0 引言

若f是区间[a，b]上的凸函数，则对f在区间[a，b]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)

双边不等式(1)被称为Hermite-Hadamard不等式.关于Hermite-Hadamard不等式的各种改进、加细和推广，可参见文献[1-9].

最近，作为通常凸函数的推广，文献[10]引入了η凸函数的概念.

定义1 ^[10]  设区间$ I \subseteq R$，二元函数η：R×R→R，f：I→R，若对任意x，y∈I，t∈[0, 1]，有

$ 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), $

则称f是区间I上的η凸函数.

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

定理1 ^[11] (η凸函数的Hermite-Hadamard型不等式)若f：[a，b]→R是η凸函数，η在$f\left( {\left[ {a, b} \right]} \right) \times f([a, b]) $上有上界M_η，则有

$ \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} $

定理2 ^[11] (η凸函数的Hermite-Hadamard-Fejér型不等式)若f：[a，b]→R是η凸函数，η在$ f\left( {\left[ {a, b} \right]} \right) \times f([a, b])$上有上界，$ g:{\rm{ }}\left[ {a, b} \right] \to \left( {0, \infty } \right)$是可积函数且关于$\frac{{a + b}}{2} $对称，则有

$ \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} $

近年来，国内外研究者利用分数阶积分建立了分数阶的Hermite-Hadamard型不等式^[12-18].

定义2  设α>0，f在[a，b]上勒贝格可积，则函数f的α阶左Riemann-Liouville分数阶积分和α阶右Riemann-Liouville分数阶积分分别定义为

$ \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} $

其中Γ(α)是Gamma函数，即

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

方便起见，在下文的引理和定理中均假设

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

并且记

$ {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} $

定理3 ^[19] (η凸函数的Hermite-Hadamard型分数阶积分不等式)若f：[a，b]→R是η凸函数，η在$f\left( {\left[ {a, b} \right]} \right) \times f([a, b]) $上有上界M_η，α>0，则有

$ \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)

引理1 ^[13]   设f：[a，b]→R在(a，b)可微，f′在[a，b]上勒贝格可积，α>0，则有

$ \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} $

利用引理1，由文献[19]可得η凸函数的Hermite-Hadamard型分数阶积分不等式.

定理4 ^[19]  设f：[a，b]→R在(a，b)可微，|f′|是[a，b]上的η凸函数，α>0，则有

$ \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} $

当η(x, y)=x-y，即当f′是[a，b]上的凸函数时，由定理4得到文献[13]中凸函数的Hermite-Hadamard型分数阶积分不等式.

引理2 ^[14]  设f：[a，b]→R在(a，b)可微，f′在[a，b]上勒贝格可积，α>0，则有

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

其中，

$ {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} . $

应用引理2，由文献[19]可得到η凸函数的Hermite-Hadamard型分数阶积分不等式.

定理5 ^[19]   设f：[a，b]→R在(a，b)可微，|f′|是[a，b]上的η凸函数，0 < α≤1，则有

$ \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)

由引理2并利用积分变量代换，有

$ {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} , $

可得到以下引理：

引理3  设f：[a，b]→R在(a，b)可微，f′在[a，b]上勒贝格可积，α>0，则有

$ \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} $

目前，国内研究η凸函数的文献并不多^[20-21].本文建立了新的η凸函数Hermite-Hadamard型分数阶积分不等式.对定理3和定理5的结果进行了一定改进.对由凸函数分数阶的Hermite-Hadamard型不等式的右边部分生成的差值，给出了不同于定理4的估计.为证明主要结论，除引理1和引理3外，还需要引理4和引理5，此两引理用分部积分法易证之.

引理4  设f：[a，b]→R在(a，b)可微，f′在[a，b]上勒贝格可积，α>0，则有

$ \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} $

引理5   设f：[a，b]→R在(a，b)上可微，f′在[a，b]上勒贝格可积，α>0，则有

$ \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

定理6  若f：[a，b]→R是η凸函数，η在$ f\left( {\left[ {a, b} \right]} \right) \times f([a, b])$上有上界M_η，α>0，则有

$ \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)

证明   对任意x∈[a, b]，有$ x = \frac{{b - x}}{{b - a}}a + \frac{{x - a}}{{b - a}}b$，由η凸函数的定义，有

$ 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)

将式(5)与式(6)乘以${\left( {x - a} \right)^{\alpha - 1}} + {\left( {b - x} \right)^{\alpha - 1}} $，再分别在$ \left[ {a, {\rm{ }}\frac{{a + b}}{2}} \right]$和$\left[ {\frac{{a + b}}{2}{\rm{ }}, b} \right] $上对x积分，将所得的2个不等式相加得

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

(7)

将式(7)乘以$\frac{\alpha }{{2{{\left( {b - a} \right)}^\alpha }}} $，则式(4)的第3个不等式得证.

由η凸函数的定义，有

$ \begin{array}{l} f\left( {\frac{{a + b}}{2}} \right) = f\left( {\frac{{x + \left( {a + b - x} \right)}}{2}} \right) \le \\ \;\;\;\;\;\;\;\;\;\;f\left( x \right) + \frac{1}{2}\eta \left( {f\left( {a + b - x} \right),f\left( x \right)} \right), \end{array} $

(8)

将式(8)乘以$ {\left( {x - a} \right)^{\alpha - 1}} + {\left( {b - x} \right)^{\alpha - 1}}$，然后在[a, b]上对x积分，得

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

(9)

将式(9)乘以$\frac{\alpha }{{2{{\left( {b - a} \right)}^\alpha }}}{\rm{ }} $，则式(4)的第2个不等式得证.

最后，容易证明$ \frac{1}{{\alpha + 1}}\left( {{\rm{ }}1 - \frac{1}{{{2^\alpha }}}} \right)$是[0, 1]上关于α的单调递增函数，故式(4)右边的不等式得证.

注1   因α>0时有$ 1 - \frac{1}{{{2^\alpha }}} < \alpha $，故式(4)是式(2)的改进.

注2  在定理6中，取η(x, y)=x-y，则得到凸函数的Hermite-Hadamard型分数阶不等式^[13]：

$ f\left( {\frac{{a + b}}{2}} \right) \le {K_1} \le \frac{{f\left( a \right) + f\left( b \right)}}{2}. $

文献[3]利用积分给出了凸函数的Jensen不等式的隔离，受此启发，给出以下定理：

定理7   若f：[a，b]→R是η凸函数，α>0，则有

$ \begin{array}{l} f\left( \xi \right) - \frac{{p\left( {1 - p} \right){\rm{\Gamma }}\left( {\alpha + 1} \right)}}{{{{\left( {b - a} \right)}^\alpha }}}\left[ {J_a^{{\alpha _ + }}{f_1}\left( b \right) + J_b^{{\alpha _ - }}{f_2}\left( a \right)} \right] \le \\ \;\;\;\;\;\;{K_2} \le pf\left( a \right) + \left( {1 - p} \right)f\left( b \right) + \frac{{p\left( {1 - p} \right)}}{{\alpha + 1}} \times \\ \;\;\;\;\;\;\left[ {\eta \left( {f\left( a \right),f\left( b \right)} \right) + \eta \left( {f\left( b \right),f\left( a \right)} \right)} \right], \end{array} $

(10)

其中，

$ {f_1}\left( x \right) = \eta \left( {f\left( {b - p\left( {x - a} \right)} \right),f\left( {pa + \left( {1 - p} \right)x} \right)} \right), $

$ {f_2}\left( x \right) = \eta \left( {f\left( {a + \left( {1 - p} \right)\left( {b - x} \right)} \right),f\left( {px + \left( {1 - p} \right)b} \right)} \right). $

证明   将式(5)乘以(ξ-x)^α-1，然后在[a, ξ]上对x积分，得

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

(11)

将式(6)乘以(x-ξ)^α-1，然后在[ξ，b]上对x积分，得

$ \begin{array}{*{20}{c}} {\Gamma \left( \alpha \right)J_b^{{\alpha _ - }}f\left( \xi \right) \le \frac{{{p^\alpha }{{\left( {b - a} \right)}^\alpha }}}{\alpha }f\left( b \right) + }\\ {\frac{{{p^{\alpha + 1}}{{\left( {b - a} \right)}^\alpha }}}{{\alpha \left( {\alpha + 1} \right)}}\eta \left( {f\left( a \right),f\left( b \right)} \right),} \end{array} $

(12)

将式(11)和(12)分别乘以$ \frac{{p\alpha }}{{{{\left( {1 - p} \right)}^\alpha }{{\left( {b - a} \right)}^\alpha }}}$和$\frac{{\left( {1 - p} \right)\alpha }}{{{p^\alpha }{{\left( {b - a} \right)}^\alpha }}} $，再将所得不等式相加，则式(10)右边的不等式得证.

对任意x∈[a, ξ]，若取$ y = \frac{{\xi - px{\rm{ }}}}{{1 - p}}$，则有y∈[ξ, b]，且$ \xi = px + \left( {1 - p} \right)y$，由定义有

$ 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)

将式(13)乘以(ξ-x)^α-1，然后在[a, ξ]上对x积分，得

$ \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)

对任意x∈[ξ, b]，若取$ y\prime = \frac{{\xi - \left( {1 - p} \right)x}}{p}$，则有y′∈[a, ξ]，且$ \xi = \left( {1 - p} \right)x + py\prime $，由定义有

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

(15)

将式(15)乘以(x-ξ)^α-1，然后在[ξ, b]上对x积分，得

$ \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)

将式(14)和(16)分别乘以$ \frac{{p\alpha }}{{{{\left( {1 - p} \right)}^\alpha }{{\left( {b - a} \right)}^\alpha }}}$和$ \frac{{\left( {1 - p} \right)\alpha }}{{{p^\alpha }{{\left( {b - a} \right)}^\alpha }}}$，然后将所得不等式相加，则式(10)左边的不等式得证.

推论1  若f：[a，b]→R是η凸函数，η在$ f\left( {\left[ {a, b} \right]} \right) \times f([a, b])$上有上界M_η，α>0，则有

$ \begin{array}{l} f\left( \xi \right) - 2p\left( {1 - p} \right){M_\eta } \le {K_2} \le \\ \;\;\;\;\;\;\;\;\;\;pf\left( a \right) + \left( {1 - p} \right)f\left( b \right) + \frac{{2p\left( {1 - p} \right)}}{{\alpha + 1}}{M_\eta }. \end{array} $

定理8   设f：[a，b]→R在(a，b)上可微，f′在[a，b]上勒贝格可积，|f′|是[a，b]上的η凸函数，α>0，则有

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

(17)

证明  由引理1得

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

(18)

由|f′|的η凸性, 得

$ \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} $

(19)

$ \begin{array}{l} \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), \end{array} $

(20)

将式(19)和(20)乘以|(1-t)^α-t^α|，然后分别在$ \left[ {0, \frac{1}{2}} \right]$和$\left[ {\frac{1}{2}, 1} \right] $上对t积分，将所得的2个不等式相加得

$ \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)

综合式(18)和(21)，则式(17)获证.

注3  在定理8中，若η(x, y)=x-y，即|f′|是[a，b]上的凸函数，则可得文献[13]的凸函数的Hermite-Hadamard型分数阶积分不等式.

定理9  设f：[a，b]→R在(a，b)上可微，f′在[a，b]上勒贝格可积，|f′|是[a，b]上的η凸函数，0 < α≤1，则有

$ \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)

证明   由引理3得

$ \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)

由|f′|的η凸性得

$ \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), $

将上面4个式子相加并除以2得

$ \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)

综合式(23)和(24)，则式(22)得证.

注4   因为当0 < α≤1时，$ \frac{1}{2} - \frac{1}{{\alpha + 1}}\left( {1 - \frac{1}{{{2^\alpha }}}} \right){\rm{ }} \le \frac{1}{{{2^\alpha }\left( {\alpha + 1} \right)}}$，所以式(22)是式(3)的改进.

定理10   设f：[a，b]→R在(a，b)上可微，f′在[a，b]上勒贝格可积，|f′|是[a，b]上的η凸函数，α>0，则有

$ \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)

证明   由引理4得

$ \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)

由|f′|的η凸性得

$ \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)

综合式(26)~式(28)，则式(25)得证.

注5   在定理10中，若η(x, y)=x-y，也即|f′|是[a，b]上的凸函数，则有

$ \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} $

定理11  设f：[a，b]→R在(a，b)上可微，f′在[a，b]上勒贝格可积，|f′|是[a，b]上的η凸函数，α>0，则有

$ \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} $

证明  利用类似于引理5及定理9的证明方法可证得定理11，此证略.

推论2  若f：[a，b]→R是η凸函数，f′在[a，b]上勒贝格可积，|f′|是[a，b]上的η凸函数，α>0，则有

$ \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} $

证明   在定理11中, 取$p = \frac{1}{2} $即可得证.

注6   在定理11中，若η(x, y)=x-y，也即|f′|是[a，b]上的凸函数，则有

$ \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

建立了η凸函数的一些积分不等式，推广了通常凸函数的相应结果.寻找积分隔离η凸函数的Jensen型不等式，以及利用导函数的η凸性进行误差估计，均仿照了通常凸函数的研究方法.能对已有结果做些改进，得益于证明技巧的提升，包括分别在不同区间上对2个不等式积分，以及利用变量代换改变积分区间.有关凸函数的其他结果在η凸函数上的移植尚待进一步研究.