1 Introduction
Unless stated otherwise throughout this section I is an interval of \(\mathbb{R}\).
The following proposition gives us an alternative definition of convex functions [
1], p.2.
The following result can be deduced from Proposition
1.
Now we define the generalized convex function which can be found in [
2,
3] and [
1].
It may easily be verified that
$$[x_{i},\ldots,x_{i+n};f]=\sum_{k=0}^{n} \frac{f(x_{i+k})}{ \prod_{j=i, j\neq i+k}^{i+n}(x_{i+k}-x_{j}) }. $$
If the nth order derivative \(f^{(n)}\) exists, then f is n-convex if and only if \(f^{(n)}\geq0\).
For fixed \(m\geq2\), let \(\mathbf{x}= ( x_{1},\ldots ,x_{m} ) \) and \(\mathbf{y}= ( y_{1},\ldots,y_{m} ) \) denote two real m-tuples and \(x_{[1]} \geq x_{[2]} \geq\cdots\geq x_{[m]}\), \(y_{[1]} \geq y_{[2]} \geq\cdots\geq y_{[m]}\) their ordered components.
This notion and notation of majorization was introduced by Hardy
et al. [
4]. Now, we state the well-known majorization theorem from the same book [
4] as follows.
The following weighted version of the majorization theorem was given by Fuchs in [
5] (see also [
6], p.580 and [
1], p.323).
The following proposition is a consequence of Theorem 1 in [
7] (see also [
1], p.328) and represents an integral majorization result.
In paper [
8] the following extension of Montgomery identity via Taylor’s formula is obtained.
In case
\(n=1\) the sum
\(\sum_{k=0}^{n-2}\cdots\) is empty, so identity (
1.10) reduces to the well-known Montgomery identity (see for instance [
9])
$$f ( x ) =\frac{1}{b-a}\int_{a}^{b}f ( t )\,dt+\int_{a}^{b}P ( x,s ) f^{\prime} ( s )\,ds, $$
where
\(P ( x,s ) \) is the Peano kernel, defined by
$$P ( x,s ) = \textstyle\begin{cases} \frac{s-a}{b-a}, & a\leq s\leq x,\\ \frac{s-b}{b-a}, & x< s\leq b. \end{cases} $$
The aim of this paper is to present a new generalization of weighted majorization theorem for n-convex functions, by using generalization of Taylor’s formula. We also obtain bounds for the remainders in new majorization identities by using the Čebyšev type inequalities. We give mean value theorems and n-exponential convexity for functionals related to these new majorization identities.
2 Majorization inequality by extension of Montgomery identity via Taylor’s formula
We may state its integral version as follows:
Now we state the main generalization of the majorization inequality by using the identities just obtained.
Now we state important consequence as follows:
Its integral analogs are given as follows:
3 Bounds for identities related to generalization of majorization inequality
Let
\(g,h:[a,b]\rightarrow\mathbb{R}\) be two Lebesgue integrable functions. We consider the Čebyšev functional
$$ T(g,h)=\frac{1}{b-a}\int_{a}^{b}g(x)h(x)\,dx- \biggl( \frac{1}{b-a}\int_{a}^{b}g(x)\,dx \biggr) \biggl( \frac{1}{b-a}\int_{a}^{b}h(x)\,dx \biggr) . $$
(3.1)
The following results can be found in [
10].
Now by using aforementioned results, we are going to obtain generalizations of the results proved in the previous section.
For
m-tuples
\(w=(w_{1},\ldots,w_{m})\),
\(x=(x_{1},\ldots,x_{m})\), and
\(y=(y_{1},\ldots,y_{m})\) with
\(x_{i},y_{i}\in[ a,b]\),
\(w_{i}\in\mathbb{R}\) (
\(i=1,\ldots,m\)), and the function
\(T_{n}\) defined as in (
1.11), denote
$$ \delta(s)=\sum_{i=1}^{m}w_{i}T_{n}(y_{i},s)- \sum_{i=1}^{m}w_{i}T_{n}(x_{i},s),\quad \forall s\in [ a,b].$$
(3.4)
Similarly for continuous functions
\(x,y:[\alpha,\beta]\rightarrow [ a,b]\) and
\(w:[\alpha,\beta]\rightarrow\mathbb{R}\), denote
$$ \Delta(s)=\int_{\alpha}^{\beta}w ( t ) T_{n} \bigl(y(t),s\bigr)\,dt-\int_{\alpha}^{\beta}w ( t ) T_{n}\bigl(x(t),s\bigr)\,dt,\quad \forall s\in [ a,b].$$
(3.5)
Hence by using these notations we define Čebyšev functionals as follows:
$$\begin{aligned}& T(\delta,\delta) =\frac{1}{b-a}\int_{a}^{b} \delta^{2}(s)\,ds- \biggl( \frac{1}{b-a}\int_{a}^{b} \delta(s)\,ds \biggr) ^{2}, \\& T(\Delta,\Delta) =\frac{1}{b-a}\int_{a}^{b} \Delta^{2}(s)\,ds- \biggl( \frac{1}{b-a}\int_{a}^{b} \Delta(s)\,ds \biggr) ^{2}. \end{aligned}$$
Now, we are ready to state the main results of this section:
Here we state the integral version of the previous theorem.
By using Proposition
8 we obtain the following Grüss type inequality.
Integral version of the above theorem can be given as:
Here, the symbol
\(L_{p} [ a,b ] \) (
\(1\leq p<\infty \)) denotes the space of
p-power integrable functions on the interval
\([ a,b ] \) equipped with the norm
$$\Vert f\Vert _{p}= \biggl( \int_{a}^{b} \bigl\vert f ( t ) \bigr\vert ^{p}\,dt \biggr) ^{\frac{1}{p}}$$
and
\(L_{\infty} [ a,b ] \) denotes the space of essentially bounded functions on
\([ a,b ] \) with the norm
$$\Vert f\Vert _{\infty}=\mathop{\operatorname{ess\,sup}}_{t\in [ a,b ] }\bigl\vert f ( t ) \bigr\vert . $$
Now we state some Ostrowski-type inequalities related to the generalized majorization inequalities.
The integral case of the above theorem can be given as follows.
For our next two sections, we give here some constructions as follows. Under the assumptions of Theorem
3 using (
2.4) and Theorem
4 using (
2.7) we define the following functionals, respectively:
$$\begin{aligned}& \Lambda_{1}(f) = \sum_{i=1}^{m}w_{i}f ( y_{i} ) -\sum_{i=1}^{m}w_{i}f ( x_{i} ) -\frac{1}{b-a}\sum_{i=1}^{m}w_{i} \Biggl[ \sum_{k=0}^{n-2}\frac{1}{k! ( k+2 ) !} \bigl[ f^{ ( k+1 ) } ( a ) \\& \hphantom{\Lambda_{1}(f) =}{} \times \bigl[ ( y_{i}-a ) ^{k+2}- ( x_{i}-a ) ^{k+2} \bigr] -f^{ ( k+1 ) } ( b ) \bigl[ ( y_{i}-b ) ^{k+2}- ( x_{i}-b ) ^{k+2} \bigr] \bigr] \Biggr] , \end{aligned}$$
(A1)
$$\begin{aligned}& \Lambda_{2}(f)=\int_{\alpha}^{\beta}w ( t ) f \bigl(y(t)\bigr)\,dt-\int_{\alpha}^{\beta}w ( t ) f \bigl(x(t)\bigr)\,dt \\& \hphantom{\Lambda_{2}(f)=}{} -\frac{1}{b-a} \Biggl[ \sum_{k=0}^{n-2} \frac{1}{k! ( k+2 ) !}\int_{\alpha}^{\beta}w ( t ) \bigl[ f^{ ( k+1 ) } ( a ) \bigl[ \bigl( y(t)-a \bigr) ^{k+2}- \bigl( x(t)-a \bigr) ^{k+2} \bigr] \\& \hphantom{\Lambda_{2}(f)=}{} -f^{ ( k+1 ) } ( b ) \bigl[ \bigl( y ( t ) -b \bigr) ^{k+2}- \bigl( x ( t ) -b \bigr) ^{k+2} \bigr] \bigr]\,dt \Biggr] . \end{aligned}$$
(A2)
4 Mean value theorems
Now we give mean value theorems for \(\Lambda_{k}\), \(k\in\{1,2\}\). Here \(f_{0}(x)=\frac{x^{n}}{n!}\).
5 Log-convexity and n-exponential convexity
5.1 Logarithmically convex functions
A number of important inequalities arise from the logarithmic convexity of some functions as one can see in [
6].
Now, we recall some definitions. The following definition was originally given by Jensen in 1906 [
11]. Here
I is an interval in
\(\mathbb{R}\).
5.2 n-Exponentially convex functions
Bernstein [
12] and Widder [
13] independently introduced an important sub-class of convex functions, which is called the class of exponentially convex functions on a given open interval, and studied some properties of this newly defined class. Pečarić and Perić in [
14] introduced the notion of
n-exponentially convex functions, which is in fact a generalization of the concept of exponentially convex functions. In the present subsection, we discus the same notion of
n-exponential convexity by describing related definitions and some important results with some remarks from [
14].
Here, we get our results concerning the
n-exponential convexity and exponential convexity for our functionals
\(\Lambda_{k}\),
\(k\in\{1,2\}\), as defined in (
A1) and (
A2). Throughout the section
I is an interval in
\(\mathbb{R}\).
As a consequence of the above theorem we give the following corollaries.
Now, we give two important remarks and one useful corollary from [
15], which we will use in some examples in the next section.
Theorem
13 gives us the following corollary.
6 Examples with applications
In this section, we use various classes of functions \(\Omega=\{f_{t}:t \in I\}\) for any open interval \(I \subset\mathbb{R}\) to construct different examples of exponentially convex functions and applications to Stolarsky-type means. Let us consider some examples.
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Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
JP made the main contribution in conceiving the presented research. AAA, ARK, and JP worked jointly on each section, while AAA and ARK drafted the manuscript. All authors read and approved the final manuscript.