Abstract
We derive some properties and results for a new extended class of convex functions, generalized strongly modified h-convex functions. Moreover, we discuss Schur-type, Hermite–Hadamard-type, and Fejér-type inequalities for this class. The crucial fact is that this extended class has awesome properties similar to those of convex functions.
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1 Introduction
Nowadays, in science and modern analysis the convexity plays an important role in economics, statistics, management science, engineering, and optimization theory. For instance, Barani et al. [1] presented the Hermite–Hadamard inequality for functions with preinvex absolute values of derivatives. Characterizations of convexity via Hadamard’s inequality has been studied in [2]. In 2003, Dragomir and Pearce [3] proposed some applications of Hermite–Hadamard inequalities. In 2015, Dragomir [4] presented inequalities of Hermite–Hadamard type for h-convex functions on linear spaces. Some other interesting results can be found in books [5, 6] and research papers [7, 8]. In the recent years, generalizations and extensions were made rapidly for convex functions; for a recent generalization, see [9–11].
Convexity in the classical sense for a function \(g:L=[a_{1},a_{2}] \subset \mathbb{R} \rightarrow \mathbb{R} \) is defined as
where \(a_{1},a_{2} \in L\) and \(t \in [0,1]\).
The work on the convexity is extended day by day by using some techniques; see [12–14]. The strongly extended convexity is widely used in optimization, economics, and nonlinear programming.
Convex functiosn satisfy several inequalities in which famous inequalities are of Schur type, Hermite–Hadamard-type, and Fejér-type inequalities. The Hermite–Hadamard-type inequality introduced by Jaques Hadamard for classical convex functions \(g:L=[a_{1},a_{2}]\subset \mathbb{R} \rightarrow \mathbb{R} \) as
For extended versions of this inequality, see [12] and [13]. For further reading, see [15–19].
Lipot Fejér presented an extended version of the Hermite–Hadamard inequality, known as the Fejér inequality or a weighted version of the Hermite–Hadamard inequality. If \(f:I\rightarrow \mathbb{R} \) is a convex function, then
where \(a_{1}\leq a_{2} \), and \(w:I\rightarrow \mathbb{R} \) is nonnegative, integrable, and symmetric about \(\frac{a+b}{2} \). For further extended versions and development, see [20] and [8].
In this paper, we first present some preliminaries and basic results. In the next section, we investigate Schur-type, Hermite–Hadamard-type, and Fejér-type inequalities for the newly introduced class of functions.
2 Preliminaries
In this section, we investigate a new class of convexity by using a basic result. There is no loss of generality in the extended version of convexity. To get asymptotic results, it is necessary to put some restrictions: L is an interval in \(\mathbb{R}\), and \(\eta : A \times A \rightarrow B \subseteq \mathbb{R}\) is a bifunction.
Definition 1
(h-convex function [21])
Let \(g,h:L\subset \mathbb{R}\rightarrow \mathbb{R}\) be nonnegative functions. Then g is called an h-convex function if
for all \(a_{1},a_{2}\in L\) and \(t\in [0,1]\).
Definition 2
(Modified h-convex function [13])
Let \(g,h:L\subset \mathbb{R} \rightarrow \mathbb{R} \) be nonnegative functions. Then g is called a modified h-convex function if
for all \(a_{1},a_{2}\in L\) and \(t\in [0,1]\).
Definition 3
(Generalized modified h-convex function)
Let functions g, h: \(J\subset \mathbb{R}\rightarrow \mathbb{R} \) be nonnegative functions. Then \(g:I\subset \mathbb{R}\rightarrow \mathbb{R}\) is called a generalized modified h-convex function if
for all \(a_{1},a_{2}\in I\) and \(t\in [0,1]\).
Definition 4
(Wright-convex function [20])
A function \(g:L\subset \mathbb{R}\rightarrow \mathbb{R} \) is said to be Wright-convex if
for all \(a_{1},a_{2}\in L\) and \(t\in [0,1]\).
Definition 5
(Additivity)
A function η is said to be additive if \(\eta (x_{1},y_{1})+\eta (x_{2},y_{2}) =\eta (x_{1}+x_{2},y_{1}+y _{2})\) for all \(x_{1},x_{2},y_{1},y_{2} \in \mathbb{R} \); see [22] for more detail.
Definition 6
(Nonnegative homogeneity)
A function η is said to be nonnegatively homogeneous if \(\eta ( \lambda a_{1},\lambda a_{2} )= \lambda \eta (a_{1},a_{2})\) for all \(a_{1},a_{2} \in \mathbb{R}\) and \(\lambda \geq 0 \).
Definition 7
(Supermultiplicativity [23])
A function \(g:L\subset \mathbb{R}\rightarrow \mathbb{R_{+}} \) is said to be a supermultiplicative function if \(g(a_{1}a_{2})\geq g(a_{1}) g(a_{2})\) for all \(a_{1},a_{2} \in L \), \(t \in [0,1]\).
Definition 8
(Similar-order functions [24])
Functions f and g are said to be of similar order on \(L\subseteq \mathbb{R}\) if \(\langle f(x)-f(y),g(x)-g(y)\rangle \geqslant 0\) for all \(x,y \in L\).
Now we are going to introduce a new extended definition of convexity.
Definition 9
(Generalized strongly modified h-convex function)
Let \(g,h:L\subset \mathbb{R}\rightarrow \mathbb{R} \) be nonnegative functions. Then g is called a generalized strongly modified h-convex function if
for all \(a_{1},a_{2}\in L\) and \(t\in [0,1]\).
Remark 1
-
1.
Inequality (3) reduces to inequality (1) if \(\mu =0\) and \(\eta (x,y)=x-y\).
-
2.
Definition (9) becomes the definition of a classical convex function when \(\mu =0\), \(\eta (x,y)=x-y \), and \(h(t)=t\).
- 3.
-
4.
If \(h(t)=t \), then definition (9) reduces to the definition of a strongly generalized convex function [12].
Example 1
A function \(g:L=[a_{1},a_{2}] \subset \mathbb{R}\rightarrow \mathbb{R}\) is defined by \(g(x)=x^{2} \), \(\eta (a_{1},a_{2})=2a_{1}+a _{2} \), and \(h(t)\geq t \), then g is a generalized strongly modified h-convex function.
3 Main results
This section contains some basic and straightforward results. The following proposition shows the linearity of the class of generalized strongly modified h-convex functions.
Proposition 1
Letfandgbe generalized strongly modifiedh-convex functions whereηis additive and nonnegatively homogeneous. Then for all \(a,b \in \mathbb{R} \), \(af+bg\)is also a generalized strongly modifiedh-convex function.
Proposition 2
Let \(h_{1} \), \(h_{2} \)be nonnegative functions onLsuch that \(h_{2}(t)\leqslant h_{1}(t)\). Ifgis a generalized strongly modified \(h_{2} \)-convex function, thengis also a generalized strongly modified \(h_{1} \)-convex function.
Proof
As g is generalized strongly modified h-convex function, for all \(a_{1} ,a_{2} \in L\) and \(t \in [0,1] \), we have
This completes the proof. □
Remark 2
If g is a generalized strongly modified \(h_{1} \)-convex and \(h_{1} (t)\leqslant h_{2}(t)\), then g is a generalized strongly modified \(h_{2} \)-convex function.
Proposition 3
Letfbe a linear function such that \(f(x)-f(y)=x-y \), and letgbe a generalized strongly modifiedh-convex function. Then \(g\circ f \)is also a generalized strongly modifiedh-convex function.
Proof
As f is a linear function such that \(f(x)-f(y)=x-y \) and g is a generalized strongly modified h-convex function, for all \(a_{1} ,a_{2} \in L \) and \(t \in [0,1] \), we get
This shows that \(g\circ f \) is a generalized strongly modified h-convex function. □
Proposition 4
Let functions \(g_{j} :L\subset \mathbb{R}\rightarrow \mathbb{R}\)be generalized strongly modifiedh-convex functions, \(\sum_{j=1}^{m} \lambda _{j} =1\), and letηbe additive non-negatively homogeneous function. Then their linear combination \(f:\mathbb{R}\rightarrow \mathbb{R} \)is also a generalized strongly modifiedh-convex function.
Proof
As \(g_{j}:L \subset \mathbb{R}\rightarrow \mathbb{R} \) be generalized strongly modified h-convex functions, for \(a_{1},a_{2} \in L \) and \(t\in [0,1] \), let
Set \(x=(t a_{1}+(1-t)a_{2})\). Then
This completes the proof. □
Corollary 1
Every generalized strongly modifiedh-convex function is a generalized modified convex function.
Proof
Let g be a generalized modified h-convex function. Then
for all \(a_{1} ,a_{2} \in L\subset \mathbb{R}\). □
Corollary 2
Ifgis generalized strongly convex function and \(t\leq h(t) \), thengis a generalized strongly modifiedh-convex function.
Theorem 1
(Schur-type inequality)
Let \(g :L\rightarrow \mathbb{R}\)be a generalized strongly modifiedh-convex function, lethbe a supermultiplicative function, and let \(\eta : N\times N\rightarrow M\)be a bifunction for appropriate \(A,B\subseteq \mathbb{R}\). Then for \(a_{1},a_{2},a_{3} \in L \)such that \(a_{1}< a_{2}< a_{3} \)and \(a_{3}-a_{1},a_{3} -a_{2},a_{2}-a_{1} \in L\), we have the inequality
if and only ifgis a generalized strongly modifiedh-convex function.
Proof
Let \(a_{1},a_{2},a_{3} \in L\subset \mathbb{R} \) be such that \(\frac{(a_{3}-a_{2})}{(a_{3}-a_{1})} \in (0,1)\subseteq L \), \(\frac{(a _{2}-a_{1})}{(a_{3}-a_{1})} \in (0,1)\subseteq L\), and \(\frac{(a_{3}-a _{2})}{(a_{3}-a_{1})}+\frac{(a_{2}-a_{1})}{(a_{3}-a_{1})}=1\). Then
as h is supermultiplicative.
Suppose \(h({a_{3}-a_{2}})\geq 0\). Then by the definition of g we have
Inserting \(\frac{(a_{3}-a_{2})}{(a_{3}-a_{1})}=t\), \(x=a_{1}\), and \(y= a_{3} \) into inequality (5), we obtain
Conversely, suppose inequality (4) holds and insert \(a_{1}=x \), \(a_{2}=tx+(1-t)y \), and \(a_{3}=y \) into inequality (4). Then we get
This completes the proof. □
Remark 3
-
1.
By taking \(h(t)=t\) in (4) it is reduced to aSchur-type inequality for generalized strongly convex functions.
-
2.
If \(\mu =0\) and \(\eta (x,y)=x-y\), then (4) is reduced to a Schur-type inequality for modified h-convex functions; see [13].
Further, we will discuss the Hermite–Hadamard-type inequality for generalized strongly modified h-convex functions.
Theorem 2
(Hermit–Hadamard-type inequality)
Let function \(g:L\rightarrow \mathbb{R} \)be a generalized strongly modifiedh-convex function on \([a_{1},a_{2}] \)with \(a_{1}< a_{2} \). Then
Proof
Choosing \(w=ta_{1} +(1-t)a_{2}\) and \(z=(1-t)a_{1} +ta_{2}\), we have
Now by the definition of g we have
Integrating with respect to t on [0,1], we get
Putting \(x=(1-t)a_{1} +ta_{2}\), we get
In the right-hand side of inequality (8), we set \(x=ta_{1} +(1-t)a_{2} \), and using the definition of g, we get
Now from inequalities (8) and (9) we get
This completes the proof. □
Remark 4
-
1.
If we take \(\mu =0\) and \(\eta (x,y)=x-y\), then the Hermite–Hadamard-type inequality (10) is reduced to Hermite–Hadamard-type inequality for modified h-convex functions; for details, see [13].
-
2.
If we put \(h(t)=t \) in (10), then we get a Hermite–Hadamard-type inequality for generalized strongly convex functions; see [12].
-
3.
If we take \(\mu =0 \), \(\eta (x,y)=x-y \) and \(h(t)=t \), then inequality (10) is reduced to a Hemite–Hadard-type inequality for classical convex functions.
Now we prove the following lemma by using technique of [25]. This lemma has the crucial fact that generalized strongly modified h-convex functions behave like classic convex functions.
Lemma 1
Letgbe a generalized modifiedh-convex function, and suppose that \(\eta (x,y)=-\eta (y,x)\). Then
where \(x=ta_{1}+(1-t)a_{2} \)and \(t\in [0,1]\).
Proof
As g is generalized modified h-convex function, for \(x=ta_{1}+(1-t)a_{2}\), we get
This completes the proof. □
Lemma 2
Letgbe q the generalized strongly modifiedh-convex function, and suppose that \(\eta (x,y)=-\eta (y,x)\). Then
where \(x=ta_{1}+(1-t)a_{2} \)and \(t\in [0,1]\).
Proof
Let g be a generalized strongly modified h-convex function. Then for \(x=ta_{1}+(1-t)a_{2}\), we get
This completes the proof. □
It is very interesting that when g is a modified h-convex function [13], generalized modified h-convex, or generalized strongly modified h-convex function, then inequality (11) holds.
Theorem 3
(Fejér-type inequality)
Let \(g:[a_{1},a _{2}]\rightarrow \mathbb{R} \)be a generalized strongly modifiedh-convex, and let \(w:[a_{1},a_{2}]\rightarrow \mathbb{R}\)be nonnegative, integrable, and symmetric with respect to \(\frac{a_{1}+a _{2}}{2}\). Then
where
Proof
Let g be a generalized strongly modified h-convex function. Then
In the right hand-side of inequality (13), put \(x=ta_{1}+(1-t)a _{2}\). Then
Similarly, if we put \(x=ta_{2}+(1-t)a_{1} \) in the right-hand side of inequality (13), then we get the inequality
Adding inequalities (14) and (15), where w is symmetric, we get
Putting \(x=ta_{1}+(1-t)a_{2}\) in the right-hand side of inequality (16), we have
Now from inequalities (13) and (17) we get Fejér-type inequality (12) for generalized strongly modified h-convex functions. □
Remark 5
-
1.
If \(h(t)=t \), then inequality (12) reduced to Fejér type inequality for generalized strongly convex functions, see [12].
-
2.
If we put \(\mu =0 \) and \(\eta (x,y)=x-y\) then inequality (12) becomes a Fejér-type inequality for modified h-convex functions; see [13].
-
3.
If we put \(\mu =0\), \(\eta (x,y)=x-y\), and \(h(t)=t \), then inequality (12) is reduced to a Fejér-type inequality for classical convex functions.
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Zhao, T., Saleem, M.S., Nazeer, W. et al. On generalized strongly modified h-convex functions. J Inequal Appl 2020, 11 (2020). https://doi.org/10.1186/s13660-020-2281-6
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DOI: https://doi.org/10.1186/s13660-020-2281-6