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 浙江大学学报(理学版)  2018, Vol. 45 Issue (5): 549-554, 561  DOI:10.3785/j.issn.1008-9497.2018.05.006 0

### 引用本文 [复制中英文]

[复制中文]
SHI Tongye, ZENG Zhihong, CAO Junfei. Hermite-Hadamard type inequalities involving Riemann-Liouville fractional integrals for η-convex functions[J]. Journal of Zhejiang University(Science Edition), 2018, 45(5): 549-554, 561. DOI: 10.3785/j.issn.1008-9497.2018.05.006.
[复制英文]

### 文章历史

1. 海军指挥学院, 江苏 南京 211800;
2. 广东第二师范学院 学报编辑部, 广东 广州 510303;
3. 广东第二师范学院 数学系, 广东 广州 510303

Hermite-Hadamard type inequalities involving Riemann-Liouville fractional integrals for η-convex functions
SHI Tongye1, ZENG Zhihong2, CAO Junfei3
1. PLA Naval Command College, Nanjing 211800, China;
2. Editorial Department of Journal, Guangdong University of Education, Guangzhou 510303, China;
3. Department of Mathematics, Guangdong University of Education, Guangzhou 510303, China
Abstract: Two existing Hermite-Hadamard type inequalities involving fractional integrals for η-convex functions are improved. By using the fractional integral identities embedding the first order derivative function, new Hermite-Hadamard type inequalities involving fractional integrals are obtained provided that the absolute value of the first derivative function is η-convex function.
Key words: η-convex function    Hermite-Hadamard type inequality    fractional integral
0 引言

f是区间[ab]上的凸函数，则对f在区间[ab]上的算术平均值有以下估计：

 $f\left( {\frac{{a + b}}{2}} \right) \le \frac{1}{{b - a}}\int_a^b {f\left( x \right){\rm{d}}x} \le \frac{{f\left( a \right) + f\left( b \right)}}{2},$ (1)

 $f\left( {tx + \left( {1 - t} \right)y} \right) \le f\left( y \right) + t\eta \left( {f\left( x \right),f\left( y \right)} \right),$

$\eta \left( {x, y} \right) = x - y$时，η凸函数即为通常的凸函数.

 $\begin{array}{l} f\left( {\frac{{a + b}}{2}} \right) - \frac{1}{2}{M_\eta } \le \frac{1}{{b - a}}\int_a^b {f\left( x \right){\rm{d}}x} \le \frac{{f\left( a \right) + f\left( b \right)}}{2} + \\ \;\;\;\;\;\;\;\;\;\frac{1}{4}\left[ {\eta \left( {f\left( a \right),f\left( b \right)} \right) + \eta \left( {f\left( b \right),f\left( a \right)} \right)} \right] \le \\ \;\;\;\;\;\;\;\;\;\frac{{f\left( a \right) + f\left( b \right)}}{2} + \frac{1}{2}{M_\eta }. \end{array}$

 $\begin{array}{l} f\left( {\frac{{a + b}}{2}} \right)\int_a^b {g\left( x \right){\rm{d}}x} - \\ \;\;\;\;\;\;\;\;\frac{1}{2}\int_a^b {\eta \left( {f\left( {a + b - x} \right),f\left( x \right)} \right)g\left( x \right){\rm{d}}x} \le \\ \;\;\;\;\;\;\;\;\int_a^b {f\left( x \right)g\left( x \right){\rm{d}}x} \le \frac{{f\left( a \right) + f\left( b \right)}}{2}\int_a^b {g\left( x \right){\rm{d}}x} + \\ \;\;\;\;\;\;\;\;\frac{{\eta \left( {f\left( a \right),f\left( b \right)} \right) + \eta \left( {f\left( b \right),f\left( a \right)} \right)}}{{2\left( {b - a} \right)}}\int_a^b {\left( {b - x} \right)g\left( x \right){\rm{d}}x} . \end{array}$

 $\begin{array}{*{20}{c}} {J_a^{{\alpha _ + }}f\left( x \right) = \frac{1}{{\Gamma \left( \alpha \right)}}\int_a^x {{{\left( {x - t} \right)}^{\alpha - 1}}f\left( t \right){\rm{d}}t} ,}&{x > a,} \end{array}$
 $\begin{array}{*{20}{c}} {J_b^{{\alpha _ - }}f\left( x \right) = \frac{1}{{\Gamma \left( \alpha \right)}}\int_x^b {{{\left( {x - t} \right)}^{\alpha - 1}}f\left( t \right){\rm{d}}t} ,}&{x > b,} \end{array}$

 $\Gamma \left( \alpha \right) = \int_0^{ + \infty } {{{\rm{e}}^{ - t}}{t^{\alpha - 1}}{\rm{d}}t} .$

 $\begin{array}{*{20}{c}} {p \in \left( {0,1} \right),}&{\xi = pa + \left( {1 - p} \right)b,} \end{array}$

 ${K_1} = \frac{{\Gamma \left( {\alpha + 1} \right)}}{{2{{\left( {b - a} \right)}^\alpha }}}\left[ {J_a^{{\alpha _ + }}\left( b \right) + J_b^{{\alpha _ - }}\left( a \right)} \right],$
 $\begin{array}{l} {K_2} = \frac{{p\Gamma \left( {\alpha + 1} \right)}}{{{{\left( {1 - p} \right)}^\alpha }{{\left( {b - a} \right)}^\alpha }}}J_a^{{\alpha _ + }}f\left( \xi \right) + \\ \;\;\;\;\;\;\;\;\frac{{\left( {1 - p} \right)\Gamma \left( {\alpha + 1} \right)}}{{{p^\alpha }{{\left( {b - a} \right)}^\alpha }}}J_a^{{\alpha _ - }}f\left( \xi \right). \end{array}$

 $\begin{array}{l} f\left( {\frac{{a + b}}{2}} \right) - {M_\eta } \le {K_1} \le \frac{{f\left( a \right) + f\left( b \right)}}{2} + \\ \;\;\;\;\;\;\;\frac{\alpha }{{2\left( {\alpha + 1} \right)}}\left[ {\eta \left( {f\left( a \right),f\left( b \right)} \right) + \eta \left( {f\left( b \right),f\left( a \right)} \right)} \right] \le \\ \;\;\;\;\;\;\;\frac{{f\left( a \right) + f\left( b \right)}}{2} + \frac{\alpha }{{\alpha + 1}}{M_\eta }. \end{array}$ (2)

 $\begin{array}{l} \frac{{f\left( a \right) + f\left( b \right)}}{2} - {K_1} = \\ \;\;\;\;\;\;\frac{{b - a}}{2}\int_0^1 {\left[ {{{\left( {1 - t} \right)}^\alpha } - {t^\alpha }} \right]f'\left( {ta + \left( {1 - t} \right)b} \right){\rm{d}}t} . \end{array}$

 $\begin{array}{l} \left| {\frac{{f\left( a \right) + f\left( b \right)}}{2} - {K_1}} \right| \le \\ \;\;\;\;\;\;\frac{{b - \alpha }}{{2\left( {\alpha + 1} \right)}}\left( {1 - \frac{1}{{{2^\alpha }}}} \right)\left( {2\left| {f'\left( b \right)} \right| + \eta \left( {\left| {f'\left( a \right)} \right|,\left| {f'\left( b \right)} \right|} \right)} \right). \end{array}$

 $f\left( {\frac{{a + b}}{2}} \right) - {K_1} = \frac{{b - a}}{2}\sum\limits_{k = 1}^4 {{I_k}} ,$

 ${I_1} = \int_0^{\frac{1}{2}} {{t^\alpha }f'\left( {tb + \left( {1 - t} \right)a} \right){\rm{d}}t} ,$
 ${I_2} = \int_0^{\frac{1}{2}} {\left( { - {t^\alpha }} \right)f'\left( {ta + \left( {1 - t} \right)b} \right){\rm{d}}t} ,$
 ${I_3} = \int_{\frac{1}{2}}^1 {\left( {{t^\alpha } - 1} \right)f'\left( {tb + \left( {1 - t} \right)a} \right){\rm{d}}t} ,$
 ${I_4} = \int_{\frac{1}{2}}^1 {\left( {1 - {t^\alpha }} \right)f'\left( {ta + \left( {1 - t} \right)b} \right){\rm{d}}t} .$

 $\begin{array}{l} \left| {f\left( {\frac{{a + b}}{2}} \right) - {K_1}} \right| \le \frac{{b - a}}{{{2^{\alpha + 1}}\left( {\alpha + 1} \right)}}\left[ {\left| {f'\left( a \right)} \right| + \left| {f'\left( b \right)} \right| + } \right.\\ \left. {\eta \left( {\left| {f'\left( a \right)} \right|,\left| {f'\left( b \right)} \right|} \right) + \eta \left( {\left| {f'\left( b \right)} \right|,\left| {f'\left( a \right)} \right|} \right)} \right]. \end{array}$ (3)

 ${I_3} = \int_0^{\frac{1}{2}} {\left[ {{{\left( {1 - t} \right)}^\alpha } - 1} \right]f'\left( {ta + \left( {1 - t} \right)b} \right){\rm{d}}t} ,$
 ${I_4} = \int_0^{\frac{1}{2}} {\left[ {1 - {{\left( {1 - t} \right)}^\alpha }} \right]f'\left( {\left( {1 - t} \right)a + tb} \right){\rm{d}}t} ,$

 $\begin{array}{l} f\left( {\frac{{a + b}}{2}} \right) - {K_1} = \frac{{b - a}}{2}\int_0^{\frac{1}{2}} {\left[ {{t^\alpha } - {{\left( {1 - t} \right)}^\alpha } + 1} \right] \times } \\ \;\;\;\;\;\;\;\left[ {f'\left( {\left( {1 - t} \right)a + tb} \right) - f'\left( {ta + \left( {1 - t} \right)b} \right)} \right]{\rm{d}}t. \end{array}$

 $\begin{array}{l} {K_2} - f\xi = \frac{{\left( {1 - p} \right)\left( {b - a} \right)}}{{{p^\alpha }}} \times \\ \int_0^p {\left[ {{p^\alpha } - {{\left( {p - t} \right)}^\alpha }} \right]f'\left( {ta + \left( {1 - t} \right)b} \right){\rm{d}}t} + \frac{{p\left( {b - a} \right)}}{{{{\left( {1 - p} \right)}^\alpha }}} \times \\ \int_p^1 {\left[ {{{\left( {t - p} \right)}^\alpha }{{\left( {1 - p} \right)}^\alpha }} \right]f'\left( {ta + \left( {1 - t} \right)b} \right){\rm{d}}t} . \end{array}$

 $\begin{array}{l} pf\left( a \right) + \left( {1 - p} \right)f\left( b \right) - {K_2} = \\ \frac{{\left( {1 - p} \right)\left( {b - a} \right)}}{{{p^\alpha }}}\int_0^p {{{\left( {p - t} \right)}^\alpha }f'\left( {ta + \left( {1 - t} \right)b} \right){\rm{d}}t} - \\ \frac{{p\left( {b - a} \right)}}{{{{\left( {1 - p} \right)}^\alpha }}}\int_p^1 {{{\left( {t - p} \right)}^\alpha }f'\left( {ta + \left( {1 - t} \right)b} \right){\rm{d}}t} . \end{array}$
1 主要结果及证明

 $\begin{array}{l} f\left( {\frac{{a + b}}{2}} \right) - \frac{{{M_\eta }}}{2} \le f\left( {\frac{{a + b}}{2}} \right) - \\ \frac{\alpha }{{4{{\left( {b - a} \right)}^\alpha }}}\int_a^b {\left[ {{{\left( {x - a} \right)}^{\alpha - 1}} + {{\left( {b - x} \right)}^{\alpha - 1}}} \right] \times } \\ \eta \left( {f\left( {b + a - x} \right),f\left( x \right)} \right){\rm{d}}x \le {K_1} \le \frac{{f\left( a \right) + f\left( b \right)}}{2} + \\ \frac{1}{{2\left( {\alpha + 1} \right)}}\left( {1 - \frac{1}{{{2^\alpha }}}} \right)\left[ {\eta \left( {f\left( a \right),f\left( b \right)} \right) + \eta \left( {f\left( b \right),f\left( a \right)} \right)} \right] \le \\ \frac{{f\left( a \right) + f\left( b \right)}}{2} + \frac{1}{{\alpha + 1}}\left( {1 - \frac{1}{{{2^\alpha }}}} \right){M_\eta } \le \\ \frac{{f\left( a \right) + f\left( b \right)}}{2} + \frac{{{M_\eta }}}{4}. \end{array}$ (4)

 $f\left( x \right) \le f\left( a \right) + \frac{{x - a}}{{b - a}}\eta \left( {f\left( b \right),f\left( a \right)} \right),$ (5)
 $f\left( x \right) \le f\left( b \right) + \frac{{b - x}}{{b - a}}\eta \left( {f\left( a \right),f\left( b \right)} \right),$ (6)

 $f\left( \xi \right) \le f\left( x \right) + \left( {1 - p} \right)\eta \left( {f\left( y \right),f\left( x \right)} \right),$ (13)

 $\begin{array}{l} \frac{{{{\left( {1 - p} \right)}^\alpha }{{\left( {b - a} \right)}^\alpha }}}{\alpha }f\left( \xi \right) \le \\ \;\;\;\;\;\;{\rm{\Gamma }}\left( \alpha \right)J_a^{{\alpha _ + }}f\left( \xi \right) + {\left( {1 - p} \right)^{\alpha + 1}}{\rm{\Gamma }}\left( \alpha \right)J_a^{{\alpha _ + }}{f_1}\left( b \right), \end{array}$ (14)

 $f\left( \xi \right) \le f\left( x \right) + p\eta \left( {f'\left( y \right),f\left( x \right)} \right),$ (15)

 $\begin{array}{l} \frac{{{p^\alpha }{{\left( {b - a} \right)}^\alpha }}}{\alpha }f\left( \xi \right) \le \\ \;\;\;\;\;\;\;{\rm{\Gamma }}\left( \alpha \right)J_b^{{\alpha _ - }}f\left( \xi \right) + {p^{\alpha + 1}}{\rm{\Gamma }}\left( \alpha \right)J_b^{{\alpha _ - }}{f_2}\left( a \right), \end{array}$ (16)

 $\begin{array}{l} \int_0^1 {\left| {{{\left( {1 - t} \right)}^\alpha } - {t^\alpha }} \right|\left| {f'\left( {ta + \left( {1 - t} \right)b} \right)} \right|{\rm{d}}t} \le \\ \left| {f'\left( a \right)} \right|\int_0^{\frac{1}{2}} {\left| {{{\left( {1 - t} \right)}^\alpha } - {t^\alpha }} \right|{\rm{d}}t} + \\ \left| {f'\left( b \right)} \right|\int_{\frac{1}{2}}^1 {\left| {{{\left( {1 - t} \right)}^\alpha } - {t^\alpha }} \right|{\rm{d}}t} + \\ \eta \left( {\left| {f'\left( b \right)} \right|,\left| {f'\left( a \right)} \right|} \right)\int_0^{\frac{1}{2}} {\left( {1 - t} \right)\left| {{{\left( {1 - t} \right)}^\alpha } - {t^\alpha }} \right|{\rm{d}}t} + \\ \eta \left( {\left| {f'\left( a \right)} \right|,\left| {f'\left( b \right)} \right|} \right)\int_{\frac{1}{2}}^1 {t\left| {{{\left( {1 - t} \right)}^\alpha } - {t^\alpha }} \right|{\rm{d}}t} = \\ \frac{1}{{\alpha + 1}}\left\{ {\left( {1 - \frac{1}{{{2^\alpha }}}} \right)\left[ {\left| {f'\left( a \right)} \right| + \left| {f'\left( b \right)} \right|} \right] + } \right.\\ \frac{1}{{\alpha + 2}}\left( {1 - \frac{{\alpha + 2}}{{{2^{\alpha + 1}}}}} \right)\left[ {\eta \left( {\left| {f'\left( a \right)} \right|,\left| {f'\left( b \right)} \right|} \right) + } \right.\\ \left. {\left. {\eta \left( {\left| {f'\left( b \right)} \right|,\left| {f'\left( a \right)} \right|} \right)} \right]} \right\}, \end{array}$ (21)

 $\begin{array}{l} \left| {\frac{{f\left( a \right) + f\left( b \right)}}{2} - {K_1}} \right| \le \frac{{b - a}}{2}\left[ {\frac{1}{2} - \frac{1}{{\alpha + 1}}\left( {1 - \frac{1}{{{2^\alpha }}}} \right)} \right] \times \\ \left[ {\left| {f'\left( a \right)} \right| + \left| {f'\left( b \right)} \right| + \frac{1}{2}\left( {\eta \left( {\left| {f'\left( a \right)} \right|,} \right.} \right.} \right.\\ \left. {\left. {\left. {\left| {f'\left( b \right)} \right|} \right) + \eta \left( {\left| {f'\left( b \right)} \right|,\left| {f'\left( a \right)} \right|} \right)} \right)} \right]. \end{array}$ (22)

 $\begin{array}{l} \left| {f\left( {\frac{{a + b}}{2}} \right) - {K_1}} \right| \le \frac{{b - a}}{2}\int_0^{\frac{1}{2}} {\left[ {{t^\alpha } - {{\left( {1 - t} \right)}^\alpha } + 1} \right] \times } \\ \;\;\;\;\;\left[ {\left| {f'\left( {\left( {1 - t} \right)a + tb} \right)} \right| + \left| {f'\left( {ta + \left( {1 - t} \right)b} \right)} \right|} \right]{\rm{d}}t. \end{array}$ (23)

 $\begin{array}{l} \left| {f'\left( {\left( {1 - t} \right)a + tb} \right)} \right| \le \\ \;\;\;\;\;\;\left| {f'\left( a \right)} \right| + t\eta \left( {\left| {f'\left( b \right)} \right|,\left| {f'\left( a \right)} \right|} \right), \end{array}$
 $\begin{array}{l} \left| {f'\left( {\left( {1 - t} \right)a + tb} \right)} \right| \le \\ \;\;\;\;\;\;\left| {f'\left( b \right)} \right| + \left( {1 - t} \right)\eta \left( {\left| {f'\left( a \right)} \right|,\left| {f'\left( b \right)} \right|} \right), \end{array}$
 $\begin{array}{l} \left| {f'\left( {ta + \left( {1 - t} \right)b} \right)} \right| \le \\ \;\;\;\;\;\;\left| {f'\left( a \right)} \right| + \left( {1 - t} \right)\eta \left( {\left| {f'\left( b \right)} \right|,\left| {f'\left( a \right)} \right|} \right), \end{array}$
 $\left| {f'\left( {ta + \left( {1 - t} \right)b} \right)} \right| \le \left| {f'\left( b \right)} \right| + t\eta \left( {\left| {f'\left( a \right)} \right|,\left| {f'\left( b \right)} \right|} \right),$

 $\begin{array}{l} \left| {f'\left( {\left( {1 - t} \right)a + tb} \right)} \right| + \left| {f'\left( {ta + \left( {1 - t} \right)b} \right)} \right| \le \\ \left| {f'\left( a \right)} \right| + \left| {f'\left( b \right)} \right| + \frac{1}{2}\left( {\eta \left( {\left| {f'\left( a \right)} \right|,\left| {f'\left( b \right)} \right|} \right) + } \right.\\ \left. {\eta \left( {\left| {f'\left( b \right)} \right|,\left| {f'\left( a \right)} \right|} \right)} \right). \end{array}$ (24)

 $\begin{array}{l} \left| {{K_2} - f\left( \xi \right)} \right| \le \frac{{\alpha p\left( {1 - p} \right)\left( {b - a} \right)}}{{\alpha + 1}}\left\{ {\left| {f'\left( a \right)} \right| + \left| {f'\left( b \right)} \right| + } \right.\\ \;\;\;\;\;\frac{{\alpha + 3}}{{2\left( {\alpha + 2} \right)}}\left[ {p\eta \left( {\left| {f'\left( a \right)} \right|,\left| {f'\left( b \right)} \right|} \right) + } \right.\\ \;\;\;\;\;\left. {\left. {\left( {1 - p} \right)\eta \left( {\left| {f'\left( b \right)} \right|,\left| {f'\left( a \right)} \right|} \right)} \right]} \right\}. \end{array}$ (25)

 $\begin{array}{l} \left| {{K_2} - f\left( \xi \right)} \right| \le \frac{{\left( {1 - p} \right)\left( {b - a} \right)}}{{{p^\alpha }}} \times \\ \int_0^p {\left[ {{p^\alpha } - {{\left( {p - t} \right)}^\alpha }} \right]\left| {f'\left( {ta + \left( {1 - t} \right)b} \right)} \right|{\rm{d}}t} + \\ \frac{{p\left( {b - a} \right)}}{{{{\left( {1 - p} \right)}^\alpha }}}\int_0^p {\left[ {\left( {1 - {p^\alpha }} \right) - {{\left( {t - p} \right)}^\alpha }} \right]\left| {f'\left( {ta + \left( {1 - t} \right)b} \right)} \right|{\rm{d}}t} . \end{array}$ (26)

 $\begin{array}{l} \int_0^p {\left[ {{p^\alpha } - {{\left( {p - t} \right)}^\alpha }} \right]\left| {f'\left( {ta + \left( {1 - t} \right)b} \right)} \right|{\rm{d}}t} \le \\ \int_0^p {\left[ {{p^\alpha } - {{\left( {p - t} \right)}^\alpha }} \right]\left[ {\left| {f'\left( b \right)} \right| + t\eta \left( {\left| {f'\left( a \right)} \right|,\left| {f'\left( b \right)} \right|} \right)} \right]{\rm{d}}t} = \\ \frac{\alpha }{{\alpha + 1}}{p^{\alpha + 1}}\left| {f'\left( b \right)} \right| + \\ \frac{{{p^{\alpha + 2}}\alpha \left( {\alpha + 3} \right)}}{{2\left( {\alpha + 1} \right)\left( {\alpha + 2} \right)}}\eta \left( {\left| {f'\left( a \right)} \right|,\left| {f'\left( b \right)} \right|} \right), \end{array}$ (27)
 $\begin{array}{l} \int_p^1 {\left[ {\left( {1 - {p^\alpha }} \right) - {{\left( {t - p} \right)}^\alpha }} \right]\left| {f'\left( {ta + \left( {1 - t} \right)b} \right)} \right|{\rm{d}}t} \le \\ \int_p^1 {\left[ {\left( {1 - {p^\alpha }} \right) - {{\left( {t - p} \right)}^\alpha }} \right]\left[ {\left| {f'\left( a \right)} \right| + \left( {1 - t} \right) \times } \right.} \\ \left. {\eta \left( {\left| {f'\left( b \right)} \right|,\left| {f'\left( a \right)} \right|} \right)} \right]{\rm{d}}t = \\ \frac{\alpha }{{\alpha + 1}}{\left( {1 - p} \right)^{a + 1}}\left| {f'\left( a \right)} \right| + \\ \frac{{{{\left( {1 - p} \right)}^{a + 2}}\alpha \left( {\alpha + 3} \right)}}{{2\left( {\alpha + 1} \right)\left( {\alpha + 2} \right)}}\eta \left( {\left| {f'\left( b \right)} \right|,\left| {f'\left( a \right)} \right|} \right). \end{array}$ (28)

 $\begin{array}{l} \left| {{K_2} - f\left( \xi \right)} \right| \le \frac{{\alpha p\left( {1 - p} \right)\left( {b - a} \right)}}{{2\left( {\alpha + 1} \right)\left( {\alpha + 2} \right)}} \times \\ \;\;\;\;\left\{ {\left[ {\alpha + 1 + 2\left( {\alpha + 3} \right)p} \right]\left| {f'\left( a \right)} \right| + } \right.\\ \;\;\;\;\left. {\left[ {3\alpha + 7 - 2\left( {\alpha + 3} \right)p} \right]\left| {f'\left( b \right)} \right|} \right\}. \end{array}$

 $\begin{array}{l} \left| {pf\left( a \right) + \left( {1 - p} \right)f\left( b \right) - {K_2}} \right| \le \frac{{p\left( {1 - p} \right)\left( {b - a} \right)}}{{\alpha + 1}} \times \\ \left\{ {\left| {f'\left( a \right)} \right| + \left| {f'\left( b \right)} \right| + \frac{1}{{\alpha + 2}}\left[ {p\eta \left( {\left| {f'\left( a \right)} \right|,} \right.} \right.} \right.\\ \left. {\left. {\left. {\left| {f'\left( b \right)} \right|} \right) + \left( {1 - p} \right)\eta \left( {\left| {f'\left( b \right)} \right|,\left| {f'\left( a \right)} \right|} \right)} \right]} \right\}. \end{array}$

 $\begin{array}{l} \left| {\frac{{f\left( a \right) + f\left( b \right)}}{2} - \frac{\alpha }{2}\left[ {\int_a^b {{{\left( {b - x} \right)}^{\alpha - 1}}f\left( {\frac{{a + x}}{2}} \right){\rm{d}}x} + } \right.} \right.\\ \left. {\left. {\int_a^b {{{\left( {x - a} \right)}^{\alpha - 1}}f\left( {\frac{{x + b}}{2}} \right){\rm{d}}x} } \right]} \right| \le \\ \frac{{b - a}}{{4\left( {\alpha + 1} \right)}}\left[ {\left| {f'\left( a \right)} \right| + \left| {f'\left( b \right)} \right|} \right] + \frac{{b - a}}{{8\left( {\alpha + 1} \right)\left( {\alpha + 2} \right)}} \times \\ \left[ {\eta \left( {\left| {f'\left( a \right)} \right|,\left| {f'\left( b \right)} \right|} \right) + \eta \left( {\left| {f'\left( b \right)} \right|,\left| {f'\left( a \right)} \right|} \right)} \right], \end{array}$

 $\begin{array}{l} \left| {pf\left( a \right) + \left( {1 - p} \right)f\left( b \right) - {K_2}} \right| \le \\ \;\;\;\;\;p\left( {1 - p} \right)\left( {b - a} \right)\frac{{\left| {f'\left( a \right)} \right| + \left| {f'\left( b \right)} \right|}}{{\alpha + 1}}. \end{array}$
2 结束语