GENERALIZATION OF JENSEN’S AND JENSEN-STEFFENSEN’S INEQUALITIES AND THEIR CONVERSES BY LIDSTONE’S POLYNOMIAL AND MAJORIZATION THEOREM

. In this paper, using majorization theorems and Lidstone’s interpolating polynomials we obtain results concerning Jensen’s and Jensen-Steﬀensen’s inequalities and their converses in both the integral and the discrete case. We give bounds for identities related to these inequalities by using ˇCebyˇsev functionals. We also give Gr¨uss type inequalities and Ostrowsky type inequalities for these functionals. Also we use these generalizations to construct a linear functionals and we present mean value theorems and n -exponential convexity which leads to exponential convexity and then log-convexity for these functionals. We give some families of functions which enable us to construct a large families of functions that are exponentially convex and also give Stolarsky type means with their monotonicity.


INTRODUCTION
Majorization makes precise the vague notion that the components of a vector x are "less spread out" or "more nearly equal" than the components of a vector y. For fixed m ≥ 2 let be their ordered components.
Majorization: (see [12, p. 319]) x is said to majorize y (or y is said to be majorized by x), in symbol, x y, if x [i] holds for l = 1, 2, ..., m − 1 and Note that (1) is equivalent to m i=m−l+1 x (i) holds for l = 1, 2, ..., m − 1. There are several equivalent characterizations of the majorization relation x y in addition to the conditions given in definition of majorization. One is actually the answer of the question posed and answered in 1929 by Hardy, Littlewood and Polya in [7] and [8]: x majorizes y if for every continuous convex function φ. Another interesting characterization of x y, also by Hardy, Littlewood and Polya in [7] and [8], is that y = Px for some double stochastic matrix P. In fact, the previous characterization implies that the set of vectors x that satisfy x y is the convex hull spanned by the n! points formed from the permutations of the elements of x.
The following theorem is well-known as the majorization theorem and a convenient reference for its proof is given by Marshall and Olkin in [11, p. 14] (see also [12, p.
holds for every continuous convex function φ : [a, b] → R iff x y holds.
The following theorem can be regarded as a generalization of Theorem 1 known as Weighted Majorization Theorem and is proved by Fuchs in [6] (see also [11, p. 580] and [12, p. 323]).
Theorem 2. Let x = (x 1 , ..., x m ) , y = (y 1 , ..., y m ) be two decreasing real m-tuples with x i , y i ∈ [a, b] , i = 1, ..., m, let w = (w 1 , ..., w m ) be a real m-tuple such that Then for every continuous convex function φ : [a, b] → R, we have Bernstein has proved that if all the even derivatives are at least 0 in (a, b), then f has an analytic continuation into the complex plane. Boas suggested to Widder that this might be proved by use of the Lidstone series. This seemed plausible because the Lidstone series, a generalization of the Taylor's series, approximates a given function in the neighborhood of two points instead of one by using the even derivatives. Such series have been studied by G. J. Lidstone (1929), H. Poritsky (1932), J. M. Wittaker (1934) and others (see [3]).
In [15], Widder proved the fundamental lemma: is the homogeneous Green's function of the differential operator d 2 ds 2 on [0, 1], and with the successive iterates of G(t, s) The Lidstone polynomial can be expressed, in terms of G n (t, s) as The divided difference of order n of the function φ at distinct points x 0 , . . . , x n ∈ [a, b] is defined recursively (see [4], [12]) by The value φ[x 0 , . . . , x n ] is independent of the order of the points x 0 , . . . , x n . The definition may be extended to include the case that some (or all) of the points coincide. Assuming that φ (j−1) (x) exists, we define The notion of n-convexity goes back to Popoviciu [13]. We follow the definition given by Karlin [9]: In fact, Popoviciu proved that each continuous n-convex function on [a, b] is the uniform limit of the sequence of n-convex polynomials. Many related results, as well as some important inequalities due to Favard, Berwald and Steffensen can be found in [10].
In [3] using Lidstone's interpolating polynomials and conditions on Green's functions, the authors present results for Jensen's inequality and converses of Jensen's inequality for signed measure. In this paper we give generalized results of Jensen's and Jensen-Steffensen's inequalities and their converses by using majorization theorem and Lidstone's polynomial for (2n)-convex functions. Then we give bounds for identities related to these inequalities by usinǧ Cebyšev functionals. We give Grüss type inequalities and Ostrowsky type inequalities for these functionals. We also use these generalizations to construct a linear functionals and we present mean value theorems and n-exponential convexity which leads to exponential convexity and then log-convexity. Finally, we present several families of functions which construct to a large families of functions that are exponentially convex. We give classes of Cauchy type means and prove their monotonicity.

GENERALIZATION OF JENSEN'S INEQUALITY
We will use the following notation for composition of functions: Proof. Consider By Widder's lemma we can represent every function φ ∈ C (2n) ([a, b]) in the form: where Λ k is a Lidstone polynomial. Using (19) we calculate φ(x i ) and φ(x) and from (18) we obtain (17) Using Theorem 7 we give generalization of Jensen's inequality for (2n)convex function: If F is convex function, then the right hand side of (21) is non-negative and Now, we put x = (x 1 , . . . , x m ) and y = (x, . . . ,x) in Theorem 7 to get inequality (21).
For inequality (23) we use fact that for convex function F we have , for x(γ) ≥ x, we have: obviously holds.
So, If n ∈ N is odd, then for every (2n)-convex function φ : [a, b] → R, we obtain integral version of the inequality (21) from the above theorem which is result proved in [3]. Moreover, for the convex function F defined in (22) the right hand side of (29) is non-negative and If n is even, then for every (2n)-convex function φ : [a, b] → R the reverse inequality in (29) holds. Moreover, if F is concave function, then the reverse inequality in (30) is also valid.
Remark 11. Motivated by the inequalities (21) and (29), we define functionals Θ 1 (φ) and Θ 2 (φ) by Similarly as in [3] we can construct new families of exponentially convex function and Cauchy type means by looking at these linear functionals. The monotonicity property of the generalized Cauchy means obtained via these functionals can be prove by using the properties of the linear functionals associated with this error representation, such as n-exponential and logarithmic convexity.

GENERALIZATION OF JENSEN-STEFFENSEN'S INEQUALITY
Using majorization theorem for (2n)-convex function we give generalization of Jensen-Steffensen's inequality: Proof. For l = 1, ..., k, such that x k ≥ x we have So, similarly as in Theorem 9, we get that conditions (4) and (5)  , for x(γ) ≥ x, we have: Similarly as in the Remark 10 we get that conditions for majorization are satisfied, so inequalities (29) and (30) are valid. Proof. Using inequality (21) we have

Moreover, for the convex function F defined in (22), we have
Hence, for any odd n and (2n)-convex function φ we get (36).
For inequality (37) we use the fact that for convex function F we have (ii) Similar to the part (i) If n is even, reverse inequality in (38) is valid.
Proof. We use inequality (36) for m = a and M = b and (7).
, similarly as in Theorem 14 we get integral version of converse of Jensen's inequality.
For odd n ∈ N and for every (2n)-convex function φ : [a, b] → R we have: which is result proved in [3]. Moreover, for the convex function F defined in (22) we have If n is even, then for every (2n)-convex function φ : [a, b] → R, the reverse inequality in (39) holds. Moreover, for the concave function F defined in (22) the reverse inequality in (40) is also valid.

BOUNDS FOR IDENTITIES RELATED TO GENERALIZATION OF MAJORIZATION INEQUALITY
For two Lebesgue integrable functions f, h : [a, b] → R we considerČebyšev functional In [5], the authors proved the following theorems: The constant 1 The constant 1 2 in (43) is the best possible. In the sequel we use the above theorems to obtain generalizations of the results proved in the previous sections.  (10), we denote We have theČebyšev functionals defined as: where the remainder H 1 n (φ; a, b) satisfies the estimation (49) Proof. If we apply Theorem 18 for f → Υ and h → φ (2n) we obtain where the remainder H 1 n (φ; a, b) satisfies the estimation (49). Now from identity (17) and fact that Λ n (1 − t) = 1 0 G n (t, s)(1 − s)ds (see [2]) we obtain (48).
Integral case of the above theorem can be given: . Let the functions G n ,Υ and Ω be defined in (10), (45) and (47). Then where the remainderH 1 n (φ; a, b) satisfies the estimation Using Theorem 19 we also get the following Grüss type inequality.
Integral version of the above theorem can be given as: Theorem 23. Let φ : [a, b] → R be such that φ ∈ C (2n) [a, b] for n ∈ N and φ (2n+1) ≥ 0 on [a, b] and let the functionΥ be defined in (45). Then we have the representation (50) and the remainderH 1 n (φ; a, b) satisfies the bound We also give the Ostrowsky type inequality related to the generalization of majorization inequality.
Let φ ∈ C (2n) [a, b] be such that φ (2n) p : [a, b] → R is an R-integrable function for some N. Then we have The constant on the right hand side of (53) is sharp for 1 < p ≤ ∞ and the best possible for p = 1.