f SJ c[b], then f SJ n c[b] for all n 0,. Proof. For all x, y [0, b] x x + y c f = f n + y c fx + fy = fx + fy c x c y f + f c x y. c n c x y c x y c Proposition 8. Let r > 0, k 0 and f : [0, b] R be given. Define g : [0, b/r] R by x frx and h : [0, b] R by x fx + k. If f SJ c[b], then g SJ r c[b/r] in particular, if r 1, g SJ c[b/r] and h SJ c[b]. Proof. For all x, y [0, b/r] x + y x + y g = f r frx + fry gx + gy = r x y. c On the other hand, if x, y [0, b] are arbitrary, x + y x + y fx + fy h = f + k = fx + k + fy + k hx + hy x y. c rx ry c x y + k c x y + k c

5 On strongly Jensen -convex functions 91 Proposition 9. Let f SJ c[+ and α 0 be given. SJ αc[+. In particular, if α 1, αf SJ c[+. Proof. For all x, y [0, + x + y fx + fy αf α αfx + αfy c α c x y αc x y. c Then αf Proposition 10. If f, g : [0, + R are functions in SJ c 1 [+ and SJ c [+ respectively, then f +g SJ c 1 +c [+. In particular, f +g SJ c 1 [+ SJ c [+. Proof. For any x, y [0, + x + y fx + fy f + g 1 x y + = c gx + gy x y c f + gx + f + gy 1 + c x y. Proposition 11. Let f : [0, b] R and g : [0, b ] R be functions such that Rangef [0, b ] and g is nondecreasing. Suppose also that the functions f id and f + id id denotes the identity function are siilarly ordered on [0, b] [6]. If f J [b] and g SJ c[b ], then g f SJ c[b]. Proof. By taking x, y [0, b] arbitrary x + y x + y gfx + gfy g f = g f c fx fy. c Since f id and f +id are siilarly ordered on [0, b], fx fy x y. Therefore, x + y g fx + g fy g f x y. c Reark 1. The function f : [0, b] R, defined by fx = x is strongly -convex with odulus 1. Therefore, f SJ 1[b]. If gx = [fx] = x 4 were in SJ c[b], then c 0 which is not possible. This fact perits us to c 6 conclude that the ultiplication of functions in SJ c[b] is not necessarily in SJ c[b].

6 9 Teodoro Lara et al. 3. More Properties In this section we show soe results principally based on [1] and [7], at the sae tie a Jensen-type inequality is also obtained. We begin with a result siilar to one given in [1]. The set of positive integers will be denoted by N. Lea 13. Let c > 0 and f SJ c[+, then 3.1 k f x+ 1 k y k fx+ 1 k fy c k 1 k x y for all x, y [0, +, k, n N and k < n. 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 < n and show the inequality for n + 1. Let x, y 0, without loss of generality we ay assue that x y and k < n. Then k f x + k 1 f x + Now by hypothesis, k f 1 = k x + [ k fx + fx + y [ 1 k = f y [ + 1 ] x + y + 1 fy c [ k fy k c fy k x + ] y + 1 x y ] [ k x + c But + 1 = we get, by taking into account the assuption x y, k f x + c n+1 k and proof is coplete. y fx + [ k y y y]. + 1 fy x y y y ] x y + y y]. and k x y + y y k x y, fy k x y,

7 On strongly Jensen -convex functions 93 Definition 14. Let t be a fixed nuber in 0, 1, c > 0 a constant and as before. A function f : [0, + R is called strongly t convex with odulus c if ftx + 1 ty tfx + 1 tfy t1 tx y ; x, y [0, +. { } 1 Theore 15. Let c > 0 as in definition 3.1, : k N and f : k [0, + R be a continuous function. Then for any t 0,, f is strongly t convex with odulus c if and only if f SJ c[+. Proof. If f is strongly t convex with odulus c for all t 0,, then 0, f SJ c[+ since 1. In the other direction, if f SJ c[+, { } k 3.1 holds and because is a positive integer, the set : k, n N, k < n [ is a dense set in 0, ]. Thus by continuity of f the result is true for any t 0,. Theore 16. Let f SJ c[+ and starshaped, c > 0, n N, n and x 1, x,..., x n [0,. Then 1 n [ f x k 1 1 n 1 f 1 f k x k n 1 n n 1 x 1 + c 1 k k a k x k+1 ], where a k = k j=1 x j, for k = 1,,..., n. Proof. Proof runs by induction; we only need to check that n f x k 1 n 1 f 1 f k x k n 1 1 x k c n k a k x k+1, since for being f starshaped, 1 n f x k 1 n n n f x k. The case n = holds because function f is strongly Jensen -convex with odulus c. We assue inequality true for n and show for n + 1. n+1 n f x k = f c x k + x n+1 f n x k + fc x n+1 n c c x k x n+1 1 = 1 n f x k + f x n+1 a n x n+1

8 94 Teodoro Lara et al. and by the inductive hypothesis, 1 [ 1 f 1 1 x 1 + n 1 f k x k+1 k n 1 ] c 1 k c a k x k+1 + fx n+1 a n x n+1 = 1 f n c x n 1 + f k x k+1 k n k a k x k+1. References [1] A. Azócar, J. Giénez, K. Nikode and J. L. Sánchez, On strongly idconvex functions, Opuscula Matheatica, , no. 1, [] S. S. Dragoir, On Soe New inequalities of Herite-Hadaard Type for -Convex Functions, Takang J. of Math., 33 00, no. 1, [3] S. S. Dragoir and Gh. Toader, Soe Inequalities for -Convex Functions, Studia Univ. Babes-Bolyai, Math., , no. 1, 1 8. [4] M.V. Jovanovič, A note on strongly convex and strongly quasiconvex functions, Math. Notes, , no. 5, in Russian. [5] T. Lara, N. Merentes, R. Quintero and E. Rosales, On strongly -convex functions, Matheatica Aeterna, 5 015, no. 3, [6] T. Lara, R. Quintero, E. Rosales and J. L. Sáncez, On a generalization of the class of Jensen convex functions, Aequat. Math., , [7] N. Merentes and K. Nikode, Rearks on strongly convex functions, Aequat. Math., , [8] K. Nikode and Zs. Páles, Characterizations of inner product spaces by strongly convex functions, Banach J. Math. Anal., 5 011, no. 1, [9] E.S. Polovinkin, Strongly convex analysis, Sb. Matheatics, , no., [10] A. W. Roberts and D. E. Varberg, Convex Functions, Acadeic Press, New York, [11] G. Toader, The hierarchy of convexity and soe classic inequalities, J. Math. Inequalities, 3 009, Received: October 1, 016; Published: August 1, 017