Abstract
In this paper, we establish the Hermite–Hadamard-type inequalities for the generalized s-convex functions in the second sense on real linear fractal set \(\mathbb {R}^{\alpha }(0<\alpha <1).\)
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Introduction
The convex function plays an important role in the class mathematical analysis course and other fields. In [1], Hudzik and Maligranda introduced two kinds of s-convex functions in the space of European space \(\mathbb {R}\). In addition, many important inequalities are established for the s-convex functions in \(\mathbb {R}\). For example, the Hermite–Hadamard's inequality is one of the best known results in the literature, see [2,3,4] and so on.
In recent years, the fractal theory has received significantly remarkable attention [5]. The calculus on fractal set can lead to better comprehension for the various real-world models from the engineering and science [6].
On the fractal set, Mo et al. [7, 8] introduced the definition of the generalized convex function and established Hermite–Hadamard-type inequality. In [9], the authors introduced two kinds of generalized s-convex functions on fractal sets \(\mathbb {R}^{\alpha }(0<\alpha <1).\)
The definitions of the generalized s-convex functions are as follows:
Definition 1.1
[9] Suppose that \(\mathbb {R_+}=[0,\infty ).\) If the function \(f:\mathbb {R_+}\rightarrow \mathbb {R^{\alpha }}\) satisfies the following inequality:
for all \(u,v\in \mathbb {R_+}\) and all \(\lambda _1,\lambda _2\ge 0\) with \(\lambda _1^ s+\lambda _2^s=1\) and \(0<s<1,\) then f is said to be a generalized s-convex function in the first sense. We denote this by \(f\in GK_s^1.\)
Definition 1.2
[9] Suppose that \(\mathbb {R_+}=[0,\infty ).\) If the function \(f:\mathbb {R_+}\rightarrow \mathbb {R^{\alpha }}\) satisfies the following inequality:
for all \(u,v\in \mathbb {R_+}\) and all \(\lambda _1,\lambda _2\ge 0\) with \(\lambda _1+\lambda _2=1\) and \(0<s<1,\) then f is said to be a generalized s-convex function in the second sense. We denote this by \(f\in GK_s^2.\)
Note that the generalized s-convex function in both sense is generalized convex function [9] for \(s=1\).
Inspired by [2, 3, 8], in this paper, we will establish the Hermite–Hadamard-type inequalities for the generalized s-convex functions.
Preliminaries
Now, let us review the operations with real line number on fractal space. In addition, we will use the Gao–Yang–Kang’s idea to describe the definitions of the local fractional derivative and local fractional integral [10,11,12,13,14].
Let \(a^\alpha ,b^\alpha \) and \(c^\alpha \) belong to the set \(\mathbb {R}^\alpha (0<\alpha <1)\) of real line numbers, then
-
1.
\(a^\alpha b^\alpha \) and \(a^\alpha +b^\alpha \) belong to the set \(\mathbb {R}^\alpha \);
-
2.
\(a^\alpha +b^\alpha =(a+b)^\alpha =b^\alpha +a^\alpha =(b+a)^\alpha \);
-
3.
\(a^\alpha +(b^\alpha +c^\alpha )=(a^\alpha +b^\alpha )+c^\alpha \);
-
4.
\(a^\alpha b^\alpha =(ab)^\alpha =b^\alpha a^\alpha =(ba)^\alpha \);
-
5.
\(a^\alpha (b^\alpha c^\alpha )=(a^\alpha b^\alpha )c^\alpha \);
-
6.
\(a^\alpha (b^\alpha +c^\alpha )=a^\alpha b^\alpha +a^\alpha c^\alpha \);
-
7.
\(0^\alpha +a^\alpha =a^\alpha +0^\alpha =a^\alpha \) and \(1^\alpha \cdot a^\alpha =a^\alpha \cdot 1^\alpha =a^\alpha \).
Definition 2.1
([10]) If the function \(f:[a,b]\rightarrow \mathbb {R^\alpha }\) satisfies the inequality
for \(c>0\) and \(\alpha (0<\alpha \le 1),\) then f is called a Hölder continuous function. In this case, we think that f is in the space \(C_\alpha [a,b].\)
Definition 2.2
[10] Let \(\triangle ^\alpha (f(x)-f(x_0))\cong \Gamma (1+a)(f(x)-f(x_0)).\) Then, the local fractional derivative of f of order \(\alpha \) at \(x=x_0\) is defined by
If there exists \(f^{((k+1)\alpha )}(x)=\mathop {\overbrace{D_x^{\alpha } \ldots D_x^\alpha }}\limits ^{k+1\; {times}}f(x)\) for any \(x\in I\subseteq \mathbb {R},\) then we denoted \(f\in D_{(k+1)\alpha }(I)\), where \(k=0,1,2\dots .\)
Definition 2.3
[10] For \(f\in C_{\alpha }[a,b],\) the local fractional integral of the function f is defined by
where \(\triangle t_j=t_{j+1}-t_j,\;\triangle t=\max \{\triangle t_1,\triangle t_2,\triangle t_j,\ldots \}\) and \([t_j,t_j+1],\) \(j=0,\ldots ,N-1,\) \(t_0=a,\) \(t_N=b,\) is a partition of the interval [a, b].
Lemma 2.1
[10] Let \(f\in C_\alpha [g(a), g(b)]\) and \(g\in C_1[a,b],\) then
Lemma 2.2
[10]
-
1.
Let \(f(x)=g^{(\alpha )}(x)\in C_{\alpha }[a,b]\), then we have
$$\begin{aligned} _{a}I_{b}^{(\alpha )}f(x)=g(b)-g(a). \end{aligned}$$ -
2.
Let \(f(x), g(x)\in D_{\alpha }[a,b]\) and \(f^{(\alpha )}(x), g^{(\alpha )}(x)\in C_{\alpha }[a,b]\), then we have
$$\begin{aligned} _{a}I_{b}^{\alpha }f(x)g^{(\alpha )}(x)=f(x)g(x)\bigg |_{a}^{b}-_aI_{b}^{(\alpha )}f^{(\alpha )}(x)g(x). \end{aligned}$$
Lemma 2.3
[10]
From the above formula and Lemma 2.2 , we have
Lemma 2.4
[10] (Generalized Hölder’s inequality) Let \(f,g\in C_{\alpha }[a,b]\) and \(p,q>1\) with \(1/p+1/q=1.\) Then, it follows that
Main results
Theorem 3.1
Let \(f:\mathbb {R_+}\rightarrow \mathbb {R^{\alpha }}\) be a generalized s-convex function in the second sense for \(0<s<1\) and \(a,b\in [0,\infty )\) with \(a<b.\) Then, for \(f\in C_{\alpha }[a,b],\) the following inequalities hold:
Proof
Let \(x=a+b-t.\) Then, from Lemma 2.1, we have \(_{\frac{a+b}{2}}I_b^{(\alpha )}f(x)={_a}I_{\frac{a+b}{2}}^{(\alpha )}f(a+b-t).\)
Since f is a generalized s-convex function in the second sense, then
In the other hand, let \(x=b-(b-a)t\), \(0\le t\le 1,\) then we get
From Lemma 2.3, it is easy to see that
and
Therefore
Combining the above estimates, we obtain
\(\square \)
Theorem 3.2
Let \(I\subset \mathbb {R}\) be an interval, and \(I^0\) be the interior of I. Suppose that \(f: I\rightarrow \mathbb {R^{\alpha }}\) is a differentiable function on \(I^0\) such that \(f^{(\alpha )}\in C_{\alpha }[a,b]\) , where \(a,b\in I^0\) with \(a<b.\) If \(|f^{(\alpha )}|^q\) is a generalized s-convex function in the second sense on [a, b] for some fixed \(s\in (0,1)\) and \(q\ge 1\), then
To show Theorem 3.2 is right, we need the following Lemma.
Lemma 3.1
([8]) Let f : \(I\rightarrow \mathbb {R^{\alpha }},\) \(I\subset [0,\infty ).\) If \(f\in D_\alpha (I^0)\) and \(f^{(\alpha )}\in C_\alpha [a,b]\) for \(a,b\in I^0\) with \(a<b,\) then the following equality holds:
Now, let us give the proof of Theorem 3.2.
Proof
From Lemma 3.1, it is obvious that
Let us estimate
for \(q=1\) and \(q>1.\)
Case I. \(q=1.\)
Since \(|f^{(\alpha )}|\) is generalized sconvex on [a, b] in the second sense, we can know that for any \(t\in [0,1]\)
Then, we have
From Lemmas 2.2 and 2.3, it is easy to see that
In addition, let \(1-t=x,\) then by Lemma 2.1 and (3.4), we have
Thus, substituting (3.4) and (3.5) into (3.3), we have
Thus, from (3.2), we obtain
Case II. \(q>1.\)
Using the generalized Hölder’s inequality (Lemma 2.4), we obtain
It is obvious that
Moreover, since \(|f^{(\alpha )}|^q\) is generalized s convex in the second sense on [a, b], then
From (3.3) and (3.4), it is easy to see that
Therefore
Thus, substituting (3.8) and (3.9) into (3.7), we have
Therefore, from (3.2), it follows that
Thus, we complete the proof of Theorem 3.2. \(\square \)
Theorem 3.3
Suppose that f : \(I\rightarrow \mathbb {R^{\alpha }},\) \(I\subset [0,\infty )\) is a differentiable function on \(I^0\), such that \(f^{(\alpha )}\in C_{\alpha }[a,b]\) , where \(a,b\in I\) with \(a<b\). If \(|f^{(\alpha )}|^q\) is a generalized s-convex function in the second sense on [a, b] for some fixed \(s\in (0,1)\) and \(q>1\), then
Proof
From Lemma 3.1, we have
Let us estimate
and
respectively.
Using the generalized Hölder’s inequality(Lemma 2.4), we obtain
It is easy to see that
Let \(|f^{(\alpha )}(ta+(1-t)b)|^q=U(t).\) It is easy to see that U(t) is a generalized sconvex function in the second sense. Thus, from the right-hand side of (3.1), it follows that
Thus, substituting (3.12) and (3.13) into (3.11), we get
Moreover
In addition, similar to the estimate of (3.13), we have
Therefore, it is analogues to the estimate of (3.11), we have
Thus, combining (3.10), (3.14), and (3.15), we obtain
\(\square \)
Therefore, we complete the proof of Theorem 3.3.
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The authors would like to express their gratitude to the Editors and referees for some very valuable suggestion.
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The research is supported by the NNSF of China (11601035, 11471050).
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Mo, H., Sui, X. Hermite–Hadamard-type inequalities for generalized s-convex functions on real linear fractal set \(\mathbb {R}^{\alpha }(0<\alpha <1)\) . Math Sci 11, 241–246 (2017). https://doi.org/10.1007/s40096-017-0227-z
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DOI: https://doi.org/10.1007/s40096-017-0227-z