Majorization Type Inequalities Via Green Function and Hermite's Polynomial

The Hermite polynomial and Green function are used to constructthe identities related to majorization type inequalities for convex function. By using Cebysev functional the bounds for the new identities are found to develop the Gruss and Ostrowski type inequalities. Further more exponential convexity together with Cauchy means is presented for linear functionals associated with the obtained inequalities.DOI : http://dx.doi.org/10.22342/jims.22.1.251.1-25

The following theorem is well-known as the majorization theorem given by Marshall and Olkin [19, p. 14] (see also [21, p.
holds for every continuous convex function φ : [α, β] → R if and only if x ≻ y holds.
the H ij are fundamental polynomials of the Hermite basis defined by and the remainder is given by for all a l ≤ s ≤ a l+1 ; l = 0, . . . , r with a 0 = α and a r+1 = β.
Remark 1.1. In particular cases, for type (m, n − m) conditions, from Theorem 1.5 we have and and the remainder R (m,n) (φ, t) is given by For Type Two-point Taylor conditions, from Theorem 1.5 we have where ρ 2T (t)is the two-point Taylor interpolating polynomial i.e, (20) and the remainder R 2T (φ, t) is given by where p(t, s) = (s−α)(β−t) The following Lemma describes the positivity of Green's function (14) see (Beesack [11] and [Levin [24]). Lemma 1.6. The Green's function G H,n (t, s) has the following properties: In order to recall the definition of n−convex function, first we write the definition of divided difference.
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) the points coincide. Assuming that φ (j−1) (x) exists, we define We arrange the paper in this manner, in section 2, we use Hermite interpolating polynomial and Green function to establish identities for majorization inequalities. We present generalized majorization inequalities and in particular we discuss the results for (m, n − m) interpolating polynomial, two-point Taylor interpolating polynomial. In section 3, we give bounds for the identities related to the generalizations of majorization inequalities by usingČebyšev functionals. We also give Grüss type inequalities and Ostrowski-type inequalities for these functionals. In section 4, we present Lagrange and Cauchy type mean value theorems related to the defined functionals and also give n-exponential convexity which leads to exponential convexity and then log-convexity. At the end, in section 5, we discuss 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.

Generalization of Majorization Inequalities
We begin this section with the proof of some identities related to generalizations of majorization inequality.
Integral version of the above theorem can be stated as: Also let H ij , G H,n and G be as defined in (12), (14) and (9) respectively. Then ., x m ) and y = (y 1 , ..., y m ) be m-tuples such that x l , y l ∈ [α, β], w l ∈ R (l = 1, ..., m) and H ij , G be as defined in (12) and (9) respectively. Let φ : Consider the inequality (i) If k j is odd for each j = 2, .., r, then the inequality (28) holds.
(ii) If k j is odd for each j = 2, .., r−1 and k r is even, then the reverse inequality in (28) holds. Proof.
(i) Since the function φ is n−convex, therefore without loss of generality we can assume that φ is n−times differentiable and φ (n) ≥ 0 see [21, p. 16 and p. 293]. Also as it is given that k j is odd for each j = 2, .., r, therefore we have ω(t) ≥ 0 and by using Lemma 1.6(i) we have G H,n−2 (t, s) ≥ 0. Hence, we can apply Theorem 2.1 to obtain (28).
if k j is odd for each j = 2, .., r − 1, therefore combining all these we have and by using Lemma 1.6(i) we have G H,n−2 (t, s) ≤ 0. Hence, we can apply Theorem 2.1 to obtain reverse inequality in (28).
Integral version of the above theorem can be stated as: → R be continuous functions and H ij and G be as defined in (12) and (9) respectively. Let φ : (29) (i) If k j is odd for each j = 2, .., r, then the inequality (30) holds.
(ii) If k j is odd for each j = 2, .., r−1 and k r is even, then the reverse inequality in (30) holds.
By using type (m, n − m) conditions we can give the following result.
(ii) If n − m is odd, then the reverse inequality in (31) holds.
By using Two-point Taylor conditions we can give the following result.
(ii) If m is odd, then the reverse inequality in (32) holds.
Remark 2.1. Similarly we can give integral version of Corollaries 2.5,2.6.
The following generalization of majorization theorem is valid.
(ii) If k j is odd for each j = 2, .., r−1 and k r is even, then the reverse inequality in (33) holds.
If the inequality (reverse inequality) in (33) holds and the function F (.) = r j=1 kj i=0 φ (i+2) (a j )H ij (.) is non negative ( non positive), then the right hand side of (33) will be non negative (non positive) that is the inequality (reverse inequality) in (2) will holds.
Proof. (i) Since the function G is convex and y ≺ x therefore by Theorem 1.2, the inequality (27) holds for w l = 1. Hence by Theorem 2.3(i) the inequality (33) holds. Also if the function F is convex then by using F in (2) instead of φ we get that the right hand side of (33) is non negative. Similarly we can prove part (ii).
In the following theorem we give generalization of Fuch's majorization theorem.
(ii) If k j is odd for each j = 2, .., r−1 and k r is even, then the reverse inequality in (34) holds.
If the inequality (reverse inequality) in (34) holds and the function F (.) = r j=1 kj i=0 φ (i+2) (a j )H ij (.) is non negative (non postive), then the right hand side of (34) will be non negative (non positive) that is the inequality (reverse inequality) in (5) will hold.
Proof. Similar to the proof of Theorem 2.7.
In the following theorem we give generalized majorization integral inequality.
(ii) If k j is odd for each j = 2, .., r−1 and k r is even, then the reverse inequality in (35) holds.
If the inequality (reverse inequality) in (35) holds and the function is non negative (non positive), then the right hand side of (35) will be non negative (non positive) that is the inequality (reverse inequality) in (8) will hold.
(ii) If n − m is odd, then the reverse inequality in (36) holds.

If the inequality (reverse inequality) in (36) holds and the function
is non negative (non positive), then the right hand side of (36) will be non negative (non positive) that is the inequality (reverse inequality) in (2) will hold. By using Two-point Taylor conditions we can give generalization of majorization inequality for majorized tuples: Corollary 2.11. Let [α, β] be an interval and x = (x 1 , ..., x p ), y = (y 1 , ..., y p ) be decreasing p-tuples such that y ≺ x with x l , y l ∈ [α, β] (l = 1, ..., p). where (i) If m is even, then the inequality (37) holds.
(ii) If m is odd, then the reverse inequality in (37) holds.
If the inequality (reverse inequality) in (37) holds and the function F (.) is non negative (non positive), then the right hand side of (37) will be non negative (non positive) that is the inequality (reverse inequality) in (2) will hold.
By using type (m, n − m) conditions we can give the following weighted majorization inequality.
(ii) If n − m is odd, then the reverse inequality in (38) holds.

If the inequality (reverse inequality) in (38) holds and the function
is non negative (non positive), then the right hand side of (38) will be non negative (non positive) that is the inequality (reverse inequality) in (5) will hold. By using Two-point Taylor conditions we can give the following weighted majorization inequality.
(ii) If m is odd, then the reverse inequality in (39) holds.
If the inequality (reverse inequality) in (39) holds and the function F (.) is non negative (non positive), then the right hand side of (39) will be non negative (non positive) that is the inequality (reverse inequality) in (5) will hold.
The integral version of the above Corollaries can be stated as:  (7) hold. Let τ i and η i be as defined in (16) and (17) respectively and φ : (i) If n − m is even, then the inequality (40) holds.
(ii) If n − m is odd, then the reverse inequality in (40) holds.
If the inequality (reverse inequality) in (40) holds and the function is non negative (non positive), then the right hand side of (40) will be non negative (non positive) that is the inequality (reverse inequality) in (8) will hold.  (6) and (7) hold. Let φ : where (i) If m is even, then the inequality (41) holds.
(ii) If m is odd, then the reverse inequality in (41) holds.
If the inequality (reverse inequality) in (41) holds and the function F (.) is non negative (non positive), then the right hand side of (41) will be non negative (non positive) that is the inequality (reverse inequality) in (8) will hold.

Bounds for Identities Related to Generalizations of Majorization Inequality
For two Lebesgue integrable functions f, h : [α, β] → R we consider thě Cebyšev functional In [13] the authors proved the following theorems: The constant 1 √ 2 in (42) is the best possible.
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 section.
Theorem 3.7. Suppose that all assumptions of Theorem 2.1 hold. Assume (p, q) is a pair of conjugate exponents, that is 1 ≤ p, q ≤ ∞, 1/p + 1/q = 1. Let φ (n) p : [α, β] → R be an R-integrable function for some n ∈ N. Then we have: where L is defined in (44). The constant on the right-hand side of (54) is sharp for 1 < p ≤ ∞ and the best possible for p = 1.
Proof. The proof is similar to the proof of Theorem 19 in [5].
Integral version of the above theorem can be given as: Theorem 3.8. Suppose that all assumptions of Theorem 2.2 hold. Assume (p, q) is a pair of conjugate exponents, that is 1 ≤ p, q ≤ ∞, 1/p + 1/q = 1. Let φ (n) p : [α, β] → R be an R-integrable function for some n ∈ N. Then we have: where J is defined in (45). The constant on the right-hand side of (55) is sharp for 1 < p ≤ ∞ and the best possible for p = 1.

n−Exponential Convexity and Exponential Convexity
We begin this section by giving some definitions and notions which are used frequently in the results. For more details see e.g. [12], [15] and [22].
hold for all choices ξ 1 , . . . , ξ n ∈ R and all choices x 1 , . . . , x n ∈ I. A function φ : I → R is n-exponentially convex if it is n-exponentially convex in the Jensen sense and continuous on I.   Motivated by inequalities (28) and (30), under the assumptions of Theorems 2.3 and 2.4 we define the following linear functionals: Lagrange and Cauchy type mean value theorems related to defined functionals are given in the following theorems.
provided that the denominators are non-zero and ̥ H i , i = 1, 2, are defined by (56) and(57).
Proof. Similar to the proof of Theorem 4.2 in [16]. Now we will produce n−exponentially and exponentially convex functions applying defined functionals. We use an idea from [22]. In the sequel J will be interval in R.