On approximation and energy estimates for delta 6-convex functions

The smooth approximation and weighted energy estimates for delta 6-convex functions are derived in this research. Moreover, we conclude that if 6-convex functions are closed in uniform norm, then their third derivatives are closed in weighted \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{2}$\end{document}L2-norm.


Introduction
In the recent decade, the study of convex functions and convex sets has developed rapidly because of its use in applied mathematics, specially in non-linear programming and optimization theory. Furthermore, the elegance shape and properties of a convex function develop interest in studying this branch of mathematics. But the classical definitions of convex function and convex set are not enough to overcome advanced applied problems.
The function f (x) ∈ C n (I) is said to be n-convex if f (n) (x) ≥ 0, x ∈ I. The weighted energy estimates for the convex function and 4-convex function are derived in [15] and [16].
These estimates are important in hedging strategies in finance [17]. Throughout the paper, w(x) is the non-negative weight function which satisfies the following axiom: In the present paper, we deal with a delta 6-convex function. We derive some basic properties of the delta 6-convex function under certain conditions. Moreover, we approximate an arbitrary delta 6-convex function by smooth ones and derive weighted energy estimates for the derivative of delta 6-convex function.
The following proposition gives some basic properties of the delta 6-convex function. Proposition 1.3 Let f and g be the two delta 6-convex functions, and let α ≥ 0 be real. Then (i) f + g is also a delta 6-convex function.
(ii) αf is also a delta 6-convex function.
(iii) Let g be increasing and f be a delta 6-convex function, then f • g is also a delta 6-convex function.
Proof The proof of the proposition is straightforward.

Approximation of a smooth delta 6-convex function and the statement of the main result
First we define the mollification of an arbitrary delta 6-convex function in [a, b]. The mollification of an arbitrary function is very well explained in the book by Evans [19]. Let f (x) be an arbitrary delta 6-convex, 4-convex, and also 2-convex function. Then, by the property of the differentiability of the 6-convex, 4-convex, and 2-convex functions, Let θ ∈ C ∞ (R) have support on the interval I = I(x 0 , r ). The θ is called approximation identity or mollifier. Take Now, using θ as a kernel, we define the convolution of f as follows: If f is continuous, then f converges uniformly to f in any compact subset K ⊆ I, Let f be a delta 2-convex function, 4-convex function, and 6-convex function. We have to show that f is also a delta 2-convex, delta 4-convex, and delta 6-convex function. So, It is sufficient to prove that each f i, (x) is convex, 2-convex, 4-convex, and 6-convex, where i = 1, 2. Since so, f i, , i = 1, 2, is convex, which gives f is a delta convex function. Similarly, the delta 2convexity of f (2) and f (4) gives delta 4-convexity and 6-convexity of f . Now we give the statement of our main theorem.
Theorem 2.1 Let f(x) be an arbitrary delta 6-convex function over the interval I. Also, let f(x) be delta 4-convex and delta 2-convex, then the following holds:

1)
where w(x) is a non-negative weight function which satisfies (1.1). And f 1 and f 2 are such that Remark 2.2 Let f i (x), i = 1, 2, be continuous arbitrary 6-convex functions over the interval I. Also, let f i (x), i = 1, 2, be 4-convex and 2-convex functions, then the following holds: where w(x) is a non-negative weight function which satisfies (1.1).
Proof By substituting f = f 2f 1 in Theorem 2.1, we get the required result.

Some basic results and proof of the main result
Let w(x) be the weight function which is non-negative, twice continuously differentiable, and satisfying We come to the following result of Hussain, Pecaric, and Shashiashvili [15].

Lemma 3.1 For the smooth convex function f (x) and the non-negative weight function w(x) defined on the interval I, satisfying (3.1), we have
The results of 4-convex functions are established in [16].

Lemma 3.2 Let f (x) be both 4-convex and 2-convex functions. Let w(x) be the non-negative smooth weight function as defined in (3.1) and satisfying the condition
Then the following estimate holds:

4)
We will start by the following theorem.

Theorem 3.3 Let f (x) ∈ C 6 (I) be a delta 6-convex function. Also f(x) is delta 4-convex as well as delta 2-convex. Then the following energy estimate is valid:
where w(x) is the weight function satisfying (1.1).
To prove Theorem 3.3, we first prove the proposition. b], and F be 6-convex, 4-convex, as well as 2-convex function such that the condition is fulfilled. Let w(x) be a non-negative 2-concave, 4-convex weight function satisfying (1.1), then the following energy estimate is valid: Proof Take Using the integration by parts formula and making use of condition (1.1), we get Now take the first integral of (3.8) on the right-hand side. Using the integration by parts formula and condition (1.1), we have Now take the first and the second integrals on the right-hand side of the latter expression. Using the integration by parts formula and making use of condition (1.1), we get = - Proceeding in the similar way and using condition (1.1) and the definition of weight function, we obtain = - Now we take Using Theorem 2.1 from [16], we have Using the above conditions, we obtain Using the integration by parts formula, we obtain Hence, as the required proof.
The following weighted energy inequality for the smooth 6-convex function can be obtained simply by taking F = f in (3.16), where f ∈ C 6 [a, b] and f and w satisfy the conditions of the last theorem. Then we have The next result describes the energy estimate for the difference of two 6-convex functions.
Proof of Theorem 3.3 Take f = f 2f 1 and F = f 1 + f 2 in Proposition (3.4) to get We conclude the section with the following remark.
Remark 3.5 Let f 1 , f 2 , and w(x) be the same as in the latter theorem. Then, by using Holder's inequality, we have where 1 p + 1 q = 1 and where c k+l = I k+l |w (vi) (x)| dx. Now, taking limit m → ∞, we obtain Now, writing the left-hand integral for the smaller interval I k ⊂ I k+l and also taking limit l → ∞, we obtain Since we have taking limit as k → ∞, we obtain the required result (2.1).

Conclusion
From the result (2.2) we conclude that, if 6-convex functions are closed in uniform norms, then their third derivatives are also closed in weighted L 2 -norm.