一个新的Grüss型不等式

A New Grüss Type Inequality

摘要: 基于分形分析应用需要，本文讨论了Riemann-Liouville分数阶积分的Grüss型不等式，得到了一个改进的结果。文章最后还证明了[1]中的定理2和[2]中的定理9是本文所得结论的特殊形式。

Abstract: Based on the application of fractal analysis, this paper discusses the Grüss type inequalities in-cluding the Riemann-Liouville fractional integral and obtains an improved result. In the end, this paper also proves that Theorem 2 of [1] and Theorem 9 of [2] are a special form of the conclusion of this paper.

1. 引言

积分不等式在研究积分与微分方程中起着非常重要的作用。本文讨论Grüss不等式。

1935年，G. Grüss [3] 提出了Grüss不等式：

命题1.1 [3] ：设 $f\left(t\right)$ 和 $g\left(t\right)$ 是 $\left[a,b\right]$ 上两个可积函数。若存在 $m,M,n,N\in ℝ$，使得当 $t\in \left[a,b\right]$ 时，有 $m\le f\left(t\right)\le M,n\le g\left(t\right)\le N$，则

$\left|\frac{1}{b-a}{\displaystyle {\int }_a^{b}f\left(t\right)g\left(t\right)\text{d}t}-\frac{1}{{\left(b-a\right)}^2}{\displaystyle {\int }_a^{b}f\left(t\right)\text{d}t{\displaystyle {\int }_a^{b}g\left(t\right)\text{d}t}}\right|\le \frac{1}{4}\left(M-m\right)\left(N-n\right).$ (1.1)

Grüss不等式在1935年提出后备受关注。学者们基于经典Grüss不等式(即1.1式)建立了大量的Grüss型不等式，并将其应用于一些分析问题中。

1993年，Mitrinovic [4] 考察了Grüss不等式的离散形式以及行列式形式，并将其用于估计集合 $A=\left\{f\left(t\right)|f\left(t\right)\in AC\left[a,b\right],f^{\prime }\left(t\right)\in L_2\left[a,b\right]\right\}$ ( $AC\left[a,b\right]$ 表示区间 $\left[a,b\right]$ 上的绝对连续函数)中元素的范数。1998年，Dragomir [5] 将经典Grüss不等式中的积分推广到加权积分，得到了一个Grüss型不等式。然后又讨论被积函数 $f\left(t\right)$ 和 $g\left(t\right)$ 是否满足Lipschitz条件和Hölder条件，从而给出一系列Grüss型不等式。1999年，Dragomir [6] 将Grüss不等式应用于内积空间，从而得到了一个新的Grüss型不等式。2002年，Dragomir [7] 将Grüss型不等式 [5] 中的加权积分推广为Riemann-Stieltjes积分，并讨论被积函数的各种情形，从而得到一系列新的Grüss型不等式。2010年，Moslehian [8] 将Grüss不等式推广到了线性算子空间。

Grüss不等式在发展初期，人们主要考虑在不同空间中的Grüss型不等式。而随着空间理论的成熟，人们开始将关注点转向Grüss不等式中积分(整数阶积分)的变化。例：1998年，Dragomir [5] 将经典Grüss不等式中的积分推广到加权积分；2002年，Dragomir [7] 将Grüss型不等式 [5] 中的加权积分推广为Riemann-Stieltjes积分等。随着积分理论的发展，一些学者开始考虑分数阶积分与不等式的结合，例 [9] [10] [11] 。Grüss不等式主要考虑与Riemann-Liouville分数阶积分结合。

2010年，Dahmani [12] 第一次将Riemann-Liouville分数阶积分与Grüss不等式结合在一起得到了分数阶积分不等式：

命题1.2：(见 [12] 的定理3.1)设 $f\left(t\right)$， $g\left(t\right)$ 是 $\left[0,\infty \right)$ 上两个可积函数。若存在常数 $m,M,n,N$，使得 $f\left(t\right)$ 和 $g\left(t\right)$ 满足 $m\le f\left(t\right)\le M,n\le g\left(t\right)\le N,t\in \left[0,\infty \right)$。那么当 $t>0,\alpha >0$ 时，有

$\left|\frac{t^{\alpha }}{\Gamma \left(\alpha +1\right)}{J}^{\alpha }f\left(t\right)g\left(t\right)-J^{\alpha }f\left(t\right)J^{\alpha }g\left(t\right)\right|\le {\left(\frac{t^{\alpha }}{2\Gamma \left(\alpha +1\right)}\right)}^2\left(M-m\right)\left(N-n\right),$ (1.2)

这里 $J^{\alpha }f\left(t\right)$ 是 $f\left(t\right)$ 的Riemann-Liouville分数阶积分(将在第二部分给出它的定义)。2016年，Erden [1] 建立一个新的分数阶积分，并得到一个与此积分有关的Grüss型不等式：

命题1.3：(见 [1] 的定理2)设 $h\left(t\right):\left[0,\infty \right)\to \left(0,\infty \right)$ 是 $\left[0,\infty \right)$ 上的单调增函数并在 $\left(0,\infty \right)$ 上有连续导函数 $h^{\prime }\left(t\right)$。若 $f\left(t\right)$ 和 $g\left(t\right)$ 是 $\left[0,\infty \right)$ 上两个可积函数并且满足

$m\le f\left(t\right)\le M,n\le g\left(t\right)\le N;m,M,n,N\in ℝ,t\in \left[0,\infty \right).$

那么当 $\alpha >0$ 时，有

$\left|\frac{\left(h\left(t\right)-h\right){\left(0\right)}^{\alpha }}{\Gamma \left(\alpha +1\right)}{J}_h^{\alpha }f\left(t\right)g\left(t\right)-J_h^{\alpha }f\left(t\right)J_h^{\alpha }g\left(t\right)\right|\le {\left(\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }}{2\Gamma \left(\alpha +1\right)}\right)}^2\left(M-m\right)\left(N-n\right),$ (1.3)

这里 $J_h^{\alpha }f\left(t\right)$ 是 $f\left(t\right)$ 推广的Riemann-Liouville分数阶积分(将在第二部分给出其定义)。

在过去的几年中，人们只考虑Grüss型不等式中被积函数的限制条件是常数的情况。而近些年，人们开始关注其限制条件是可积函数的情形。2014年，Tariboon [2] 将 [12] 中被积函数 $f\left(t\right),g\left(t\right)$ 的四个常边界 $m,M,n,N$ 用四个可积函数 ${\phi }_1\left(t\right),{\phi }_2\left(t\right),{\psi }_1\left(t\right),{\psi }_2\left(t\right)$ 取代，从而得到新的Grüss型不等式：

命题1.4：(见 [2] 的定理9)设 $f\left(t\right),g\left(t\right)$ 是 $\left[0,\infty \right)$ 上两个可积函数。若

(C₁) 在 $\left[0,\infty \right)$ 上存在两个可积函数 ${\phi }_1\left(t\right),{\phi }_2\left(t\right)$ 且满足

${\phi }_1\left(t\right)\le f\left(t\right)\le {\phi }_2\left(t\right),t\in \left[0,\infty \right).$ (1.4)

(C₂) 在 $\left[0,\infty \right)$ 上存在两个可积函数 ${\psi }_1\left(t\right),{\psi }_2\left(t\right)$ 且满足

${\psi }_1\left(t\right)\le g\left(t\right)\le {\psi }_2\left(t\right),t\in \left[0,\infty \right).$ (1.5)

那么当 $t>0,\alpha >0$ 时，有

$\left|\frac{t^{\alpha }}{\Gamma \left(\alpha +1\right)}{J}^{\alpha }f\left(t\right)g\left(t\right)-J^{\alpha }f\left(t\right)J^{\alpha }g\left(t\right)\right|\le \sqrt{S\left(f,{\phi }_1,{\phi }_2\right)S\left(g,{\psi }_1,{\psi }_2\right)},$ (1.6)

这里

$\begin{array}{c}S\left(x,y,z\right)=\left(J^{\alpha }z\left(t\right)-J^{\alpha }x\left(t\right)\right)\left(J^{\alpha }x\left(t\right)-J^{\alpha }y\left(t\right)\right)\\ +\frac{t^{\alpha }}{\Gamma \left(\alpha +1\right)}{J}^{\alpha }y\left(t\right)x\left(t\right)-J^{\alpha }y\left(t\right)J^{\alpha }x\left(t\right)\\ +\frac{t^{\alpha }}{\Gamma \left(\alpha +1\right)}{J}^{\alpha }z\left(t\right)x\left(t\right)-J^{\alpha }z\left(t\right)J^{\alpha }x\left(t\right)\\ -\frac{t^{\alpha }}{\Gamma \left(\alpha +1\right)}{J}^{\alpha }y\left(t\right)z\left(t\right)+J^{\alpha }y\left(t\right)J^{\alpha }z\left(t\right)\end{array}$ (1.7)

从Grüss不等式的发展历程来看，我们可从两个方面来研究它。一方面，我们考虑在Grüss不等式中使用何种类型的分数阶积分。另一方面，考虑不等式中被积函数的限制条件。由于Rieamm-Liouville分数阶积分在分形分析和相关理论中有着广泛的应用，例 [13] [14] 。因此本文讨论有关推广的Riemann-Liouville分数阶积分 $J_h^{\alpha }$ 的Grüss型不等式。而从命题1.3和命题1.4中，我们又可以将被积函数的限制条件推广到函数的情形。以上就是本文研究的主要内容。

我们将在第二部分给出本文必要的定义及引理，在第三部分给出本文结论。最后可以看出本文结论是新的。并且本文结论可以用于无界函数(将在例子中展出)。

2. 定义和引理

定义2.1 [1] ：设 $f\left(t\right)\in L_1\left(0,\infty \right)$，则 $f\left(t\right)$ 的 $\alpha \ge 0$ 阶Riemann-Liouville分数阶积分是

$J^{\alpha }f\left(t\right)=\frac{1}{\Gamma \left(\alpha \right)}{\displaystyle {\int }_0^t{\left(t-s\right)}^{\alpha -1}f\left(s\right)\text{d}s},t>0,$ (2.1)

这里 $\Gamma \left(\alpha \right)={\displaystyle {\int }_0^{\infty }{u}^{\alpha -1}{\text{e}}^{-u}\text{d}u}$。

定义2.2 [1] ：设 $h\left(t\right):\left[0,\infty \right)\to \left(0,\infty \right)$ 是 $\left[0,\infty \right)$ 上的单调增函数并在 $\left[0,\infty \right)$ 上有连续导函数 $h^{\prime }\left(t\right)$。若 $f\left(t\right)$ 是 $\left[0,\infty \right)$ 上可积函数，则 $f\left(t\right)$ 的 $\alpha \ge 0$ 阶推广的Riemann-Liouville分数阶积分是

$J_h^{\alpha }f\left(t\right)=\frac{1}{\Gamma \left(\alpha \right)}{\displaystyle {\int }_0^t\frac{h^{\prime }\left(s\right)}{{\left(h\left(t\right)-h\left(s\right)\right)}^{1-\alpha }}f\left(s\right)\text{d}s},t>0,$ (2.2)

这里 $\Gamma \left(\alpha \right)={\displaystyle {\int }_0^{\infty }{u}^{\alpha -1}}{\text{e}}^{-u}\text{d}u$。

下面给出一个基本引理，以便后面的应用。

引理2.3：设 $h\left(t\right):\left[0,\infty \right)\to \left(0,\infty \right)$ 是 $\left[0,\infty \right)$ 上的单调增函数并在 $\left(0,\infty \right)$ 上有连续导函数 $h^{\prime }\left(t\right)$。若 $f\left(t\right)$ 是 $\left[0,\infty \right)$ 上可积函数并满足条件(C₁)。那么当 $t>0,\alpha >0$ 时，有

$\begin{array}{l}\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }}{\Gamma \left(\alpha +1\right)}{J}_h^{\alpha }{f}^2\left(t\right)-{\left(J_h^{\alpha }f\left(t\right)\right)}^2=\left(J_h^{\alpha }{\phi }_2\left(t\right)-J_h^{\alpha }f\left(t\right)\right)\left(J_h^{\alpha }f\left(t\right)-J_h^{\alpha }{\phi }_1\left(t\right)\right)\\ -\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }}{\Gamma \left(\alpha +1\right)}{J}_h^{\alpha }\left({\phi }_2\left(t\right)-f\left(t\right)\right)\left(f\left(t\right)-{\phi }_1\left(t\right)\right)+\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }}{\Gamma \left(\alpha +1\right)}{J}_h^{\alpha }\left({\phi }_1\left(t\right)f\left(t\right)\right)-J_h^{\alpha }{\phi }_1\left(t\right)J_h^{\alpha }f\left(t\right)\\ +\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }}{\Gamma \left(\alpha +1\right)}{J}_h^{\alpha }\left({\phi }_2\left(t\right)f\left(t\right)\right)-J_h^{\alpha }{\phi }_2\left(t\right)J_h^{\alpha }f\left(t\right)-\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }}{\Gamma \left(\alpha +1\right)}{J}_h^{\alpha }\left({\phi }_1\left(t\right){\phi }_2\left(t\right)\right)+J_h^{\alpha }{\phi }_1\left(t\right)J_h^{\alpha }{\phi }_2\left(t\right)\end{array}$ (2.3)

证：当 $x>0,y>0$ 时，有

$\begin{array}{l}\left({\phi }_2\left(y\right)-f\left(y\right)\right)\left(f\left(x\right)-{\phi }_1\left(x\right)\right)+\left({\phi }_2\left(x\right)-f\left(x\right)\right)\left(f\left(y\right)-{\phi }_1\left(y\right)\right)\\ -\left({\phi }_2\left(x\right)-f\left(x\right)\right)\left(f\left(x\right)-{\phi }_1\left(x\right)\right)-\left({\phi }_2\left(y\right)-f\left(y\right)\right)\left(f\left(y\right)-{\phi }_1\left(y\right)\right)\\ =f^2\left(x\right)+f^2\left(y\right)-2f\left(x\right)f\left(y\right)+f\left(x\right){\phi }_2\left(y\right)+{\phi }_1\left(x\right)f\left(y\right)-{\phi }_1\left(x\right){\phi }_2\left(y\right)\\ +{\phi }_2\left(x\right)f\left(y\right)+f\left(x\right){\phi }_1\left(y\right)-{\phi }_2\left(x\right){\phi }_1\left(y\right)-{\phi }_2\left(x\right)f\left(x\right)+{\phi }_1\left(x\right){\phi }_2\left(x\right)\\ -{\phi }_1\left(x\right)f\left(x\right)-{\phi }_2\left(y\right)f\left(y\right)-{\phi }_1\left(y\right)f\left(y\right)+{\phi }_1\left(y\right){\phi }_2(y)\end{array}$

上式两边同乘以

$\frac{h^{\prime }\left(x\right)}{\Gamma \left(\alpha \right){\left(h\left(t\right)-h\left(x\right)\right)}^{1-\alpha }},x\in \left(0,t\right),t>0$

并对等式两边关于x在 $\left(0,t\right)$ 上积分，得到

$\begin{array}{l}\left({\phi }_2\left(y\right)-f\left(y\right)\right)\left(J_h^{\alpha }f\left(t\right)-J_h^{\alpha }{\phi }_1\left(t\right)\right)+\left(J_h^{\alpha }{\phi }_2\left(t\right)-J_h^{\alpha }f\left(t\right)\right)\left(f\left(y\right)-{\phi }_1\left(y\right)\right)\\ -\left(J_h^{\alpha }{\phi }_2\left(t\right)-f\left(t\right)\right)\left(f\left(t\right)-{\phi }_1\left(t\right)\right)-\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }}{\Gamma \left(\alpha +1\right)}\left({\phi }_2\left(y\right)-f\left(y\right)\right)\left(f\left(y\right)-{\phi }_1\left(y\right)\right)\\ =J_h^{\alpha }{f}^2\left(t\right)+\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }}{\Gamma \left(\alpha +1\right)}{f}^2\left(y\right)-2f\left(y\right)J_h^{\alpha }f\left(t\right)+{\phi }_2\left(y\right)J_h^{\alpha }f\left(t\right)-{\phi }_2\left(y\right)J_h^{\alpha }{\phi }_1\left(t\right)\\ +f\left(y\right)J_h^{\alpha }{\phi }_2\left(t\right)+f\left(y\right)J_h^{\alpha }{\phi }_1\left(t\right)+{\phi }_1\left(y\right)J_h^{\alpha }f\left(t\right)-{\phi }_1\left(y\right)J_h^{\alpha }{\phi }_2\left(t\right)-J_h^{\alpha }\left({\phi }_2\left(t\right)f\left(t\right)\right)\\ +J_h^{\alpha }\left({\phi }_1\left(t\right){\phi }_2\left(t\right)\right)-J_h^{\alpha }\left({\phi }_1\left(t\right)f\left(t\right)\right)-\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }}{\Gamma \left(\alpha +1\right)}{\phi }_2\left(y\right)f\left(y\right)\\ +\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }}{\Gamma \left(\alpha +1\right)}{\phi }_1\left(y\right){\phi }_2\left(y\right)-\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }}{\Gamma \left(\alpha +1\right)}{\phi }_1\left(y\right)f(y)\end{array}$

再对上式两边同乘上 $h^{\prime }\left(y\right)/\left(\Gamma \left(\alpha \right){\left(h\left(t\right)-h\left(y\right)\right)}^{1-\alpha }\right),y\in \left(0,t\right),t>0$，并对y在 $\left(0,t\right)$ 上积分就可以得到(2.3)。

3. 定理及证明

现给出有关推广的Riemann-Liouville分数阶积分 $J_h^{\alpha }$ 的Grüss型不等式。

定理3.1：设 $h\left(t\right):\left[0,\infty \right)\to \left(0,\infty \right)$ 是 $\left[0,\infty \right)$ 上的单调增函数并在 $\left(0,\infty \right)$ 上有连续导函数 $h^{\prime }\left(t\right)$。若 $f\left(t\right)$ 和 $g\left(t\right)$ 是 $\left[0,\infty \right)$ 上两个可积函数并满足条件(C₁)和(C₂)。那么当 $t>0,\alpha >0$ 时，可以得到

$\left|\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }}{\Gamma \left(\alpha +1\right)}{J}_h^{\alpha }f\left(t\right)g\left(t\right)-J_h^{\alpha }f\left(t\right)J_h^{\alpha }g\left(t\right)\right|\le \sqrt{S^{*}\left(f,{\phi }_1,{\phi }_2\right)S^{*}\left(g,{\psi }_1,{\psi }_2\right)},$ (3.1)

这里

$\begin{array}{c}{S}^{*}\left(x,y,z\right)=\left(J_h^{\alpha }z\left(t\right)-J_h^{\alpha }x\left(t\right)\right)\left(J_h^{\alpha }x\left(t\right)-J_h^{\alpha }y\left(t\right)\right)\\ +\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }}{\Gamma \left(\alpha +1\right)}{J}_h^{\alpha }y\left(t\right)x\left(t\right)-J_h^{\alpha }y\left(t\right)J_h^{\alpha }x\left(t\right)\\ +\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }}{\Gamma \left(\alpha +1\right)}{J}_h^{\alpha }z\left(t\right)x\left(t\right)-J_h^{\alpha }z\left(t\right)J_h^{\alpha }x\left(t\right)\\ -\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }}{\Gamma \left(\alpha +1\right)}{J}_h^{\alpha }y\left(t\right)z\left(t\right)+J_h^{\alpha }y\left(t\right)J_h^{\alpha }z\left(t\right)\end{array}$ (3.2)

证：令

$\begin{array}{c}T\left(x,y\right)=\left(f\left(x\right)-f\left(y\right)\right)\left(g\left(x\right)-g\left(y\right)\right)\\ =f\left(x\right)g\left(x\right)+f\left(y\right)g\left(y\right)-f\left(x\right)g\left(y\right)-f\left(y\right)g\left(x\right),x,y\in \left(0,t\right),t>0\end{array}$ (3.3)

对上式两边同乘以

$\frac{h^{\prime }\left(x\right)h^{\prime }\left(y\right)}{2{\Gamma }^2\left(\alpha \right){\left(h\left(t\right)-h\left(x\right)\right)}^{1-\alpha }{\left(h\left(t\right)-h\left(y\right)\right)}^{1-\alpha }}$

并分别对x和y在 $\left(0,t\right)$ 上积分，有

$\frac{1}{2{\Gamma }^2\left(\alpha \right)}{\displaystyle {\int }_0^t{\displaystyle {\int }_0^t\frac{h^{\prime }\left(x\right)h^{\prime }\left(y\right)}{{\left(h\left(t\right)-h\left(x\right)\right)}^{1-\alpha }{\left(h\left(t\right)-h\left(y\right)\right)}^{1-\alpha }}T\left(x,y\right)\text{d}x\text{d}y}}=\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }}{\Gamma \left(\alpha +1\right)}{J}_h^{\alpha }f\left(t\right)g\left(t\right)-J_h^{\alpha }f\left(t\right)J_h^{\alpha }g\left(t\right)$ (3.4)

现在将(3.3)式中的 $T\left(x,y\right)$ 代入(3.4)的左边，从而由Cauchy-Schwarz不等式，有

$\begin{array}{l}{\left(\frac{1}{2{\Gamma }^2\left(\alpha \right)}{\displaystyle {\int }_0^t{\displaystyle {\int }_0^t\frac{h^{\prime }\left(x\right)h^{\prime }\left(y\right)}{{\left(h\left(t\right)-h\left(x\right)\right)}^{1-\alpha }{\left(h\left(t\right)-h\left(y\right)\right)}^{1-\alpha }}T\left(x,y\right)\text{d}x\text{d}y}}\right)}^2\\ \le \frac{1}{2{\Gamma }^2\left(\alpha \right)}{\displaystyle {\int }_0^t{\displaystyle {\int }_0^t\frac{h^{\prime }\left(x\right)h^{\prime }\left(y\right)}{{\left(h\left(t\right)-h\left(x\right)\right)}^{1-\alpha }{\left(h\left(t\right)-h\left(y\right)\right)}^{1-\alpha }}{\left(f\left(x\right)-f\left(y\right)\right)}^2\text{d}x\text{d}y}}\\ \times \frac{1}{2{\Gamma }^2\left(\alpha \right)}{\displaystyle {\int }_0^t{\displaystyle {\int }_0^t\frac{h^{\prime }\left(x\right)h^{\prime }\left(y\right)}{{\left(h\left(t\right)-h\left(x\right)\right)}^{1-\alpha }{\left(h\left(t\right)-h\left(y\right)\right)}^{1-\alpha }}{\left(g\left(x\right)-g\left(y\right)\right)}^2\text{d}x\text{d}y}}\\ =\left(\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }}{\Gamma \left(\alpha +1\right)}{J}_h^{\alpha }{f}^2\left(t\right)-{\left(J_h^{\alpha }f\left(t\right)\right)}^2\right)\left(\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }}{\Gamma \left(\alpha +1\right)}{J}_h^{\alpha }{g}^2\left(t\right)-{\left(J_h^{\alpha }g\left(t\right)\right)}^2\right)\end{array}$ (3.5)

再将(3.4)式代入(3.5)有

$\begin{array}{l}{\left(\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }}{\Gamma \left(\alpha +1\right)}{J}_h^{\alpha }f\left(t\right)g\left(t\right)-J_h^{\alpha }f\left(t\right)J_h^{\alpha }g\left(t\right)\right)}^2\\ \le \left(\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }}{\Gamma \left(\alpha +1\right)}{J}_h^{\alpha }{f}^2\left(t\right)-{\left(J_h^{\alpha }f\left(t\right)\right)}^2\right)\times \left(\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }}{\Gamma \left(\alpha +1\right)}{J}_h^{\alpha }{g}^2\left(t\right)-{\left(J_h^{\alpha }g\left(t\right)\right)}^2\right)\end{array}$ (3.6)

因为 $f\left(t\right)$ 和 $g\left(t\right)$ 是两个在 $\left[0,\infty \right)$ 上满足条件(C₁)和(C₂)的可积函数，

所以有

$\left({\psi }_2\left(t\right)-g\left(t\right)\right)\left(g\left(t\right)-{\psi }_1\left(t\right)\right)\ge 0,t\ge 0$

和

$\left({\phi }_2\left(t\right)-f\left(t\right)\right)\left(f\left(t\right)-{\phi }_1\left(t\right)\right)\ge 0,t\ge 0$

从而

$\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }}{\Gamma \left(\alpha +1\right)}{J}_h^{\alpha }\left({\psi }_2\left(t\right)-g\left(t\right)\right)\left(g\left(t\right)-{\psi }_1\left(t\right)\right)\ge 0$

且

$\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }}{\Gamma \left(\alpha +1\right)}{J}_h^{\alpha }\left({\phi }_2\left(t\right)-f\left(t\right)\right)\left(f\left(t\right)-{\phi }_1\left(t\right)\right)\ge 0$

因此由引理2.3，有

$\begin{array}{l}\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }}{\Gamma \left(\alpha +1\right)}{J}_h^{\alpha }{g}^2\left(t\right)-{\left(J_h^{\alpha }g\left(t\right)\right)}^2\le \left(J_h^{\alpha }{\psi }_2\left(t\right)-J_h^{\alpha }f\left(t\right)\right)\left(J_h^{\alpha }f\left(t\right)-J_h^{\alpha }{\psi }_1\left(t\right)\right)\\ +\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }}{\Gamma \left(\alpha +1\right)}{J}_h^{\alpha }\left({\psi }_1\left(t\right)f\left(t\right)\right)-J_h^{\alpha }{\psi }_1\left(t\right)J_h^{\alpha }f\left(t\right)\\ +\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }}{\Gamma \left(\alpha +1\right)}{J}_h^{\alpha }\left({\psi }_2\left(t\right)f\left(t\right)\right)-J_h^{\alpha }{\psi }_2\left(t\right)J_h^{\alpha }f\left(t\right)\\ -\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }}{\Gamma \left(\alpha +1\right)}{J}_h^{\alpha }\left({\psi }_1\left(t\right){\psi }_2\left(t\right)\right)+J_h^{\alpha }{\psi }_1\left(t\right)J_h^{\alpha }{\psi }_2\left(t\right)\\ =S^{*}\left(g,{\psi }_1,{\psi }_2\right)\end{array}$ (3.7)

和

$\begin{array}{l}\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }}{\Gamma \left(\alpha +1\right)}{J}_h^{\alpha }{f}^2\left(t\right)-{\left(J_h^{\alpha }f\left(t\right)\right)}^2=\left(J_h^{\alpha }{\phi }_2\left(t\right)-J_h^{\alpha }f\left(t\right)\right)\left(J_h^{\alpha }f\left(t\right)-J_h^{\alpha }{\phi }_1\left(t\right)\right)\\ +\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }}{\Gamma \left(\alpha +1\right)}{J}_h^{\alpha }\left({\phi }_1\left(t\right)f\left(t\right)\right)-J_h^{\alpha }{\phi }_1\left(t\right)J_h^{\alpha }f\left(t\right)\\ +\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }}{\Gamma \left(\alpha +1\right)}{J}_h^{\alpha }\left({\phi }_2\left(t\right)f\left(t\right)\right)-J_h^{\alpha }{\phi }_2\left(t\right)J_h^{\alpha }f\left(t\right)\\ -\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }}{\Gamma \left(\alpha +1\right)}{J}_h^{\alpha }\left({\phi }_1\left(t\right){\phi }_2\left(t\right)\right)+J_h^{\alpha }{\phi }_1\left(t\right)J_h^{\alpha }{\phi }_2\left(t\right)\\ =S^{*}\left(f,{\phi }_1,{\phi }_2\right)\end{array}$ (3.8)

由(3.6)，(3.7)和(3.8)，(3.1)得证。

由下面给出的推论得到命题1.3和命题1.4是本文的一个特例。

推论3.2：若 $S^{*}\left(f,{\phi }_1,{\phi }_2\right)=S^{*}\left(f,m,M\right)$ 且 $S^{*}\left(g,{\psi }_1,{\psi }_2\right)=S^{*}\left(g,n,N\right)$，那么有

$\left|\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }}{\Gamma \left(\alpha +1\right)}{J}_h^{\alpha }f\left(t\right)g\left(t\right)-J_h^{\alpha }f\left(t\right)J_h^{\alpha }g\left(t\right)\right|\le {\left(\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }}{2\Gamma \left(\alpha +1\right)}\right)}^2\left(M-m\right)\left(N-n\right).$ (3.9)

这里的推论3.2就是命题1.3。

推论3.3：若 $h\left(t\right)=t$ (即 $J_h^{\alpha }=J^{\alpha }$ )，那么有

$\left|\frac{t^{\alpha }}{\Gamma \left(\alpha +1\right)}{J}^{\alpha }f\left(t\right)g\left(t\right)-J^{\alpha }f\left(t\right)J^{\alpha }g\left(t\right)\right|\le \sqrt{S\left(f,{\phi }_1,{\phi }_2\right)S\left(g,{\psi }_1,{\psi }_2\right)}.$ (3.10)

这里的推论3.3就是命题1.4。

对于一个无界函数来说，无法用常数来限制其边界但是可以用两个函数来限制其边界。因而积分不等式(1.2)和(1.3)对于无界函数不成立，但在一定的条件下积分不等式(3.1)对于无界函数是成立的。

例3.3：设 $h\left(t\right):\left[0,\infty \right)\to \left(0,\infty \right)$ 是 $\left[0,\infty \right)$ 上的单调增函数并在 $\left(0,\infty \right)$ 上有连续导函数 $h^{\prime }\left(t\right)$。若 $f\left(t\right)$ 和 $g\left(t\right)$ 是 $\left[0,\infty \right)$ 上两个可积函数并满足 $h\left(t\right)-1\le f\left(t\right)\le h\left(t\right)$ 和 $h\left(t\right)\le g\left(t\right)\le h\left(t\right)+1$。那么当 $t>0,\alpha >0$ 时，有

$\left|\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }}{\Gamma \left(\alpha +1\right)}{J}_h^{\alpha }f\left(t\right)g\left(t\right)-J_h^{\alpha }f\left(t\right)J_h^{\alpha }g\left(t\right)\right|\le \sqrt{S^{*}\left(f,h\left(t\right)-1,h\left(t\right)\right)S^{*}\left(g,h\left(t\right),h\left(t\right)+1\right)},$

这里

$\begin{array}{l}{S}^{*}\left(f,h-1,h\right)\\ =\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }}{\Gamma \left(\alpha +1\right)}{J}_h^{\alpha }\left(h\left(t\right)-1\right)f\left(t\right)-\left(\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }\left(h\left(t\right)+\alpha h\left(0\right)\right)}{\Gamma \left(\alpha +2\right)}-\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }}{\Gamma \left(\alpha +1\right)}\right)J_h^{\alpha }f\left(t\right)\\ -\left(\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }{\left(h\left(t\right)+\alpha h\left(0\right)\right)}^2+h^2\left(t\right)+\alpha h^2\left(0\right)}{\Gamma \left(\alpha +3\right)}-\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }\left(h\left(t\right)+\alpha h\left(0\right)\right)}{\Gamma \left(\alpha +2\right)}\right)\\ \times \frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }}{\Gamma \left(\alpha +1\right)}+\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }}{\Gamma \left(\alpha +1\right)}{J}_h^{\alpha }h\left(t\right)f\left(t\right)-\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }\left(h\left(t\right)+\alpha h\left(0\right)\right)}{\Gamma \left(\alpha +2\right)}{J}_h^{\alpha }f(t)\end{array}$

$\begin{array}{l}+\left(\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }\left(h\left(t\right)+\alpha h\left(0\right)\right)}{\Gamma \left(\alpha +2\right)}-\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }}{\Gamma \left(\alpha +1\right)}\right)\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }\left(h\left(t\right)+\alpha h\left(0\right)\right)}{\Gamma \left(\alpha +2\right)}\\ +\left(J_h^{\alpha }f\left(t\right)-\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }\left(h\left(t\right)+\alpha h\left(0\right)\right)}{\Gamma \left(\alpha +2\right)}+\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }}{\Gamma \left(\alpha +1\right)}\right)\\ \times \left(\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }\left(h\left(t\right)+\alpha h\left(0\right)\right)}{\Gamma \left(\alpha +2\right)}-J_h^{\alpha }f(t)\right)\end{array}$

和

$\begin{array}{l}{S}^{*}\left(g,h,h+1\right)\\ =\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }}{\Gamma \left(\alpha +1\right)}{J}_h^{\alpha }\left(h\left(t\right)+1\right)g\left(t\right)-\left(\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }\left(h\left(t\right)+\alpha h\left(0\right)\right)}{\Gamma \left(\alpha +2\right)}-\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }}{\Gamma \left(\alpha +1\right)}\right)J_h^{\alpha }g\left(t\right)\\ -\left(\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }{\left(h\left(t\right)+\alpha h\left(0\right)\right)}^2+h^2\left(t\right)+\alpha h^2\left(0\right)}{\Gamma \left(\alpha +3\right)}+\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }\left(h\left(t\right)+\alpha h\left(0\right)\right)}{\Gamma \left(\alpha +2\right)}\right)\\ \times \frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }}{\Gamma \left(\alpha +1\right)}+\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }}{\Gamma \left(\alpha +1\right)}{J}_h^{\alpha }h\left(t\right)g\left(t\right)-\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }\left(h\left(t\right)+\alpha h\left(0\right)\right)}{\Gamma \left(\alpha +2\right)}{J}_h^{\alpha }g(t)\end{array}$

$\begin{array}{l}+\left(\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }\left(h\left(t\right)+\alpha h\left(0\right)\right)}{\Gamma \left(\alpha +2\right)}+\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }}{\Gamma \left(\alpha +1\right)}\right)\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }\left(h\left(t\right)+\alpha h\left(0\right)\right)}{\Gamma \left(\alpha +2\right)}\\ +\left(\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }\left(h\left(t\right)+\alpha h\left(0\right)\right)}{\Gamma \left(\alpha +2\right)}+\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }}{\Gamma \left(\alpha +1\right)}-J_h^{\alpha }g\left(t\right)\right)\\ \times \left(J_h^{\alpha }g\left(t\right)-\frac{{\left(h\left(t\right)-h\left(0\right)\right)}^{\alpha }\left(h\left(t\right)+\alpha h\left(0\right)\right)}{\Gamma \left(\alpha +2\right)}\right)\end{array}$

4. 结论

本文主要是命题1.3和命题1.4的推广。从积分形式上看，本文是将命题1.4中的Riemann-Liouville分数阶积分 $J^{\alpha }$ 推广为Riemann-Liouville型分数阶积分 $J_h^{\alpha }$。而从被积函数的限制条件上看，本文是将命题1.3中被积函数的常数边界推广到函数边界的情形。