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

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 1 ( [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, 1) we have Some further properties of this class of functions can be found in [28,29,31,44,47,48].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 (1.1) is satisfied for any vectors x, y ∈ C and t ∈ (0, 1).If the function f : C ⊆ X → R is non-negative and convex, then is of Godunova-Levin type.Definition 2 ( [31]).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, 1] we have (1.2) f (tx + (1 − 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 (1.3) f (tx + (1 − t) y) ≤ max {f (x) , f (y)} for all x, y ∈ I and t ∈ [0, 1].
For some results on P -functions see [31] 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 (1.2) (or (1.3)) holds true for x, y ∈ C and t ∈ [0, 1].
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 Utilising the elementary inequality (a + b) s ≤ a s + b s that holds for any a, b ≥ 0 and s ∈ (0, 1], we have for the function g (x) = x s that for any x, y ∈ X and t ∈ [0, 1] , 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, 1) ⊆ J and functions h and f are real non-negative functions defined in J and I, respectively.
For some results concerning this class of functions see [6,42,49,[51][52][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.
Definition 5. We say that the function f : for all t ∈ (0, 1) and x, y ∈ C.
We observe that for s = 0 we obtain the class of P -functions while for s = 1 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 For different inequalities related to these classes of functions, see [1-4, 6, 9-37, 40-42, 45-52].
A function h : J → R is said to be supermultiplicative if (1.6) h (ts) ≥ h (t) h (s) for any t, s ∈ J.
If the inequality (1.6) is reversed, then h is said to be submultiplicative.If the equality holds in (1.6) then h is said to be a multiplicative function on J.
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 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.
In particular, we have the unweighted inequality is Breckner sconvex on the convex subset C of the linear space X with s ∈ (0, 1) , then for any If (X, • ) is a normed linear space, then for s ∈ (0, 1), , on the convex subset C of the linear space X, then for any This result generalizes the Jensen type inequality obtained in [44] for s = 1.
Let K be a finite non-empty set of positive integers.We can define the index set function, see also [53] (1.12) where Let M and K be finite non-empty sets of positive integers, 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.

More Jensen Type Results
Let h (z) = ∞ n=0 a n z n be a power series with complex coefficients and convergent on the open disk D (0, R) ⊂ C, R > 0. We have the following examples (2.1) Other important examples of functions as power series representations with nonnegative coefficients are: (2.2) where Γ is Gamma function.
The following result may provide many examples of supemultiplicative functions.

Theorem 3. Let h (z) =
∞ n=0 a n z 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 : 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 : Let t, s ∈ (0, 1) 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) Since the series are convergent, then by letting n → ∞ in (2.4) we get i.e. h r (ts) ≥ h r (t) h r (s) .This inequality is also obviously satisfied at the end points of the interval [0, 1] and the proof is completed.Remark 1. Utilising the above theorem, we then conclude that the functions We say that the function f : (2.5) for any x, y ∈ C and t ∈ [0, 1] .
In particular, for r = 1 2 we have 1 2 -resolvent convex functions defined by the condition for any t ∈ [0, 1] and x, y ∈ C. Since it follows that any nonnegative convex function is 1 2 -resolvent convex which, in its turn, is of Godunova-Levin type.

We say that the function
for any t ∈ [0, 1] and x, y ∈ C. Since it follows that any nonnegative convex function is r-exponential convex with r ∈ (0, ∞) .
Corollary 4. Let h (z) = ∞ n=0 a n z n be a power series with complex coefficients and convergent on the open disk for any t ∈ [0, 1] and x, y ∈ C, then for any

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 (3.1) ) on the convex subset C of the linear space X, then for any p, q ∈S + (K) we have i.e., J K is a subadditive (superadditive) functional on S + (K) . Proof.
Since h is supermultiplicative, then and 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.
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 i.e., J K is monotonic nondecreasing on S + (K) .
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 M p ≥ q ≥mp, for some M > m > 0, we have Proof.From the inequality (3.5) we have 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
t is decreasing (increasing), then for any p, q ∈S + (K) we have Proof.If h is convex on (0, ∞) , then we have for any p, q ∈S + (K) Making use of (3.9) and (3.10) we deduce the desired result (3.8).
The case when h is concave and g is increasing goes likewise and the details are omitted.
Also, for any p, q ∈S + (K) with M p ≥ q ≥mp, for some M > m > 0, we have We define the difference functional We observe that, if h is supermultiplicative and f : C ⊆ X → [0, ∞) is h-convex, then by Jensen's type inequality (1.7) we have S K (p) ≥ 0 for any p ∈S + (K) .
If p, q ∈S + (K) with p ≥ q, then we have Also, for any p, q ∈S + (K) with M p ≥ q ≥mp, for some M > m > 0, we have (3.15)h (M P K ) h (P K ) S K (p) ≥ S K (q) ≥ h (mP K ) h (P K ) 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 1 the superadditivity and monotonicity properties of the functional Je K (p) := i∈K p i f (x i ) − P K f 1 P K i∈K p i x i established in ( [32]).From (3.15) we get (3.16)M Je K (p) ≥ Je K (q) ≥ mJe K (p) that has been obtained in [24].