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On approximation and energy estimates for delta 6convex functions
Journal of Inequalities and Applications volume 2018, Article number: 46 (2018)
Abstract
The smooth approximation and weighted energy estimates for delta 6convex functions are derived in this research. Moreover, we conclude that if 6convex functions are closed in uniform norm, then their third derivatives are closed in weighted \(L^{2}\)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 nonlinear 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. In the last few years, many efforts have been made on generalization of the notion of convexity to meet the hurdles in advanced optimization theory. Among many generalizations, some are quasi convex [1], pseudo convex [2], logarithmically convex [3], nconvex [4], delta convex [5], sconvex [6], hconvex [7], mid convex [8] and [9–14]. The function \(f:I\rightarrow\mathbb{R}\) is said to be convex if \(\forall x,y\in I\) and \(\alpha,\beta\in[0,1] \) such that \(\alpha +\beta=1\),
holds. The function \(f(x)\in C^{n}(I)\) is said to be nconvex if \(f^{(n)}(x)\geq0\), \(x\in I\). The weighted energy estimates for the convex function and 4convex function are derived in [15] and [16]. These estimates are important in hedging strategies in finance [17]. Throughout the paper, we will use the following notations over I, where \(I=[a,b]\).
 \(C(I)\) :

space of continuous functions over I.
 \(C^{6}(I)\) :

space of six times continuously differentiable functions on I.
 \(w(x)\) :

is the nonnegative weight function which satisfies the following axiom:
$$ \left . \textstyle\begin{array}{l@{\quad}l} w^{(\mathit{iv})}(x)\geq0 &\text{if } x \in I,\\ w{''}(x)\leq0 &\text{if } x \in I,\\ w(x)=w{'}(x)=w{''}(x)=w{'''}(x)=w^{(\mathit{iv})}(x)=w^{(v)}(x)=0 &\forall x\in\partial I . \end{array}\displaystyle \right \} $$(1.1)
In the present paper, we deal with a delta 6convex function. We derive some basic properties of the delta 6convex function under certain conditions. Moreover, we approximate an arbitrary delta 6convex function by smooth ones and derive weighted energy estimates for the derivative of delta 6convex function.
Definition 1.1
([18] Delta convex function)
The function \(f:I\rightarrow\mathbb{R}\) is said to be delta convex function (or DC) over I if there exist continuous convex functions \(f_{1}\) and \(f_{2}\) on I such that
Definition 1.2
(Delta 6convex function)
The function \(f:I\rightarrow\mathbb{R}\) is said to be delta 6convex function over I if there exist continuous 6convex functions \(f_{1}\) and \(f_{2}\) on I such that
The following proposition gives some basic properties of the delta 6convex function.
Proposition 1.3
Let f and g be the two delta 6convex functions, and let \(\alpha\geq 0\) be real. Then

(i)
\(f+g\) is also a delta 6convex function.

(ii)
αf is also a delta 6convex function.

(iii)
Let g be increasing and f be a delta 6convex function, then \(f\circ g\) is also a delta 6convex function.
Proof
The proof of the proposition is straightforward. □
Approximation of a smooth delta 6convex function and the statement of the main result
First we define the mollification of an arbitrary delta 6convex 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 6convex, 4convex, and also 2convex function. Then, by the property of the differentiability of the 6convex, 4convex, and 2convex functions, \(f \in C^{3}[a,b]\). Let \(\theta _{\epsilon}\in C^{\infty}(\mathbb{R})\) have support on the interval \(I_{\epsilon}=I (x_{0},r_{\epsilon})\). The θ is called approximation identity or mollifier. Take
where c is a constant such that
Now, using \(\theta_{\epsilon}\) as a kernel, we define the convolution of f as follows:
Since \(\theta\epsilon\in C^{\infty}(\mathbb{R})\), so \(f_{\epsilon}\in C^{\infty}(\mathbb{R})\).
If f is continuous, then \(f_{\epsilon}\) converges uniformly to f in any compact subset \(K\subseteq I\),
if \(\epsilon=\frac{1}{m}\) then \(\vert f_{m}f \vert \underset{m\to \infty}{\longrightarrow}0\).
Let f be a delta 2convex function, 4convex function, and 6convex function. We have to show that \(f_{\epsilon}\) is also a delta 2convex, delta 4convex, and delta 6convex function.
So,
It is sufficient to prove that each \(f_{i,\epsilon}(x)\) is convex, 2convex, 4convex, and 6convex, where \(i=1,2\).
Since
so, \(f_{i,\epsilon}\), \(i=1,2\), is convex, which gives \(f_{\epsilon}\) is a delta convex function. Similarly, the delta 2convexity of \(f^{(2)}\) and \(f^{(4)}\) gives delta 4convexity and 6convexity of \(f_{\epsilon}\). Now we give the statement of our main theorem.
Theorem 2.1
Let f(x) be an arbitrary delta 6convex function over the interval I. Also, let f(x) be delta 4convex and delta 2convex, then the following holds:
where \(w(x)\) is a nonnegative 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 6convex functions over the interval I. Also, let \(f_{i}(x)\), \(i=1,2\), be 4convex and 2convex functions, then the following holds:
where \(w(x)\) is a nonnegative weight function which satisfies (1.1).
Proof
By substituting \(f=f_{2}f_{1} \) 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 nonnegative, twice continuously differentiable, and satisfying
with \(a\leq x\leq b\). We come to the following result of Hussain, Pecaric, and Shashiashvili [15].
Lemma 3.1
For the smooth convex function \(f(x)\) and the nonnegative weight function \(w(x)\) defined on the interval I, satisfying (3.1), we have
The results of 4convex functions are established in [16].
Lemma 3.2
Let \(f(x)\) be both 4convex and 2convex functions. Let \(w(x)\) be the nonnegative smooth weight function as defined in (3.1) and satisfying the condition
Then the following estimate holds:
We will start by the following theorem.
Theorem 3.3
Let \(f(x)\in C^{6}(I)\) be a delta 6convex function. Also f(x) is delta 4convex as well as delta 2convex. 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.
Proposition 3.4
Let \(f, F \in C^{6}[a,b]\), and F be 6convex, 4convex, as well as 2convex function such that the condition
is fulfilled. Let \(w(x)\) be a nonnegative 2concave, 4convex 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 righthand side. Using the integration by parts formula and condition (1.1), we have
Now take the first and the second integrals on the righthand 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
Now, using (2.6) of [15], we have
Substituting (3.12) and (3.13) in (3.11), we have
Here,
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 6convex function can be obtained simply by taking \(F=f\) in (3.16), where \(f\in C^{6}[a,b]\) 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 6convex functions.
Proof of Theorem 3.3
Take \(f=f_{2}f_{1}\) 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 \(\frac{1}{p}+\frac{1}{q}=1\) and
Proof of Theorem 2.1
For the continuous arbitrary 6convex functions \(f_{i}(x)\), \(i=1,2\), consider the smooth approximation \(f_{m,i}(x)\), \(i=1,2\).
For the interval \(I_{k+l}\), there exists an integer \(m_{k+l}\) such that \(f_{m,i}(x)\) converges uniformly to \(f_{i}(x)\), \(i=1,2\), and also \(f_{m,i}(x)\) is smooth for \(m\geq m_{k+l}\).
Now, writing inequality (3.18) for the functions \(f_{m,1}\) and \(f_{m,2}\) over the interval \(I_{k+l}\), we get
where \(c_{k+l}=\int_{I_{k+l}} \vert w^{(\mathit{vi})}(x) \vert \,dx\).
Now, taking limit \(m\rightarrow\infty\), we obtain
Now, writing the lefthand integral for the smaller interval \(I_{k} \subset I_{k+l}\) and also taking limit \(l \rightarrow\infty\), we obtain
Since we have
taking limit as \(k\rightarrow\infty\), we obtain the required result (2.1). □
Conclusion
From the result (2.2) we conclude that, if 6convex functions are closed in uniform norms, then their third derivatives are also closed in weighted \(L^{2}\)norm.
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The authors express their sincere thanks to the referee(s) and editor of the journal for the careful and detailed reading of the manuscript and very helpful suggestions that improved the manuscript substantially.
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Saleem, M.S., Pečarić, J., Rehman, N. et al. On approximation and energy estimates for delta 6convex functions. J Inequal Appl 2018, 46 (2018). https://doi.org/10.1186/s1366001816377
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Keywords
 Convex function
 Delta 6convex functions
 Mollification
 Energy estimates