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 浙江大学学报(理学版)  2017, Vol. 44 Issue (5): 531-537  DOI:10.3785/j.issn.1008-9497.2017.05.006 0

### 引用本文 [复制中英文]

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SUN Wenbing. New Hermite-Hadamard-type inequalities for s-convex functions via fractional integrals[J]. Journal of Zhejiang University(Science Edition), 2017, 44(5): 531-537. DOI: 10.3785/j.issn.1008-9497.2017.05.006.
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### 文章历史

New Hermite-Hadamard-type inequalities for s-convex functions via fractional integrals
SUN Wenbing
Department of Science and Information Science, Shaoyang University, Shaoyang 422000, Hunan Province, China
Abstract: In this paper, we establish a new identity for Riemann-Liouville fractional integrals. Using the established identity, some new Hermite-Hadamard type inequalities for differentiable s-convex mappings that are connected with the Riemann-Liouville fractional integrals are obtained. Also, some results are deduced for differentiable s-concave functions. Our results extend some proved results in the existing researches. Finally, we give an example to illustrate the applications of the results.
Key words: Hadamard's inequality    s-convex function    Hölder inequality    Riemann-Liouville fractional integral
0 引言

f:IRR是一个凸函数且a, bI, a < b, 则有

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

KIRMACI[12]证明了以下与不等式(1) 左端有关联的一些结果:

 $\begin{array}{l} \frac{1}{{b - a}}\int_a^b {f\left( x \right){\rm{d}}x} - f\left( {\frac{{a + b}}{2}} \right) = \\ \;\;\;\;\;\;\;\left( {b - a} \right)\left[ {\int_0^{1/2} {tf'\left( {ta + \left( {1 - t} \right)b} \right){\rm{d}}t} } \right. + \\ \;\;\;\;\;\;\;\left. {\int_{1/2}^0 {\left( {t - 1} \right)f'\left( {ta + \left( {1 - t} \right)b} \right){\rm{d}}t} } \right]. \end{array}$ (2)

 $\begin{array}{l} \left| {\frac{1}{{b - a}}\int_a^b {f\left( x \right){\rm{d}}x} - f\left( {\frac{{a + b}}{2}} \right)} \right| \le \\ \;\;\;\;\;\;\;\frac{{\left( {b - a} \right)}}{8}\left( {\left| {f'\left( a \right)} \right| + \left| {f'\left( b \right)} \right|} \right). \end{array}$ (3)

 $\begin{array}{l} \left| {\frac{1}{{b - a}}\int_a^b {f\left( x \right){\rm{d}}x} - f\left( {\frac{{a + b}}{2}} \right)} \right| \le \frac{{b - a}}{{16}}{\left( {\frac{4}{{p + 1}}} \right)^{1/p}} \times \\ \left[ {{{\left( {{{\left| {f'\left( a \right)} \right|}^{p/\left( {p - 1} \right)}} + 3{{\left| {f'\left( b \right)} \right|}^{p/\left( {p - 1} \right)}}} \right)}^{\left( {p - 1} \right)/p}} + } \right.\\ \left. {{{\left( {3{{\left| {f'\left( a \right)} \right|}^{p/\left( {p - 1} \right)}} + {{\left| {f'\left( b \right)} \right|}^{p/\left( {p - 1} \right)}}} \right)}^{\left( {p - 1} \right)/p}}} \right]. \end{array}$ (4)

 $\begin{array}{l} \left| {\frac{1}{{b - a}}\int_a^b {f\left( x \right){\rm{d}}x} - f\left( {\frac{{a + b}}{2}} \right)} \right| \le \\ \;\;\;\;\;\frac{{b - a}}{4}{\left( {\frac{4}{{p + 1}}} \right)^{1/p}}\left( {\left| {f'\left( a \right)} \right| + \left| {f'\left( b \right)} \right|} \right). \end{array}$ (5)

KIRMACI等[13]还证明了对于凹函数，有

 $\begin{array}{l} \left| {\frac{1}{{b - a}}\int_a^b {f\left( x \right){\rm{d}}x} - f\left( {\frac{{a + b}}{2}} \right)} \right| \le \\ \;\;\;\;\;\;\left( {\frac{{b - a}}{8}} \right)\left( {f'\left( {a + b} \right)} \right). \end{array}$ (6)

 $f\left( {\lambda x + \left( {1 - \lambda } \right)y} \right) \le {\lambda ^s}f\left( x \right) + {\left( {1 - \lambda } \right)^s}f\left( y \right)$

 ${2^{s - 1}}f\left( {\frac{{a + b}}{2}} \right) \le \frac{1}{{b - a}}\int_a^b {f\left( x \right){\rm{d}}x} \le \frac{{f\left( a \right) + f\left( b \right)}}{{s + 1}},$ (7)

 $J_{a + }^\alpha f\left( x \right) = \frac{1}{{\Gamma \left( \alpha \right)}}\int_a^x {{{\left( {x - t} \right)}^{\alpha - 1}}f\left( t \right){\rm{d}}t} ,x > a,$
 $J_{b - }^\alpha f\left( x \right) = \frac{1}{{\Gamma \left( \alpha \right)}}\int_x^b {{{\left( {t - x} \right)}^{\alpha - 1}}f\left( t \right){\rm{d}}t} ,x < b,$

 ${B_x}\left( {a,b} \right) = \int_0^x {{t^{a - 1}}{{\left( {1 - t} \right)}^{b - 1}}{\rm{d}}t} .$

 $\begin{array}{l} \frac{{\Gamma \left( {\alpha + 1} \right)}}{{{{\left( {b - a} \right)}^\alpha }}}\left[ {{{\left( { - 1} \right)}^\alpha }J_{\frac{{a + b}}{2} - }^\alpha f\left( b \right) - J_{\frac{{a + b}}{2} + }^\alpha f\left( a \right)} \right] + \\ \;\;\;\;\;\;\;\;\;\left[ { - 1 + {{\left( { - 1} \right)}^\alpha }} \right]\frac{1}{{{2^\alpha }}}f\left( {\frac{{a + b}}{2}} \right) = \\ \;\;\;\;\;\;\;\;\;\left( {b - a} \right)\left[ {\int_0^{1/2} {{t^\alpha }f'\left( {ta + \left( {1 - t} \right)b} \right){\rm{d}}t} + } \right.\\ \;\;\;\;\;\;\;\;\;\left. {\int_{1/2}^1 {{{\left( {t - 1} \right)}^\alpha }f'\left( {ta + \left( {1 - t} \right)b} \right){\rm{d}}t} } \right]. \end{array}$ (8)

 $\begin{array}{l} \int_0^{1/2} {{t^\alpha }f'\left( {ta + \left( {1 - t} \right)b} \right){\rm{d}}t} = \\ {t^\alpha }\frac{{f\left( {ta + \left( {1 - t} \right)b} \right)}}{{a - b}}\left| {_0^{1/2}} \right. - \int_0^{1/2} {\alpha {t^{\alpha - 1}}\frac{{f\left( {ta + \left( {1 - t} \right)b} \right)}}{{a - b}}{\rm{d}}t} = \\ - \frac{1}{{{2^\alpha }\left( {b - a} \right)}}f\left( {\frac{{a + b}}{2}} \right) + \frac{\alpha }{{b - a}}\int_b^{\frac{{a + b}}{2}} {{{\left( {\frac{{x - b}}{{a - b}}} \right)}^{\alpha - 1}}\frac{{f\left( x \right)}}{{a - b}}{\rm{d}}x} = \\ - \frac{1}{{{2^\alpha }\left( {b - a} \right)}}f\left( {\frac{{a + b}}{2}} \right) + {\left( { - 1} \right)^\alpha }\frac{{\Gamma \left( {\alpha + 1} \right)}}{{{{\left( {b - a} \right)}^{\alpha + 1}}}}J_{\frac{{a + b}}{2} - }^\alpha f\left( b \right) \end{array}$ (9)

 $\begin{array}{l} \int_{1/2}^1 {{{\left( {t - 1} \right)}^\alpha }f'\left( {ta + \left( {1 - t} \right)b} \right){\rm{d}}t} = \\ {\left( {t - 1} \right)^\alpha }\frac{{f\left( {ta + \left( {1 - t} \right)b} \right)}}{{a - b}}\left| {_{1/2}^1} \right. - \\ \int_{1/2}^1 {\alpha {{\left( {t - 1} \right)}^{\alpha - 1}}\frac{{f\left( {ta + \left( {1 - t} \right)b} \right)}}{{a - b}}{\rm{d}}t} = \\ {\left( { - 1} \right)^\alpha }\frac{1}{{{2^\alpha }\left( {b - a} \right)}}f\left( {\frac{{a + b}}{2}} \right) + \\ \frac{\alpha }{{b - a}}\int_{\frac{{a + b}}{2}}^a {{{\left( {\frac{{x - a}}{{a - b}}} \right)}^{\alpha - 1}}\frac{{f\left( x \right)}}{{a - b}}{\rm{d}}x} = \\ {\left( { - 1} \right)^\alpha }\frac{1}{{{2^\alpha }\left( {b - a} \right)}}f\left( {\frac{{a + b}}{2}} \right) - \frac{{\Gamma \left( {\alpha + 1} \right)}}{{{{\left( {b - a} \right)}^{\alpha + 1}}}}J_{\frac{{a + b}}{2} + }^\alpha f\left( a \right), \end{array}$ (10)

 $\begin{array}{l} \left( {b - a} \right)\int_0^{1/2} {{t^\alpha }f'\left( {ta + \left( {1 - t} \right)b} \right){\rm{d}}t} = \\ - \frac{1}{{{2^\alpha }}}f\left( {\frac{{a + b}}{2}} \right) + {\left( { - 1} \right)^\alpha }\frac{{\Gamma \left( {\alpha + 1} \right)}}{{{{\left( {b - a} \right)}^\alpha }}}J_{\frac{{a + b}}{2} - }^\alpha f\left( b \right) \end{array}$ (11)

 $\begin{array}{l} \left( {b - a} \right)\int_{1/2}^1 {{{\left( {t - 1} \right)}^\alpha }f'\left( {ta + \left( {1 - t} \right)b} \right){\rm{d}}t} = \\ {\left( { - 1} \right)^\alpha }\frac{1}{{{2^\alpha }}}f\left( {\frac{{a + b}}{2}} \right) - \frac{{\Gamma \left( {\alpha + 1} \right)}}{{{{\left( {b - a} \right)}^\alpha }}}J_{\frac{{a + b}}{2} + }^\alpha f\left( a \right). \end{array}$ (12)

 $\begin{array}{l} \left| {\frac{{\Gamma \left( {\alpha + 1} \right)}}{{{{\left( {b - a} \right)}^\alpha }}}\left[ {{{\left( { - 1} \right)}^\alpha }J_{\frac{{a + b}}{2} - }^\alpha f\left( b \right) - J_{\frac{{a + b}}{2} + }^\alpha f\left( a \right)} \right] + \left[ { - 1 + } \right.} \right.\\ \left. {\left. {{{\left( { - 1} \right)}^\alpha }} \right]\frac{1}{{{2^\alpha }}}f\left( {\frac{{a + b}}{2}} \right)} \right| \le \left( {b - a} \right)\left[ {\frac{1}{{{2^{\alpha + s + 1}}\left( {\alpha + s + 1} \right)}} + } \right.\\ \left. {{B_{\frac{1}{2}}}\left( {\alpha + 1,s + 1} \right)} \right]\left( {\left| {f'\left( a \right)} \right| + \left| {f'\left( b \right)} \right|} \right). \end{array}$ (13)

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

 $\int_0^{1/2} {{t^{\alpha + s}}{\rm{d}}t} = \int_{1/2}^1 {{{\left( {1 - t} \right)}^{\alpha + s}}{\rm{d}}t} = \frac{1}{{{2^{\alpha + s + 1}}\left( {\alpha + s + 1} \right)}}$

 $\begin{array}{*{20}{c}} {\int_0^{1/2} {{t^\alpha }{{\left( {1 - t} \right)}^s}{\rm{d}}t} = \int_{1/2}^1 {{{\left( {1 - t} \right)}^\alpha }{t^s}{\rm{d}}t} = }\\ {{B_{\frac{1}{2}}}\left( {\alpha + 1,s + 1} \right).} \end{array}$

 $\begin{array}{l} \left| {\frac{{\Gamma \left( {\alpha + 1} \right)}}{{{{\left( {b - a} \right)}^\alpha }}}\left[ {{{\left( { - 1} \right)}^\alpha }J_{\frac{{a + b}}{2} - }^\alpha f\left( b \right) - J_{\frac{{a + b}}{2} + }^\alpha f\left( a \right)} \right] + \left[ { - 1 + } \right.} \right.\\ \left. {\left. {\;\;\;\;\;\;\;\;{{\left( { - 1} \right)}^\alpha }} \right]\frac{1}{{{2^\alpha }}}f\left( {\frac{{a + b}}{2}} \right)} \right| \le \\ \;\;\;\;\;\;\;\;\frac{{\left( {b - a} \right)}}{{{2^{\alpha + 1}}\left( {\alpha + 1} \right)}}\left( {\left| {f'\left( a \right)} \right| + \left| {f'\left( b \right)} \right|} \right). \end{array}$ (14)

 $\begin{array}{l} {B_{\frac{1}{2}}}\left( {\alpha + 1,2} \right) = \int_0^{1/2} {{t^\alpha }\left( {1 - t} \right){\rm{d}}t} = \\ \;\;\;\;\;\int_{1/2}^1 {{{\left( {1 - t} \right)}^\alpha }{t^s}{\rm{d}}t} = \frac{1}{{{2^{\alpha + 2}}}}\frac{{\alpha + 3}}{{\left( {\alpha + 1} \right)\left( {\alpha + 2} \right)}}, \end{array}$

 $\begin{array}{l} \left| {\frac{1}{{b - a}}\int_a^b {f\left( x \right){\rm{d}}x} - f\left( {\frac{{a + b}}{2}} \right)} \right| \le \\ \left( {b - a} \right)\left[ {\frac{{{2^{s + 1}} - 1}}{{{2^{s + 1}}\left( {s + 1} \right)\left( {s + 2} \right)}}} \right]\left( {\left| {f'\left( a \right)} \right| + \left| {f'\left( b \right)} \right|} \right). \end{array}$ (15)

 $\begin{array}{l} {B_{\frac{1}{2}}}\left( {2,s + 1} \right) = \int_0^{1/2} {t{{\left( {1 - t} \right)}^s}{\rm{d}}t} = \\ \;\;\;\;\;\int_{1/2}^1 {\left( {1 - t} \right){t^s}{\rm{d}}t} = \frac{1}{{{2^{s + 2}}}}\frac{{{2^{s + 2}} - s - 3}}{{\left( {s + 1} \right)\left( {s + 2} \right)}}. \end{array}$

 $\begin{array}{l} \left| {\frac{{\Gamma \left( {\alpha + 1} \right)}}{{{{\left( {b - a} \right)}^\alpha }}}\left[ {{{\left( { - 1} \right)}^\alpha }J_{\frac{{a + b}}{2} - }^\alpha f\left( b \right) - J_{\frac{{a + b}}{2} + }^\alpha f\left( a \right)} \right]} \right. + \left[ { - 1 + } \right.\\ \left. {\left. {{{\left( { - 1} \right)}^\alpha }} \right]\frac{1}{{{2^\alpha }}}f\left( {\frac{{a + b}}{2}} \right)} \right| \le \frac{{b - a}}{{{2^{\alpha + s + 1}}\left( {s + 1} \right)}}{\left( {\frac{{{2^s}\left( {s + 1} \right)}}{{\alpha p + 1}}} \right)^{\frac{1}{p}}} \times \\ \left[ {{{\left( {{{\left| {f'\left( a \right)} \right|}^{p/\left( {p - 1} \right)}} + \left( {{2^{s + 1}} - 1} \right){{\left| {f'\left( b \right)} \right|}^{p/\left( {p - 1} \right)}}} \right)}^{\left( {p - 1} \right)/p}} + } \right.\\ \left. {{{\left( {\left( {{2^{s + 1}} - 1} \right){{\left| {f'\left( a \right)} \right|}^{p/\left( {p - 1} \right)}} + {{\left| {f'\left( b \right)} \right|}^{p/\left( {p - 1} \right)}}} \right)}^{\left( {p - 1} \right)/p}}} \right]. \end{array}$ (16)

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

 $\frac{1}{p} + \frac{1}{q} = 1.$

 $\begin{array}{l} \int_0^{1/2} {{{\left| {f'\left( {ta + \left( {1 - t} \right)b} \right)} \right|}^q}{\rm{d}}t} \le \\ \int_0^{1/2} {\left( {{t^s}{{\left| {f'\left( a \right)} \right|}^q} + {{\left( {1 - t} \right)}^s}{{\left| {f'\left( b \right)} \right|}^q}} \right){\rm{d}}t} = \\ \frac{1}{{{2^{s + 1}}\left( {s + 1} \right)}}\left( {{{\left| {f'\left( a \right)} \right|}^q} + \left( {{2^{s + 1}} - 1} \right){{\left| {f'\left( b \right)} \right|}^q}} \right) \end{array}$

 $\begin{array}{l} \int_{1/2}^1 {{{\left| {f'\left( {ta + \left( {1 - t} \right)b} \right)} \right|}^q}{\rm{d}}t} \le \\ \int_{1/2}^1 {\left( {{t^s}{{\left| {f'\left( a \right)} \right|}^q} + {{\left( {1 - t} \right)}^s}{{\left| {f'\left( b \right)} \right|}^q}} \right){\rm{d}}t} = \\ \frac{1}{{{2^{s + 1}}\left( {s + 1} \right)}}\left( {\left( {{2^{s + 1}} - 1} \right){{\left| {f'\left( a \right)} \right|}^q} + {{\left| {f'\left( b \right)} \right|}^q}} \right). \end{array}$

 $\int_0^{1/2} {{t^{\alpha p}}{\rm{d}}t} = \int_{1/2}^1 {\left| {{{\left( {t - 1} \right)}^{\alpha p}}} \right|{\rm{d}}t} = \frac{1}{{{2^{\alpha p + 1}}\left( {\alpha p + 1} \right)}}.$

 $\begin{array}{l} \left| {\frac{{\Gamma \left( {\alpha + 1} \right)}}{{{{\left( {b - a} \right)}^\alpha }}}\left[ {{{\left( { - 1} \right)}^\alpha }J_{\frac{{a + b}}{2} - }^\alpha f\left( b \right) - J_{\frac{{a + b}}{2} + }^\alpha f\left( a \right)} \right]} \right. + \\ \left. {\left[ { - 1 + {{\left( { - 1} \right)}^\alpha }} \right]\frac{1}{{{2^\alpha }}}f\left( {\frac{{a + b}}{2}} \right)} \right| \le \\ \frac{{b - a}}{{{2^{\alpha + s + 1}}\left( {s + 1} \right)}}{\left( {\frac{{{2^s}\left( {s + 1} \right)}}{{\alpha p + 1}}} \right)^{\frac{1}{p}}} \times \\ \left[ {{{\left( {{{\left| {f'\left( a \right)} \right|}^{p/\left( {p - 1} \right)}} + \left( {{2^{s + 1}} - 1} \right){{\left| {f'\left( b \right)} \right|}^{p/\left( {p - 1} \right)}}} \right)}^{\left( {p - 1} \right)/p}} + } \right.\\ \left. {{{\left( {\left( {{2^{s + 1}} - 1} \right){{\left| {f'\left( a \right)} \right|}^{p/\left( {p - 1} \right)}} + {{\left| {f'\left( b \right)} \right|}^{p/\left( {p - 1} \right)}}} \right)}^{\left( {p - 1} \right)/p}}} \right], \end{array}$

 $\begin{array}{l} \left| {\frac{{\Gamma \left( {\alpha + 1} \right)}}{{{{\left( {b - a} \right)}^\alpha }}}\left[ {{{\left( { - 1} \right)}^\alpha }J_{\frac{{a + b}}{2} - }^\alpha f\left( b \right) - J_{\frac{{a + b}}{2} + }^\alpha f\left( a \right)} \right]} \right. + \left[ { - 1 + } \right.\\ \left. {\left. {{{\left( { - 1} \right)}^\alpha }} \right]\frac{1}{{{2^\alpha }}}f\left( {\frac{{a + b}}{2}} \right)} \right| \le \\ \frac{{b - a}}{{{2^{\alpha + 3}}}}{\left( {\frac{4}{{\alpha p + 1}}} \right)^{\frac{1}{p}}}\left[ {\left( {{{\left| {f'\left( a \right)} \right|}^{p/\left( {p - 1} \right)}} + } \right.} \right.\\ {\left. {3{{\left| {f'\left( b \right)} \right|}^{p/\left( {p - 1} \right)}}} \right)^{\left( {p - 1} \right)/p}} + \\ \left. {{{\left( {3{{\left| {f'\left( a \right)} \right|}^{p/\left( {p - 1} \right)}} + {{\left| {f'\left( b \right)} \right|}^{p/\left( {p - 1} \right)}}} \right)}^{\left( {p - 1} \right)/p}}} \right]. \end{array}$ (17)

 $\begin{array}{l} \left| {\frac{1}{{b - a}}\int_a^b {f\left( x \right){\rm{d}}x - f\left( {\frac{{a + b}}{2}} \right)} } \right| \le \\ \frac{{b - a}}{{{2^{s + 2}}\left( {s + 1} \right)}}{\left( {\frac{{{2^s}\left( {s + 1} \right)}}{{p + 1}}} \right)^{\frac{1}{p}}} \times \\ \left[ {{{\left( {{{\left| {f'\left( a \right)} \right|}^{p/\left( {p - 1} \right)}} + \left( {{2^{s + 1}} - 1} \right){{\left| {f'\left( b \right)} \right|}^{p/\left( {p - 1} \right)}}} \right)}^{\left( {p - 1} \right)/p}} + } \right.\\ \left. {{{\left( {\left( {{2^{s + 1}} - 1} \right){{\left| {f'\left( a \right)} \right|}^{p/\left( {p - 1} \right)}} + {{\left| {f'\left( b \right)} \right|}^{p/\left( {p - 1} \right)}}} \right)}^{\left( {p - 1} \right)/p}}} \right]. \end{array}$ (18)

 $\begin{array}{l} \left| {\frac{{\Gamma \left( {\alpha + 1} \right)}}{{{{\left( {b - a} \right)}^\alpha }}}\left[ {{{\left( { - 1} \right)}^\alpha }J_{\frac{{a + b}}{2} - }^\alpha f\left( b \right) - J_{\frac{{a + b}}{2} + }^\alpha f\left( a \right)} \right]} \right. + \left[ { - 1 + } \right.\\ \left. {\left. {{{\left( { - 1} \right)}^\alpha }} \right]\frac{1}{{{2^\alpha }}}f\left( {\frac{{a + b}}{2}} \right)} \right| \le \\ \frac{{b - a}}{{{2^\alpha }\left( {s + 1} \right)}}{\left( {\frac{{{2^s}\left( {s + 1} \right)}}{{\alpha p + 1}}} \right)^{\frac{1}{p}}}\left( {\left| {f'\left( a \right)} \right| + \left| {f'\left( b \right)} \right|} \right). \end{array}$ (19)

 $\sum\limits_{i = 1}^n {{{\left( {{a_i} + {b_i}} \right)}^r}} \le \sum\limits_{i = 1}^n {a_i^r} + \sum\limits_{i = 1}^n {b_i^r} .$

 $\begin{array}{l} \left| {\frac{{\Gamma \left( {\alpha + 1} \right)}}{{{{\left( {b - a} \right)}^\alpha }}}\left[ {{{\left( { - 1} \right)}^\alpha }J_{\frac{{a + b}}{2} - }^\alpha f\left( b \right) - J_{\frac{{a + b}}{2} + }^\alpha f\left( a \right)} \right]} \right. + \\ \left. {\left[ { - 1 + {{\left( { - 1} \right)}^\alpha }} \right]\frac{1}{{{2^\alpha }}}f\left( {\frac{{a + b}}{2}} \right)} \right| \le \\ \frac{{b - a}}{{{2^{\alpha + s + 1}}\left( {s + 1} \right)}}{\left( {\frac{{{2^s}\left( {s + 1} \right)}}{{\alpha p + 1}}} \right)^{\frac{1}{p}}} \times \\ \left[ {{{\left( {{{\left| {f'\left( a \right)} \right|}^{p/\left( {p - 1} \right)}} + \left( {{2^{s + 1}} - 1} \right){{\left| {f'\left( b \right)} \right|}^{p/\left( {p - 1} \right)}}} \right)}^{\left( {p - 1} \right)/p}} + } \right.\\ \left. {{{\left( {\left( {{2^{s + 1}} - 1} \right){{\left| {f'\left( a \right)} \right|}^{p/\left( {p - 1} \right)}} + {{\left| {f'\left( b \right)} \right|}^{p/\left( {p - 1} \right)}}} \right)}^{\left( {p - 1} \right)/p}}} \right] \le \\ \frac{{b - a}}{{{2^{\alpha + s + 1}}\left( {s + 1} \right)}}{\left( {\frac{{{2^s}\left( {s + 1} \right)}}{{\alpha p + 1}}} \right)^{\frac{1}{p}}} \times \\ \left[ {1 + {{\left( {{2^{s + 1}} - 1} \right)}^{\left( {p - 1} \right)/p}}} \right]\left( {\left| {f'\left( a \right)} \right| + \left| {f'\left( b \right)} \right|} \right) \le \\ \frac{{b - a}}{{{2^\alpha }\left( {s + 1} \right)}}{\left( {\frac{{{2^s}\left( {s + 1} \right)}}{{\alpha p + 1}}} \right)^{\frac{1}{p}}}\left( {\left| {f'\left( a \right)} \right| + \left| {f'\left( b \right)} \right|} \right), \end{array}$

 $\begin{array}{l} \left| {\frac{{\Gamma \left( {\alpha + 1} \right)}}{{{{\left( {b - a} \right)}^\alpha }}}\left[ {{{\left( { - 1} \right)}^\alpha }J_{\frac{{a + b}}{2} - }^\alpha f\left( b \right) - J_{\frac{{a + b}}{2} + }^\alpha f\left( a \right)} \right]} \right. + \\ \left. {\left[ { - 1 + {{\left( { - 1} \right)}^\alpha }} \right]\frac{1}{{{2^\alpha }}}f\left( {\frac{{a + b}}{2}} \right)} \right| \le \\ \frac{{b - a}}{{{2^{\alpha + 1}}}}{\left( {\frac{4}{{\alpha p + 1}}} \right)^{\frac{1}{p}}}\left( {\left| {f'\left( a \right)} \right| + \left| {f'\left( b \right)} \right|} \right). \end{array}$ (20)

 $\begin{array}{l} \left| {\frac{1}{{b + a}}\int_a^b {f\left( x \right){\rm{d}}x} - f\left( {\frac{{a + b}}{2}} \right)} \right| \le \\ \frac{{b - a}}{{2\left( {s + 1} \right)}}{\left( {\frac{{{2^s}\left( {s + 1} \right)}}{{p + 1}}} \right)^{\frac{1}{p}}}\left( {\left| {f'\left( a \right)} \right| + \left| {f'\left( b \right)} \right|} \right). \end{array}$ (21)

 $\begin{array}{l} \left| {\frac{{\Gamma \left( {\alpha + 1} \right)}}{{{{\left( {b - a} \right)}^\alpha }}}\left[ {{{\left( { - 1} \right)}^\alpha }J_{\frac{{a + b}}{2} - }^\alpha f\left( b \right) - J_{\frac{{a + b}}{2} + }^\alpha f\left( a \right)} \right]} \right. + \\ \left. {\left[ { - 1 + {{\left( { - 1} \right)}^\alpha }} \right]\frac{1}{{{2^\alpha }}}f\left( {\frac{{a + b}}{2}} \right)} \right| \le \\ \frac{{b - a}}{{{2^{\alpha p + 1}}\left( {\alpha p + 1} \right)}}{2^{\frac{{s - 1}}{q}}}\left[ {\left| {f'\left( {\frac{{a + 3b}}{4}} \right)} \right| + \left| {f'\left( {\frac{{3a + b}}{4}} \right)} \right|} \right], \end{array}$ (22)

 $\begin{array}{l} \left| {\frac{{\Gamma \left( {\alpha + 1} \right)}}{{{{\left( {b - a} \right)}^\alpha }}}\left[ {{{\left( { - 1} \right)}^\alpha }J_{\frac{{a + b}}{2} - }^\alpha f\left( b \right) - J_{\frac{{a + b}}{2} + }^\alpha f\left( a \right)} \right]} \right. + \\ \left. {\left[ { - 1 + {{\left( { - 1} \right)}^\alpha }} \right]\frac{1}{{{2^\alpha }}}f\left( {\frac{{a + b}}{2}} \right)} \right| \le \\ \left( {b - a} \right)\left[ {\int_0^{1/2} {\left| {{t^\alpha }} \right|\left| {f'\left( {ta + \left( {1 - t} \right)b} \right)} \right|{\rm{d}}t + } } \right.\\ \left. {\int_{1/2}^1 {\left| {{{\left( {t - 1} \right)}^\alpha }} \right|\left| {f'\left( {ta + \left( {1 - t} \right)b} \right)} \right|{\rm{d}}t} } \right] \le \\ \left( {b - a} \right)\left[ {{{\left( {\int_0^{1/2} {{t^{\alpha p}}{\rm{d}}t} } \right)}^{\frac{1}{p}}}{{\left( {\int_0^{1/2} {{{\left| {f'\left( {ta + \left( {1 - t} \right)b} \right)} \right|}^q}{\rm{d}}t} } \right)}^{\frac{1}{q}}}} \right] + \\ \left( {b - a} \right)\left[ {{{\left( {\int_0^{1/2} {\left| {{{\left( {t - 1} \right)}^{\alpha p}}} \right|{\rm{d}}t} } \right)}^{\frac{1}{p}}} \times } \right.\\ \left. {{{\left( {\int_{1/2}^1 {{{\left| {f'\left( {ta + \left( {1 - t} \right)b} \right)} \right|}^q}{\rm{d}}t} } \right)}^{\frac{1}{q}}}} \right]. \end{array}$

 $\int_0^{1/2} {{{\left| {f'\left( {ta + \left( {1 - t} \right)b} \right)} \right|}^q}{\rm{d}}t} \le {2^{s - 1}}{\left| {f'\left( {\frac{{a + 3b}}{4}} \right)} \right|^q},$
 $\int_{1/2}^1 {{{\left| {f'\left( {ta + \left( {1 - t} \right)b} \right)} \right|}^q}{\rm{d}}t} \le {2^{s - 1}}{\left| {f'\left( {\frac{{3a + b}}{4}} \right)} \right|^q}.$

 $\int_0^{1/2} {{t^{\alpha p}}{\rm{d}}t} = \int_{1/2}^1 {\left| {{{\left( {t - 1} \right)}^{\alpha p}}} \right|{\rm{d}}t} = \frac{1}{{{2^{\alpha p + 1}}\left( {\alpha p + 1} \right)}}.$

 $\begin{array}{l} \left| {\frac{{\Gamma \left( {\alpha + 1} \right)}}{{{{\left( {b - a} \right)}^\alpha }}}\left[ {{{\left( { - 1} \right)}^\alpha }J_{\frac{{a + b}}{2} - }^\alpha f\left( b \right) - J_{\frac{{a + b}}{2} + }^\alpha f\left( a \right)} \right]} \right. + \\ \left. {\left[ { - 1 + {{\left( { - 1} \right)}^\alpha }} \right]\frac{1}{{{2^\alpha }}}f\left( {\frac{{a + b}}{2}} \right)} \right| \le \\ \frac{{b - a}}{{{2^{\alpha p + 1}}\left( {\alpha p + 1} \right)}}{2^{\frac{{s - 1}}{q}}}\left[ {\left| {f'\left( {\frac{{a + 3b}}{4}} \right)} \right| + \left| {f'\left( {\frac{{3a + b}}{4}} \right)} \right|} \right]. \end{array}$

 $\begin{array}{l} \left| {\frac{{\Gamma \left( {\alpha + 1} \right)}}{{{{\left( {b - a} \right)}^\alpha }}}\left[ {{{\left( { - 1} \right)}^\alpha }J_{\frac{{a + b}}{2} - }^\alpha f\left( b \right) - J_{\frac{{a + b}}{2} + }^\alpha f\left( a \right)} \right]} \right. + \\ \left. {\left[ { - 1 + {{\left( { - 1} \right)}^\alpha }} \right]\frac{1}{{{2^\alpha }}}f\left( {\frac{{a + b}}{2}} \right)} \right| \le \\ \frac{{b - a}}{{{2^{\alpha p + 1}}\left( {\alpha p + 1} \right)}}{2^{\frac{{s - 1}}{q}}}\left| {f'\left( {a + b} \right)} \right|. \end{array}$ (23)

2 应用举例

 $f\left( t \right) = \left\{ \begin{array}{l} {a_1},\;\;t = 0,\\ {b_1}{t^s} + {c_1},\;\;\;\;t > 0. \end{array} \right.$

 $\begin{array}{l} \frac{{\Gamma \left( {\alpha + 1} \right)}}{{{{\left( {b - a} \right)}^\alpha }}}{\left( { - 1} \right)^\alpha }J_{\frac{{a + b}}{2} - }^\alpha f\left( b \right) = {\left( { - 1} \right)^\alpha }\alpha \int_1^{\frac{1}{2}} {{{\left( {t - 1} \right)}^{\alpha - 1}}{t^{\lambda - 1}}{\rm{d}}t} = \\ \;\;\;\;\;\alpha \int_0^{\frac{1}{2}} {{x^{\alpha - 1}}{{\left( {1 - x} \right)}^{\lambda - 1}}{\rm{d}}x} = \alpha {B_{\frac{1}{2}}}\left( {\alpha ,\lambda } \right), \end{array}$

 $\frac{{\Gamma \left( {\alpha + 1} \right)}}{{{{\left( {b - a} \right)}^\alpha }}}J_{\frac{{a + b}}{2} + }^\alpha f\left( a \right) = {\left( { - 1} \right)^{\alpha - 1}}\alpha \int_{\frac{1}{2}}^0 {{t^{\alpha + \lambda - 2}}{\rm{d}}t} = \frac{{{{\left( { - 1} \right)}^\alpha }\alpha }}{{\alpha + \lambda - 1}}{\left( {\frac{1}{2}} \right)^{\alpha + \lambda - 1}}.$

 $\begin{array}{l} \left| {\alpha {B_{\frac{1}{2}}}\left( {\alpha ,\lambda } \right) + \left[ {\frac{{{{\left( { - 1} \right)}^\alpha }\alpha }}{{\alpha + \lambda - 1}} - 1 + {{\left( { - 1} \right)}^\alpha }} \right]\frac{1}{{{2^{\alpha + \lambda - 1}}}}} \right| \le \\ \left[ {\frac{1}{{{2^{\alpha + s - 1}}\left( {\alpha + s - 1} \right)}} + {B_{\frac{1}{2}}}\left( {\alpha + 1,s + 1} \right)} \right]\left( {\lambda - 1} \right). \end{array}$