Some new Hermite–Hadamard type inequalities for s-convex functions and their applications

In this paper, we establish some new integral inequalities of Hermite–Hadamard type for s-convex functions by using the Hölder–İşcan integral inequality. We also compare our new results with the known results and show that the results which we obtained are better than the known results. Finally, we give some applications to trapezoidal formula and to special means.

The classical or the usual convexity is defined as follows:

A function \(f:I\subseteq \mathbb{R} \rightarrow \mathbb{R} \) is said to be convex on interval I if

for all \(x,y\in I\) and \(t\in [ 0,1 ] \).

Let \(f:I\subseteq \mathbb{R} \longrightarrow \mathbb{R} \) be a convex function on I and \(a,b\in I\) with \(a< b\). The following inequality is known in the literature as the Hermite–Hadamard inequality for convex functions:

In [8], Hudzik and Maligranda considered the class of s-convex functions in second sense. This is defined as follows:

A function \(f: [ 0,\infty ) \rightarrow \mathbb{R} \) is said to be s-convex in second sense if the inequality

holds for all \(x,y\in [ 0,\infty ) \), \(t\in [ 0,1 ] \), and \(s\in ( 0,1 ] \).

In [4], Dragomir and Fitzpatrick established a variant of Hermite–Hadamard inequality which holds for the s-convex functions.

Theorem 1.1

Suppose that \(f: [ 0,\infty ) \rightarrow [ 0, \infty )\) is an s-convex function in second sense, where \(s\in ( 0,1 ]\). Let \(a,b\in [ 0,\infty )\) and \(a< b\). If \(f\in L [ 0,1 ]\), then the following inequality holds:

For the generalizations and applications of the Hermite–Hadamard inequalities, see [1,2,3, 5,6,7, 9, 12,13,14,15, 17, 19,20,21].

A number of studies have shown that many of the results obtained about the theory of inequalities has a close relationship with the theory of convex functions.

The celebrated inequality of Hölder is well known for its fundamental role in many branches of pure and applied sciences. It has also important applications to the theory of convex functions as well as in many disciplines of applied mathematics. The power-mean integral inequality is also one of the most famous inequalities on applications to convex functions.

Theorem 1.2

(Hölder inequality for integrals [16])

Let \(p>1\) and \(1/p+1/q=1\). If f and g are real functions defined on \([ a,b ] \) and if \(\vert f \vert ^{p}\) and \(\vert g \vert ^{q}\) are integrable on \([ a,b ] \), then

with equality holds if and only if \(A \vert f ( x ) \vert ^{p}=B \vert g ( x ) \vert ^{q}\) almost everywhere, where A and B are constants.

A different version of Hölder integral inequality is given as follows.

Theorem 1.3

(Power-mean integral inequality)

Let \(q\geq 1\). If f and g are real functions defined on \([ a,b ] \) and if \(\vert f \vert \), \(\vert f \vert \vert g \vert ^{q}\) are integrable on \([ a,b ] \), then

In [10], İşcan obtained the following inequality for integrals which gives better results than the classical Hölder inequality.

Theorem 1.4

(Hölder–İşcan integral inequality)

Let \(p>1\) and \(1/p+1/q=1\). If f and g are real functions defined on \([ a,b ] \) and if \(\vert f \vert ^{p}\) and \(\vert g \vert ^{q}\) are integrable on \([ a,b ] \), then

In [11], a different representation of Hölder–İşcan inequality was given as follows.

Theorem 1.5

(Improved power-mean integral inequality)

Let \(q\geq 1\). If f and g are real functions defined on \([ a,b ] \) and if \(\vert f \vert \), \(\vert f \vert \vert g \vert ^{q}\) are integrable functions on \([ a,b ] \), then

In this section we obtain some new results about Hermite–Hadamard inequality for s-convex functions by using Hölder–İşcan integral inequality and improved power-mean integral inequality which provide better approach than the classical Hölder and power-mean integral inequalities, respectively.

In [5], Dragomir and Pearce obtained the following equality for differentiable functions.

Lemma 2.1

Let \(f:I\subseteq \mathbb{R} \longrightarrow \mathbb{R} \) be a differentiable function on \(I^{\circ }\) where \(a,b\in I\) with \(a< b\). If \(f^{\prime }\in L [ a,b ] \), then the following equality holds:

In [17], Muddassar et al. obtained the following inequality for s-convex functions by using Lemma 2.1 and Hölder integral inequality.

Theorem 2.1

Let \(f:I^{\circ }\subseteq \mathbb{R} \rightarrow \mathbb{R} \) be a differentiable function on \(I^{\circ }\), \(a,b\in I^{\circ }\) with \(a< b \) and \(f^{\prime }\in L [ a,b ] \). If \(\vert f ^{\prime } \vert ^{q}\) is s-convex on \([ a,b ] \), \(p>1\) such that \(q=\frac{p}{p-1}\), then

In the following theorem, we will obtain a new upper bound for the right-hand side of Hermite–Hadamard inequality for s-convex functions, which is better than inequality (3).

Theorem 2.2

Let \(f:I^{\circ }\subseteq \mathbb{R} \rightarrow \mathbb{R} \) be a differentiable function on \(I^{\circ }\), \(a,b\in I^{\circ }\) with \(a< b \) and \(f^{\prime }\in L [ a,b ] \). If \(\vert f ^{\prime } \vert ^{q}\) is s-convex on \([ a,b ] \), where \(p>1\) such that \(q=\frac{p}{p-1}\), then

Proof

From Lemma 2.1 and using Hölder–İşcan integral inequality (1), we have

Since \(\vert f^{\prime } \vert ^{q}\) is s-convex on \([ a,b ] \), then

and

and also

By inequalities (5), (6), and (7), we get inequality (4). □

Corollary 2.1

Let \(f:I^{\circ }\subseteq \mathbb{R} \rightarrow \mathbb{R} \) be a differentiable function on \(I^{\circ }\), \(a,b\in I^{\circ }\) with \(a< b \) and \(f^{\prime }\in L [ a,b ] \). If \(\vert f ^{\prime } \vert ^{q}\) is s-convex on \([ a,b ] \), where \(p>1\) such that \(q=\frac{p}{p-1}\), then

Proof

Using the fact \(\sum_{i=1}^{n} ( a_{i}+b_{i} ) ^{k}\leq \sum ( a_{i} ) ^{k}+\sum ( b_{i} ) ^{k}\) for \(k\in ( 0,1 ) \) with \(p>1\) such that \(q=\frac{p}{p-1}\) completes the proof. □

Remark 2.1

Inequality (4) is better than inequality (3). Indeed, since the function \(g: [ 0,\infty ) \rightarrow \mathbb{R} \), \(g ( x ) =x^{r}\), \(r\in ( 0,1 ] \) is concave, we can write

for all \(\alpha ,\beta \geq 0\). In inequality (9), if we choose

and \(r=\frac{1}{q}\), then we have

Thus, we obtain the following inequality:

If we take \(s=1\) in Remark 2.1, we have the following result which was obtained by İşcan in [10].

Remark 2.2

Let \(f:I^{\circ }\subseteq \mathbb{R} \rightarrow \mathbb{R} \) be a differentiable function on \(I^{\circ }\), \(a,b\in I^{\circ }\) with \(a< b \) and \(f^{\prime }\in L [ a,b ] \). If \(\vert f ^{\prime } \vert ^{q}\) is convex on \([ a,b ] \), then

where \(\frac{1}{p}+\frac{1}{q}=1\).

In [17], Muddassar et al. obtained the following inequality by using Lemma 2.1 and power-mean integral inequality.

Theorem 2.3

Let \(f:I^{\circ }\subseteq \mathbb{R} \longrightarrow \mathbb{R} \) be a differentiable function on \(I^{\circ }\) where \(a,b\in I\) with \(a< b\) and \(f^{\prime }\in L [ a,b ] \). If \(\vert f^{\prime } \vert ^{q}\) is s-convex on \([ a,b ] \) for some \(p>1\) such that \(q=\frac{p}{p-1}\), then the following inequality holds:

If Theorem 2.3 is proved again by using improved power-mean integral inequality, then we get the following result.

Theorem 2.4

Let \(f:I^{\circ }\subseteq \mathbb{R} \longrightarrow \mathbb{R} \) be a differentiable function on \(I^{\circ }\) where \(a,b\in I\) with \(a< b\) and \(f^{\prime }\in L [ a,b ] \). If \(\vert f^{\prime } \vert ^{q}\) is s-convex on \([ a,b ] \) for some \(p>1\) such that \(q=\frac{p}{p-1}\), then the following inequality holds:

Proof

From Lemma 2.1 and applying improved power-mean integral inequality (2) for \(q>1\), we have

By s-convexity of \(\vert f^{\prime } \vert ^{q}\) on \([ a,b ] \), we have

and

Furthermore,

By using inequalities (12) and (13), we get inequality (11). □

Remark 2.3

Inequality (11) is better than inequality (10). Indeed, using inequality (9) in Remark 2.1, we have

Therefore, we obtain

If we take \(s=1\) in Remark 2.3, we have the following result for convex functions which is better than the inequality given in [18, Theorem 1].

Remark 2.4

Let \(f:I^{\circ }\subseteq \mathbb{R} \rightarrow \mathbb{R} \) be a differentiable function on \(I^{\circ }\), \(a,b\in I^{\circ }\) with \(a< b \) and \(f^{\prime }\in L [ a,b ] \). If \(\vert f ^{\prime } \vert ^{q}\) is convex on \([ a,b ] \), then

where \(\frac{1}{p}+\frac{1}{q}=1\).

Let D be a division of the interval \([ a,b ] \), i.e., \(D:a=x_{0}< x_{1}<\cdots<x_{n-1}<x_{n}=b\) and consider the quadrature formula

where

is the trapezoidal formula and \(R ( f,D ) \) denotes the associated approximation error of the integral I.

Proposition 3.1

Let \(f:I\subseteq \mathbb{R} \rightarrow \mathbb{R} \) be a differentiable function on \(I^{\circ }\), \(a,b\in I\) with \(a< b\) and \(f^{\prime }\in L [ a,b ] \). If \(\vert f^{\prime } \vert ^{q}\) is s-convex on \([ a,b ] \) for \(p,q>1\), for every division D of \([ a,b ] \), the following trapezoidal error estimate holds:

Proof

Applying Corollary 2.1 on the subintervals \([ x_{k},x_{k+1} ] \), \(( k=0,1,2,\ldots,n-1 ) \) of the division D, we have

Summing over k from 0 to \(n-1\) and using the generalized triangle inequality, we get

So, we have

which is the required result. □

In [6], Hudzik and Maligranda gave the following example.

Let \(a,b,c\in \mathbb{R} \) and \(s\in ( 0,1 ) \). For \(t\in [ 0,\infty ) \), define the function \(f: [ 0, \infty ) \rightarrow \mathbb{R} \) as

If \(b\geq 0\) and \(0\leq c\leq a\), then \(f\in K_{s}^{2}\). So, we have \(f: [ 0,1 ] \rightarrow [ 0,1 ] \), \(f ( t ) =t^{s}\), \(f\in K_{s}^{2}\) for \(a=c=0\) and \(b=1\).

Let us recall the following special means for arbitrary real numbers a and b \(( a\neq b )\).

1. (1)

   The arithmetic mean:

   $$\begin{aligned} A:=A ( a,b ) =\frac{a+b}{2}. \end{aligned}$$
2. (2)

   The p-logarithmic mean:

   $$\begin{aligned} L_{p}:=L_{p} ( a,b ) = \biggl[ \frac{b^{p+1}-a^{p+1}}{ ( p+1 ) ( b-a ) } \biggr] ^{\frac{1}{p}},\quad p \in \mathbb{R} \backslash \{ -1,0 \} . \end{aligned}$$

Proposition 4.1

Let \(p>1\), \(a< b\), \(s\in ( 0,1 ) \), and \(q=\frac{p}{p-1}\). Then we have

Proof

By Corollary 2.1 applied for the s-convex function \(f: [ 0,1 ] \rightarrow [ 0,1 ] \), \(f ( x ) =x ^{s}\) for \(s\in ( 0,1 ) \), the proof is completed. □

In this paper, some new results of Hermite–Hadamard type for s-convex functions are established. It is shown that the results obtained here are better than the known results. Some applications of these results to trapezoidal formula and to special means have also been presented. The results of this paper may stimulate further research for the researchers working in this field.

Acknowledgements

Authors are thankful to the editor and anonymous referees for their valuable comments and suggestions.

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Funding

There is no funding for this research article.

Affiliations

1. Department of Mathematics, Faculty of Art and Science, Kırklareli University, Kırklareli, Turkey

   • Serap Özcan

2. Department of Mathematics, Faculty of Art and Science, Giresun University, Giresun, Turkey

   • İmdat İşcan

Contributions

The authors contributed equally to the writing of this paper. The authors read and approved the final manuscript.

Corresponding author

Correspondence to Serap Özcan.

Competing interests

The authors declare that they have no competing interests.

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