Inequalities of Jensen Type for h-convex Functions on Linear Spaces

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1 Mathematica Moravica Vol , 07 2 Inequalities of Jensen Type for h-convex Functions on Linear Spaces Silvestru Sever Dragomir Abstract. Some inequalities of Jensen type for h-convex functions defined on convex subsets in real or complex linear spaces are given. Applications for norm inequalities are provided as well.. Introduction We recall here some concepts of convexity that are well known in the literature. Let I be an interval in R. Definition [38]. We say that f : I R is a Godunova-Levin function or that f belongs to the class Q I if f is non-negative and for all x, y I and t 0, we have. f tx + t y t f x + t f y. Some further properties of this class of functions can be found in [28, 29, 3,44,47,48]. Among others, its has been noted that non-negative monotone and non-negative convex functions belong to this class of functions. The above concept can be extended for functions f : C X [0, where C is a convex subset of the real or complex linear space X and the inequality. is satisfied for any vectors x, y C and t 0,. If the function f : C X R is non-negative and convex, then is of Godunova- Levin type. Definition 2 [3]. We say that a function f : I R belongs to the class P I if it is nonnegative and for all x, y I and t [0, ] we have.2 f tx + t y f x + f y. Obviously Q I contains P I and for applications it is important to note that also P I contain all nonnegative monotone, convex and quasi convex functions, i. e. nonnegative functions satisfying 99 Mathematics Subject Classification. Primary: 26D5, 25D0. Key words and phrases. Convex functions, Integral inequalities, h-convex functions. 07 c 205 Mathematica Moravica

2 08 Inequalities of Jensen Type for h-convex Functions on Linear Spaces.3 f tx + t y max {f x, f y} for all x, y I and t [0, ]. For some results on P -functions see [3] and [45] while for quasi convex functions, the reader can consult [30]. If f : C X [0,, where C is a convex subset of the real or complex linear space X, then we say that it is of P -type or quasi-convex if the inequality.2 or.3 holds true for x, y C and t [0, ]. Definition 3 [7]. Let s be a real number, s 0, ]. A function f : [0, [0, is said to be s-convex in the second sense or Breckner s-convex if f tx + t y t s f x + t s f y for all x, y [0, and t [0, ]. For some properties of this class of functions see [,2,7,8,26,27,39,4,50]. The concept of Breckner s-convexity can be similarly extended for functions defined on convex subsets of linear spaces. It is well known that if X, is a normed linear space, then the function f x = x p, p is convex on X. Utilising the elementary inequality a + b s a s + b s that holds for any a, b 0 and s 0, ], we have for the function g x = x s that g tx + t y = tx + t y s t x + t y s t x s + [ t y ] s = t s g x + t s g y for any x, y X and t [0, ], which shows that g is Breckner s-convex on X. In order to unify the above concepts for functions of real variable, S. Varošanec introduced the concept of h-convex functions as follows. Assume that I and J are intervals in R, 0, J and functions h and f are real non-negative functions defined in J and I, respectively. Definition 4 [53]. Let h : J [0, with h not identical to 0. We say that f : I [0, is an h-convex function if for all x, y I we have.4 f tx + t y h t f x + h t f y for all t 0,. For some results concerning this class of functions see [6, 42, 49, 5 53]. This concept can be extended for functions defined on convex subsets of linear spaces in the same way as above replacing the interval I be the corresponding convex subset C of the linear space X. We can introduce now another class of functions.

3 Silvestru Sever Dragomir 09 Definition 5. We say that the function f : C X [0, is of s- Godunova-Levin type, with s [0, ], if.5 f tx + t y t s f x + t s f y, for all t 0, and x, y C. We observe that for s = 0 we obtain the class of P -functions while for s = we obtain the class of Godunova-Levin. If we denote by Q s C the class of s-godunova-levin functions defined on C, then we obviously have P C = Q 0 C Q s C Q s2 C Q C = Q C for 0 s s 2. For different inequalities related to these classes of functions, see [ 4, 6, 9 37, 40 42, 45 52]. A function h : J R is said to be supermultiplicative if.6 h ts h t h s for any t, s J. If the inequality.6 is reversed, then h is said to be submultiplicative. If the equality holds in.6 then h is said to be a multiplicative function on J. In [53] it has been noted that if h : [0, [0, with h t = x + c p, then for c = 0 the function h is multiplicative. If c, then for p 0, the function h is supermultiplicative and for p > the function is submultiplicative. We observe that, if h, g are nonnegative and supermultiplicative, the same is their product. In particular, if h is supermultiplicative then its product with a power function l r t = t r is also supermultiplicative. The case of h-convex function with h supermultiplicative is of interest due to several Jensen type inequalities one can derive. The following results were obtained in [53] for functions of a real variable. However, with similar proofs they can be extended to h-convex function defined on convex subsets in linear spaces. Theorem. Let h : J [0, be a supermultiplicative function on J. If the function f : C X [0, is h-convex on the convex subset C of the linear space X, then for any w i 0, i {,..., n}, n 2 with := n w i > 0 we have wi.7 f w i x i h f x i. In particular, we have the unweighted inequality.8 f x i h f x i. n n

4 0 Inequalities of Jensen Type for h-convex Functions on Linear Spaces Corollary [27]. If the function f : C X [0, is Breckner s- convex on the convex subset C of the linear space X with s 0,, then for any x i C, w i 0, i {,..., n}, n 2 with := n w i > 0 we have.9 f w i x i W s n wi s f x i. If X, is a normed linear space, then for s 0,, x i X, w i 0, i {,..., n}, n 2 with := n w i > 0 we have the norm inequality s.0 w i x i wi s x i s. Corollary 2. If the function f : C X [0, is of s-godunova-levin type, with s [0, ], on the convex subset C of the linear space X, then for any x i C, w i > 0, i {,..., n}, n 2 we have. f w i x i Wn s f x i. w s i This result generalizes the Jensen type inequality obtained in [44] for s =. Let K be a finite non-empty set of positive integers. We can define the index set function, see also [53].2 J K := h w i f x i h W K f w i x i, W K where W K := w i > 0, x i C, i K. We notice that if h : [0, [0, is a supermultiplicative function on [0, and the function f : C X [0, is h-convex on the convex subset C of the linear space X, then [ ] wi.3 J K h W K h f x i f w i x i 0. W K W K Theorem 2. Assume that h : [0, [0, is a supermultiplicative function on [0, and the function f : C X [0, is h-convex on the convex subset C of the linear space X. Let M and K be finite non-empty sets of positive integers, w i > 0, x i C, i K M. Then.4 J K M J K + J M 0, i.e., J is a superadditive index set functional. This results was proved in an equivalent form in [53] for functions of a real variable. The proof is similar for functions defined on convex sets in linear spaces.

5 Silvestru Sever Dragomir Corollary 3. With the assumptions of Theorem 2 and if we note M k := {,..., k}, then.5 J M n J M n... J M 2 0 and.6 J M n max i<j n 0. { h w i f x i + h w j f x j } wi x i + w j x j h w i + w j f w i + w j If we consider the functional J s K := wi s x i s s w i x i for s 0,, then we have the norm inequalities.7 wi s x i s s n n w i x i wi s x i s s w i x i 2... wi s x i s 2 s w i x i 0 and.8 wi s x i s s w i x i { w s i x i s + wj s x j s w i x i + w j x j s} 0 max i<j n where w i 0, x i X, i {,..., n}, n More Jensen Type Results Let h z = n=0 a nz n be a power series with complex coefficients and convergent on the open disk D 0, R C, R > 0. We have the following

6 2 Inequalities of Jensen Type for h-convex Functions on Linear Spaces examples 2. h z = h z = h z = h z = n= n=0 n=0 n=0 n zn = ln, z D 0, ; z 2n! z2n = cosh z, z C; 2n +! z2n+ = sinh z, z C; z n =, z D 0,. z Other important examples of functions as power series representations with nonnegative coefficients are: h z = n! zn = exp z, z C, n=0 h z = 2n z2n = + z 2 ln, z D 0, ; z n= Γ n + 2 h z = z 2n+ = sin z, z D 0, ; 2.2 π 2n + n! h z = n=0 n= 2n z2n = tanh z, z D 0, h z = 2 F α, β, γ, z = z D 0, ; n=0 Γ n + α Γ n + β Γ γ n!γ α Γ β Γ n + γ zn, α, β, γ > 0, where Γ is Gamma function. The following result may provide many examples of supemultiplicative functions. Theorem 3. Let h z = n=0 a nz n be a power series with complex coefficients and convergent on the open disk D 0, R C, R > 0. Assume that 0 < r < R and define h r : [0, ] [0,, h r t := hrt hr. Then h r is supemultiplicative on [0, ]. Proof. We use the Čebyšev inequality for synchronous the same monotonicity sequences c i i N, b i i N and nonnegative weights p i i N : 2.3 p i p i c i b i p i c i p i b i,

7 Silvestru Sever Dragomir 3 for any n N. Let t, s 0, and define the sequences c i := t i, b i := s i. These sequences are decreasing and if we apply Čebyšev s inequality for these sequences and the weights p i := a i r i 0 we get 2.4 a i r i a i rts i a i rt i a i rs i for any n N. Since the series a i r i, a i rts i, a i rt i and are convergent, then by letting n in 2.4 we get i.e. h r h rts h rt h rs h r ts h r t h r s. a i rs i This inequality is also obviously satisfied at the end points of the interval [0, ] and the proof is completed. Remark. Utilising the above theorem, we then conclude that the functions h r : [0, ] [0,, h r t := r, r 0, rt and h r : [0, ] [0,, h r t := exp [ r t], r > 0 are supermultiplicative. We say that the function f : C X [0, is r-resolvent convex with r fixed in 0,, if f is h-convex with h t = r rt, i.e. [ 2.5 f tx + t y r rt f x + r + rt f y for any x, y C and t [0, ]. In particular, for r = 2 we have 2-resolvent convex functions defined by the condition 2.6 f tx + t y 2 t f x + + t f y for any t [0, ] and x, y C. Since t < 2 t < t and t < + t < for t 0, t it follows that any nonnegative convex function is 2 -resolvent convex which, in its turn, is of Godunova-Levin type. ]

8 4 Inequalities of Jensen Type for h-convex Functions on Linear Spaces We say that the function f : C X [0, is r-exponential convex with r fixed in 0,, if f is h-convex with h t = exp [ r t], i.e. 2.7 f tx + t y exp [ r t] f x + exp rt f y for any t [0, ] and x, y C. Since t exp [ r t] and t exp rt for t [0, ] it follows that any nonnegative convex function is r-exponential convex with r 0,. Corollary 4. Let h z = n=0 a nz n be a power series with complex coefficients and convergent on the open disk D 0, R C, R > 0. Assume that 0 < r < R and define h r : [0, ] [0,, h r t := hrt hr. If the function f : C X [0, is h r -convex on the convex subset C of the linear space X, namely 2.8 f tx + t y [h rt f x + h r t f y] h r for any t [0, ] and x, y C, then for any x i C, w i 0, i {,..., n}, n 2 with := n w i > 0 we have 2.9 f w i x i h r w i f x i. h r Remark 2. If the function f : C X [0, is 2-resolvent convex on C, then for any x i C, w i 0, i {,..., n}, n 2 with := n w i > 0 we have f w i x i f x i. 2 w i If the function f : C X [0, is r-exponential convex with r fixed in 0,, then for any x i C, w i 0, i {,..., n}, n 2 with := n w i > 0 we have [ f w i x i exp r w ] i f x i. 3. Some Related Functionals Let us fix K P f N the class of finite parts of N and x i C i K. Now consider the functional J K : S + K R given by 3. J K p := h f p i x i 0

9 Silvestru Sever Dragomir 5 where S + K := { p = p i i I p i 0, i K and > 0 } with h : 0, 0, and f is nonnegative on C. Theorem 4. Let h : 0, 0, be a supermultiplicative submultiplicative function on J. If the function f : C X [0, is h-convex h-concave on the convex subset C of the linear space X, then for any p, q S + K we have 3.2 J K p + q J K p + J K q, i.e., J K is a subadditive superadditive functional on S + K. Proof. If the function f : C X [0, is h -convex, then we have for any p, q S + K 3.3 J K p + q = h + Q K f p i + q i x i + PK P = h + Q K f K p ix i + Q K Q K q ix i + [ PK h + Q K h f p i x i + ] QK + h f q i x i + Q K := A. Since h is supermultiplicative, then PK h + Q K h h + and QK h + Q K h h Q K + which imply that A h f p i x i + h Q K f q i x i 3.4 Q K = J K p + J K q. Making use of 3.3 and 3.4 we deduce the desired result 3.2. The case when h is submultiplicative and f : C X [0, is h-concave goes likewise and the details are omitted.

10 6 Inequalities of Jensen Type for h-convex Functions on Linear Spaces Corollary 5. Let h : 0, 0, be a submultiplicative function on J. If the function f : C X [0, is h-concave on the convex subset C of the linear space X, then for any p, q S + K with p q, i.e. p i q i for any i K, we have 3.5 J K p J K q 0, i.e., J K is monotonic nondecreasing on S + K. The proof is obvious from 3.2 on noticing that J K p = J K p q + q J K p q + J K q J K q. We also have: Corollary 6. Let h : 0, 0, be a submultiplicative function on J. If the function f : C X [0, is h-concave on the convex subset C of the linear space X, then for any p, q S + K with Mp q mp, for some M > m > 0, we have 3.6 h M h J K p J K q h m h J K p. Proof. From the inequality 3.5 we have J K Mp J K q. However J K Mp = h M f Mp i x i M = h M f p i x i = h M h J K p, which proves the first inequality in 3.6. The second inequality can be proved similarly and the details are omitted. Further, consider the functional L K : S + K R given by 3.7 L K p := h pi h f x i 0, where S + K := { p = p i i I p i 0, i K and > 0 } with h : 0, 0, and f is nonnegative on C. Theorem 5. Let h : 0, 0, and f : C X [0,. If h is convex concave on 0, and g : 0, 0, defined by g t = ht t is decreasing increasing, then for any p, q S + K we have 3.8 L K p + q L K p + L K q.

11 Silvestru Sever Dragomir 7 Proof. If h is convex on 0,, then we have for any p, q S + K L K p + q = h + Q K pi + q i h f x i + Q K = h + Q K p PK i q P h K + Q i K Q K f x i + Q K 3.9 Since g t = ht t and Therefore 3.0 h + Q K = h + Q K + Q K + h + Q K Q K + Q K := B. is decreasing, then [ + Q K h Q K + h + Q K h pi qi h Q K h + Q K + Q K h h + Q K + Q K h Q K Q K. pi f x i qi f x i Q K ] f x i B h pi h f x i + h Q K qi h f x i Q K = L K p + L K q. Making use of 3.9 and 3.0 we deduce the desired result 3.8. The case when h is concave and g is increasing goes likewise and the details are omitted. Corollary 7. Let h : 0, 0, and f : C X [0,. If h is concave on 0, and g : 0, 0, defined by g t = ht t is increasing, then for any p, q S + K with p q we have 3. L K p L K q 0. Also, for any p, q S + K with Mp q mp, for some M > m > 0, we have h M 3.2 h L K p L K q h m h L K p.

12 8 Inequalities of Jensen Type for h-convex Functions on Linear Spaces We define the difference functional S K p := L K p J K p [ = h h pi f x i f p i x i ] We observe that, if h is supermultiplicative and f : C X [0, is h-convex, then by Jensen s type inequality.7 we have S K p 0 for any p S + K. Proposition. Let h : 0, 0, be supermultiplicative and f : C X [0, a h-convex function on C. If h is concave on 0, and g : 0, 0, defined by g t = ht t is increasing, then for any p, q S + K 3.3 S K p + q S K p + S K q 0. If p, q S + K with p q, then we have 3.4 S K p S K q 0. Also, for any p, q S + K with Mp q mp, for some M > m > 0, we have h M 3.5 h S K p S K q h m h S K p. The proof follows by Theorem 4 and Theorem 5 and we omit the details. If we take h t = t, i.e. in the case of convex functions we obtain from Proposition the superadditivity and monotonicity properties of the functional Je K p := p i f x i f p i x i established in [32]. From 3.5 we get 3.6 MJe K p Je K q mje K p that has been obtained in [24]. References [] M. Alomari and M. Darus, The Hadamard s inequality for s-convex function. Int. J. Math. Anal. Ruse , no. 3-6, [2] M. Alomari and M. Darus, Hadamard-type inequalities for s-convex functions. Int. Math. Forum , no , [3] G. A. Anastassiou, Univariate Ostrowski inequalities, revisited. Monatsh. Math., , no. 3,

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15 Silvestru Sever Dragomir 2 [42] M. A. Latif, On some inequalities for h-convex functions. Int. J. Math. Anal. Ruse 4 200, no , [43] D. S. Mitrinović and I. B. Lacković, Hermite and convexity, Aequationes Math , [44] D. S. Mitrinović and J. E. Pečarić, Note on a class of functions of Godunova and Levin. C. R. Math. Rep. Acad. Sci. Canada 2 990, no., [45] C. E. M. Pearce and A. M. Rubinov, P-functions, quasi-convex functions, and Hadamard-type inequalities. J. Math. Anal. Appl , no., [46] J. E. Pečarić and S. S. Dragomir, On an inequality of Godunova-Levin and some refinements of Jensen integral inequality. Itinerant Seminar on Functional Equations, Approximation and Convexity Cluj-Napoca, 989, , Preprint, 89-6, Univ. "Babeş-Bolyai, Cluj-Napoca, 989. [47] J. Pečarić and S. S. Dragomir, A generalization of Hadamard s inequality for isotonic linear functionals, Radovi Mat. Sarajevo 7 99, [48] M. Radulescu, S. Radulescu and P. Alexandrescu, On the Godunova-Levin-Schur class of functions. Math. Inequal. Appl , no. 4, [49] M. Z. Sarikaya, A. Saglam, and H. Yildirim, On some Hadamard-type inequalities for h-convex functions. J. Math. Inequal , no. 3, [50] E. Set, M. E. Özdemir and M. Z. Sarıkaya, New inequalities of Ostrowski s type for s-convex functions in the second sense with applications. Facta Univ. Ser. Math. Inform , no., [5] M. Z. Sarikaya, E. Set and M. E. Özdemir, On some new inequalities of Hadamard type involving h-convex functions. Acta Math. Univ. Comenian. N.S , no. 2, [52] M. Tunç, Ostrowski-type inequalities via h-convex functions with applications to special means. J. Inequal. Appl. 203, 203:326. [53] S. Varošanec, On h-convexity. J. Math. Anal. Appl , no., Address: Professor Silvestru Sever Dragomir Mathematics, College of Engineering & Science Victoria University, PO Box 4428 Melbourne City, MC 800 Australia Secondary address: School of Computational & Applied Mathematics University of the Witwatersrand Private Bag 3 Johannesburg 2050 South Africa address: sever.dragomir@vu.edu.au URL: