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Jensen and Jensen-Mercer Inequalities for Harmonic (p, s)-Convex Functions on Fractal Sets and Their Applications
DOI: 10.12677/PM.2023.1310288, PDF, HTML, XML, 下载: 274  浏览: 358  国家自然科学基金支持

Abstract: For the first time, the definition of harmonic (p, s)-convex functions on fractal sets is proposed. The generalized Jensen inequality and the generalized Jensen-Mercer inequality for the functions are established. By introducing local fractional order integrals and the constructed Jensen and Jensen-Mercer inequalities, the Hermite-Hadamard inequality for generalized harmonic (p, s)-convex functions is derived. Finally, applications of some results in probability are discussed.

1. 引言

$f\left(\frac{xy}{{\left[\left(1-t\right){x}^{p}+t{y}^{p}\right]}^{\frac{1}{p}}}\right)\le {t}^{s}f\left(x\right)+{\left(1-t\right)}^{s}f\left(y\right)$ (1)

$f\left(\underset{i=1}{\overset{n}{\sum }}{w}_{i}{x}_{i}\right)\le \underset{i=1}{\overset{n}{\sum }}{w}_{i}f\left({x}_{i}\right)$ (2)

2003年，Mercer [9] 给出Jensen不等式的推广形式。

$f\left(a+b-\underset{i=1}{\overset{n}{\sum }}{w}_{i}{x}_{i}\right)\le f\left(a\right)+f\left(b\right)-\underset{i=1}{\overset{n}{\sum }}{w}_{i}f\left({x}_{i}\right)$ (3)

2. 预备知识

${ℝ}^{\alpha }\left(0<\alpha \le 1\right)$ 是维数为 $\alpha$ 的分形集，参考文献 [10] [11] ，则下面运算律成立：

${a}^{\alpha },{b}^{\alpha },{c}^{\alpha }\in {ℝ}^{\alpha }$ ，则

1) ${a}^{\alpha }+{b}^{\alpha }\in {ℝ}^{\alpha }$${a}^{\alpha }{b}^{\alpha }\in {ℝ}^{\alpha }$

2) ${a}^{\alpha }+{b}^{\alpha }={b}^{\alpha }+{a}^{\alpha }={\left(a+b\right)}^{\alpha }={\left(b+a\right)}^{\alpha }$${a}^{\alpha }{b}^{\alpha }={b}^{\alpha }{a}^{\alpha }={\left(ab\right)}^{\alpha }={\left(ba\right)}^{\alpha }$

3) ${a}^{\alpha }+\left({b}^{\alpha }+{c}^{\alpha }\right)=\left({a}^{\alpha }+{b}^{\alpha }\right)+{c}^{\alpha }$${a}^{\alpha }\left({b}^{\alpha }{c}^{\alpha }\right)=\left({a}^{\alpha }{b}^{\alpha }\right){c}^{\alpha }$

4) ${a}^{\alpha }\left({b}^{\alpha }+{c}^{\alpha }\right)={a}^{\alpha }{b}^{\alpha }+{a}^{\alpha }{c}^{\alpha }$

5) ${a}^{\alpha }+{0}^{\alpha }={0}^{\alpha }+{a}^{\alpha }={a}^{\alpha }$${a}^{\alpha }{1}^{\alpha }={1}^{\alpha }{\alpha }^{\alpha }={a}^{\alpha }$

$f\left(x\right)\in {C}_{\alpha }\left(a,b\right)$ 表示 $f\left(x\right)$ 在区间 $\left[a,b\right]$ 上局部分数阶连续； $f\left(x\right)\in {D}_{\alpha }\left[a,b\right]$ 表示 $f\left(x\right)$ 在区间 $\left[a,b\right]$$\alpha$ 阶局部分数阶可导； ${}_{a}I{}_{b}^{\left(\alpha \right)}f\left(x\right)$ 表示 $f\left(x\right)$ 在区间 $\left[a,b\right]$$\alpha$ 阶局部分数阶积分。

${}_{a}I{}_{b}^{\left(\alpha \right)}f\left(x\right)=g\left(b\right)-g\left(a\right)$

2) 设 $f\left(x\right)$$g\left(x\right)\in {D}_{\alpha }\left[a,b\right]$ ，且 ${f}^{\left(\alpha \right)}\left(x\right),{g}^{\left(\alpha \right)}\left(x\right)\in {C}_{\alpha }\left[a,b\right]$ ，则

${}_{a}I{}_{b}^{\left(\alpha \right)}f\left(x\right){g}^{\left(\alpha \right)}\left(x\right)={f\left(x\right)g\left(x\right)|}_{a}^{b}-{}_{a}I{}_{b}^{\left(\alpha \right)}{f}^{\left(\alpha \right)}\left(x\right)g\left(x\right)$

$\frac{1}{\Gamma \left(1+\alpha \right)}{\int }_{a}^{b}{x}^{k\alpha }{\left(\text{d}x\right)}^{\alpha }=\frac{\Gamma \left(1+k\alpha \right)}{\Gamma \left(1+\left(k+1\right)\alpha \right)}\left({b}^{\left(k+1\right)\alpha }-{a}^{\left(k+1\right)\alpha }\right)$$k\in R$

$\frac{1}{\Gamma \left(1+\alpha \right)}{\int }_{a}^{b}|f\left(x\right)g\left(x\right)|{\left(\text{d}x\right)}^{\alpha }\le {\left(\frac{1}{\Gamma \left(1+\alpha \right)}{\int }_{a}^{b}{|f\left(x\right)|}^{p}{\left(\text{d}x\right)}^{\alpha }\right)}^{\frac{1}{p}}{\left(\frac{1}{\Gamma \left(1+\alpha \right)}{\int }_{a}^{b}{|g\left(x\right)|}^{q}{\left(\text{d}x\right)}^{\alpha }\right)}^{\frac{1}{q}}$

$f\left(\frac{xy}{tx+\left(1-t\right)y}\right)\le {t}^{\alpha }f\left(y\right)+{\left(1-t\right)}^{\alpha }f\left(x\right)$

$f\left({\left[\underset{i=1}{\overset{n}{\sum }}\frac{{w}_{i}}{{x}_{i}}\right]}^{-1}\right)\le \underset{i=1}{\overset{n}{\sum }}{w}_{i}^{\alpha }f\left({x}_{i}\right)$ (4)

$f\left({\left[\frac{1}{a}+\frac{1}{b}-\underset{i=1}{\overset{n}{\sum }}\frac{{w}_{i}}{{x}_{i}}\right]}^{-1}\right)\le f\left(a\right)+f\left(b\right)-\underset{i=1}{\overset{n}{\sum }}{w}_{i}^{\alpha }f\left({x}_{i}\right)$ (5)

3. 主要结论

$f\left({\left[\frac{t}{{x}^{p}}+\frac{1-t}{{y}^{p}}\right]}^{-\text{\hspace{0.17em}}\frac{1}{p}}\right)\le {t}^{s\alpha }f\left(x\right)+{\left(1-t\right)}^{s\alpha }f\left(y\right)$ (6)

$f\left({\left[\underset{i=1}{\overset{n}{\sum }}\frac{{w}_{i}}{{x}_{i}^{p}}\right]}^{-\text{\hspace{0.17em}}\frac{1}{p}}\right)\le \underset{i=1}{\overset{n}{\sum }}{w}_{i}^{s\alpha }f\left({x}_{i}\right)$ (7)

$f\left({\left[\underset{i=1}{\overset{k}{\sum }}\frac{{w}_{i}}{{x}_{i}^{p}}\right]}^{-\text{\hspace{0.17em}}\frac{1}{p}}\right)\le \underset{i=1}{\overset{k}{\sum }}{w}_{i}^{s\alpha }f\left({x}_{i}\right)$

${x}_{1},{x}_{2},\cdots ,{x}_{k},{x}_{k+1}\in I$${r}_{i}>0$$i=1,2,\cdots ,k,k+1$$\underset{i=1}{\overset{k+1}{\sum }}{r}_{i}=1$ ，取 ${w}_{i}=\frac{{r}_{i}}{1-{r}_{k+1}}$ ，则对所有的 $i=1,2,\cdots ,k$${w}_{i}$ 满足 $\underset{i=1}{\overset{k}{\sum }}{w}_{i}=1$

$\begin{array}{c}f\left({\left[\underset{i=1}{\overset{k+1}{\sum }}\frac{{r}_{i}}{{x}_{i}^{p}}\right]}^{-\text{\hspace{0.17em}}\frac{1}{p}}\right)=f\left({\left[\left(1-{r}_{k+1}\right)\frac{\frac{{r}_{1}}{{x}_{1}^{p}}+\frac{{r}_{2}}{{x}_{2}^{p}}+\cdots +\frac{{r}_{k}}{{x}_{k}^{p}}}{1-{r}_{k+1}}+\frac{{r}_{k+1}}{{x}_{k\text{+1}}^{p}}\right]}^{-\text{\hspace{0.17em}}\frac{1}{p}}\right)\\ \le {\left(1-{r}_{k+1}\right)}^{s\alpha }f\left({\left[\frac{\frac{{r}_{1}}{{x}_{1}^{p}}+\frac{{r}_{2}}{{x}_{2}^{p}}+\cdots +\frac{{r}_{k}}{{x}_{k}^{p}}}{1-{r}_{k+1}}\right]}^{-\text{\hspace{0.17em}}\frac{1}{p}}\right)+{r}_{k+1}^{s\alpha }f\left({x}_{k+1}\right)\\ ={\left(1-{r}_{k+1}\right)}^{s\alpha }f\left({\left[\frac{{w}_{1}}{{x}_{1}^{p}}+\frac{{w}_{2}}{{x}_{2}^{p}}+\cdots +\frac{{w}_{k}}{{x}_{k}^{p}}\right]}^{-\text{\hspace{0.17em}}\frac{1}{p}}\right)+{r}_{k+1}^{s\alpha }f\left({x}_{k+1}\right)\end{array}$

$\begin{array}{c}\le {\left(1-{r}_{k+1}\right)}^{s\alpha }\left[{w}_{1}^{s\alpha }f\left({x}_{1}\right)+{w}_{2}^{s\alpha }f\left({x}_{2}\right)+\cdots +{w}_{k}^{s\alpha }f\left({x}_{k}\right)\right]+{r}_{k+1}^{s\alpha }f\left({x}_{k+1}\right)\underset{{}_{}}{\overset{{}^{}}{}}\\ ={\left(1-{r}_{k+1}\right)}^{s\alpha }\left[{\left(\frac{{r}_{1}}{1-{r}_{k+1}}\right)}^{s\alpha }f\left({x}_{1}\right)+{\left(\frac{{r}_{2}}{1-{r}_{k+1}}\right)}^{s\alpha }f\left({x}_{2}\right)+\cdots +{\left(\frac{{r}_{k}}{1-{r}_{k+1}}\right)}^{s\alpha }f\left({x}_{k}\right)\right]+{r}_{k+1}^{s\alpha }f\left({x}_{k+1}\right)\\ =\underset{i=1}{\overset{k\text{+1}}{\sum }}{r}_{i}^{s\alpha }f\left( x i \right)\end{array}$

$f\left({\left[\underset{i=1}{\overset{n}{\sum }}\frac{{w}_{i}}{{x}_{i}^{p}}\right]}^{-\text{\hspace{0.17em}}\frac{1}{p}}\right)\le \underset{i=1}{\overset{n}{\sum }}{w}_{i}^{\alpha }f\left({x}_{i}\right)$ (8)

$f\left({\left[\frac{1}{{a}^{p}}+\frac{1}{{b}^{p}}-\frac{1}{{x}^{p}}\right]}^{-\text{\hspace{0.17em}}\frac{1}{p}}\right)\le \left[{t}^{s\alpha }+{\left(1-t\right)}^{s\alpha }\right]\left[f\left(a\right)+f\left(b\right)\right]-f\left(x\right)$

$\begin{array}{c}f\left({\left[\frac{1}{{a}^{p}}+\frac{1}{{b}^{p}}-\frac{1}{{x}^{p}}\right]}^{-\text{\hspace{0.17em}}\frac{1}{p}}\right)=f\left({\left[\frac{t}{{a}^{p}}+\frac{1-t}{{b}^{p}}\right]}^{-\text{\hspace{0.17em}}\frac{1}{p}}\right)\le {t}^{s\alpha }f\left(a\right)+{\left(1-t\right)}^{s\alpha }f\left(b\right)\\ =\left[{t}^{s\alpha }+{\left(1-t\right)}^{s\alpha }\right]\left[f\left(a\right)+f\left(b\right)\right]-{t}^{s\alpha }f\left(b\right)-{\left(1-t\right)}^{s\alpha }f\left(a\right)\\ \le \left[{t}^{s\alpha }+{\left(1-t\right)}^{s\alpha }\right]\left[f\left(a\right)+f\left(b\right)\right]-f\left( x \right)\end{array}$

$f\left({\left[\frac{1}{{a}^{p}}+\frac{1}{{b}^{p}}-\frac{1}{{x}^{p}}\right]}^{-\text{\hspace{0.17em}}\frac{1}{p}}\right)\le f\left(a\right)+f\left(b\right)-f\left(x\right)$ (9)

$f\left({\left[\frac{1}{{a}^{p}}+\frac{1}{{b}^{p}}-\underset{i=1}{\overset{n}{\sum }}\frac{{w}_{i}}{{x}_{i}^{p}}\right]}^{-\text{\hspace{0.17em}}\frac{1}{p}}\right)\le \underset{i=1}{\overset{n}{\sum }}{w}_{i}^{s\alpha }\left({t}^{s\alpha }+{\left(1-t\right)}^{s\alpha }\right)\left[f\left(a\right)+f\left(b\right)\right]-\underset{i=1}{\overset{n}{\sum }}{w}_{i}^{s\alpha }f\left({x}_{i}\right)$

$\begin{array}{c}f\left({\left[\frac{1}{{a}^{p}}+\frac{1}{{b}^{p}}-\underset{i=1}{\overset{n}{\sum }}\frac{{w}_{i}}{{x}_{i}^{p}}\right]}^{-\text{\hspace{0.17em}}\frac{1}{p}}\right)\le \underset{i=1}{\overset{n}{\sum }}{w}_{i}^{s\alpha }f\left({\left[\frac{1}{{a}^{p}}+\frac{1}{{b}^{p}}-\frac{1}{{x}_{i}{}^{p}}\right]}^{-\text{\hspace{0.17em}}\frac{1}{p}}\right)\\ \le \underset{i=1}{\overset{n}{\sum }}{w}_{i}^{s\alpha }\left[\left({t}^{s\alpha }+{\left(1-t\right)}^{s\alpha }\right)\left(f\left(a\right)+f\left(b\right)\right)-f\left({x}_{i}\right)\right]\\ =\underset{i=1}{\overset{n}{\sum }}{w}_{i}^{s\alpha }\left({t}^{s\alpha }+{\left(1-t\right)}^{s\alpha }\right)\left(f\left(a\right)+f\left(b\right)\right)-\underset{i=1}{\overset{n}{\sum }}{w}_{i}^{s\alpha }f\left( x i \right)\end{array}$

$f\left({\left[\frac{1}{{a}^{p}}+\frac{1}{{b}^{p}}-\underset{i=1}{\overset{n}{\sum }}\frac{{w}_{i}}{{x}_{i}^{p}}\right]}^{-\text{\hspace{0.17em}}\frac{1}{p}}\right)\le f\left(a\right)+f\left(b\right)-\underset{i=1}{\overset{n}{\sum }}{w}_{i}^{\alpha }f\left({x}_{i}\right)$ (10)

$\begin{array}{l}f\left({\left[\frac{1}{{a}^{p}}+\frac{1}{{b}^{p}}-\frac{1}{{\stackrel{¯}{a}}^{p}}\right]}^{-\text{\hspace{0.17em}}\frac{1}{p}}\right)\le \underset{i=1}{\overset{n}{\sum }}{w}_{i}^{s\alpha }f\left({\left[\frac{1}{{a}^{p}}+\frac{1}{{b}^{p}}-\left(\frac{1-t}{{\stackrel{¯}{a}}^{p}}+\frac{t}{{a}_{i}^{p}}\right)\right]}^{-\text{\hspace{0.17em}}\frac{1}{p}}\right)\\ \le \left(\underset{i=1}{\overset{n}{\sum }}{w}_{i}^{s\alpha }{\left(1-t\right)}^{s\alpha }+{t}^{s\alpha }\right)\left[\underset{i=1}{\overset{n}{\sum }}{w}_{i}^{s\alpha }\left({\left(1-t\right)}^{s\alpha }+{t}^{s\alpha }\right)\left(f\left(a\right)+f\left(b\right)\right)-\underset{i=1}{\overset{n}{\sum }}{w}_{i}^{s\alpha }f\left({a}_{i}\right)\right]\end{array}$ (11)

$\begin{array}{c}f\left({\left[\frac{1}{{a}^{p}}+\frac{1}{{b}^{p}}-\frac{1}{{\stackrel{¯}{a}}^{p}}\right]}^{-\text{\hspace{0.17em}}\frac{1}{p}}\right)=f\left({\left[\underset{i=1}{\overset{n}{\sum }}{w}_{i}\left(\frac{1}{{a}^{p}}+\frac{1}{{b}^{p}}-\left(\frac{1-t}{{\stackrel{¯}{a}}^{p}}+\frac{t}{{a}_{i}^{p}}\right)\right)\right]}^{-\text{\hspace{0.17em}}\frac{1}{p}}\right)\\ \le \underset{i=1}{\overset{n}{\sum }}{w}_{i}^{s\alpha }f\left({\left[\frac{1}{{a}^{p}}+\frac{1}{{b}^{p}}-\left(\frac{1-t}{{\stackrel{¯}{a}}^{p}}+\frac{t}{{a}_{i}^{p}}\right)\right]}^{-\text{\hspace{0.17em}}\frac{1}{p}}\right)\end{array}$ (12)

$\begin{array}{l}\underset{i=1}{\overset{n}{\sum }}{w}_{i}^{s\alpha }f\left({\left[\frac{1}{{a}^{p}}+\frac{1}{{b}^{p}}-\left(\frac{1-t}{{\stackrel{¯}{a}}^{p}}+\frac{t}{{a}_{i}^{p}}\right)\right]}^{-\text{\hspace{0.17em}}\frac{1}{p}}\right)\\ =\underset{i=1}{\overset{n}{\sum }}{w}_{i}^{s\alpha }f\left({\left[\left(1-t\right)\left(\frac{1}{{a}^{p}}+\frac{1}{{b}^{p}}-\frac{1}{{\stackrel{¯}{a}}^{p}}\right)+t\left(\frac{1}{{a}^{p}}+\frac{1}{{b}^{p}}-\frac{1}{{a}_{i}^{p}}\right)\right]}^{-\text{\hspace{0.17em}}\frac{1}{p}}\right)\\ \le \underset{i=1}{\overset{n}{\sum }}{w}_{i}^{s\alpha }\left[{\left(1-t\right)}^{s\alpha }f\left({\left[\frac{1}{{a}^{p}}+\frac{1}{{b}^{p}}-\frac{1}{{\stackrel{¯}{a}}^{p}}\right]}^{-\text{\hspace{0.17em}}\frac{1}{p}}\right)+{t}^{s\alpha }f\left({\left[\frac{1}{{a}^{p}}+\frac{1}{{b}^{p}}-\frac{1}{{a}_{i}^{p}}\right]}^{-\text{\hspace{0.17em}}\frac{1}{p}}\right)\right]\\ \le \left(\underset{i=1}{\overset{n}{\sum }}{w}_{i}^{s\alpha }{\left(1-t\right)}^{s\alpha }+{t}^{s\alpha }\right)\left[\left(\underset{i=1}{\overset{n}{\sum }}{w}_{i}^{s\alpha }{\left(1-t\right)}^{s\alpha }+{t}^{s\alpha }\right)\left(f\left(a\right)+f\left(b\right)\right)-\underset{i=1}{\overset{n}{\sum }}{w}_{i}^{s\alpha }f\left({a}_{i}\right)\right]\end{array}$ (13)

$f\left({\left[\frac{1}{{a}^{p}}+\frac{1}{{b}^{p}}-\frac{1}{{\stackrel{¯}{a}}^{p}}\right]}^{-\text{\hspace{0.17em}}\frac{1}{p}}\right)\le \underset{i=1}{\overset{n}{\sum }}{w}_{i}^{\alpha }f\left({\left[\frac{1}{{a}^{p}}+\frac{1}{{b}^{p}}-\left(\frac{1-t}{{\stackrel{¯}{a}}^{p}}+\frac{t}{{a}_{i}^{p}}\right)\right]}^{-\text{\hspace{0.17em}}\frac{1}{p}}\right)\le f\left(a\right)+f\left(b\right)-\underset{i=1}{\overset{n}{\sum }}{w}_{i}^{\alpha }f\left({a}_{i}\right)$

$\begin{array}{c}f\left({\left[\frac{1}{{a}^{p}}+\frac{1}{{b}^{p}}-\frac{1}{{\stackrel{¯}{a}}^{p}}\right]}^{-\text{\hspace{0.17em}}\frac{1}{p}}\right)\le \underset{i=1}{\overset{n}{\sum }}\frac{{w}_{i}^{s\alpha }\Gamma \left(1+\alpha \right){p}^{\alpha }}{{\left(\frac{1}{{a}_{i}^{p}}-\frac{1}{{\stackrel{¯}{a}}^{p}}\right)}^{\alpha }}{}_{{\left[\frac{1}{{a}^{p}}+\frac{1}{{b}^{p}}-\frac{1}{{\stackrel{¯}{a}}^{p}}\right]}^{-\text{\hspace{0.17em}}\frac{1}{p}}}I{}_{{\left[\frac{1}{{a}^{p}}+\frac{1}{{b}^{p}}-\frac{1}{{a}_{i}^{p}}\right]}^{-\text{\hspace{0.17em}}\frac{1}{p}}}^{\left(\alpha \right)}\frac{f\left(x\right)}{{x}^{\left(p+1\right)\alpha }}\\ \le \left(\underset{i=1}{\overset{n}{\sum }}{w}_{i}^{s\alpha }+{1}^{\alpha }\right)\left[\underset{i=1}{\overset{n}{\sum }}{w}_{i}^{s\alpha }\left(\frac{\Gamma \left(1+2s\alpha \right)\Gamma \left(1+\alpha \right)}{\Gamma \left(1+\left(2s+1\right)\alpha \right)}+\Gamma \left(1+\alpha \right){\text{B}}^{\alpha }\left(s+1,s+1\right)\right)\\ \text{\hspace{0.17em}}×\left(f\left(a\right)+f\left(b\right)\right)-\frac{\Gamma \left(1+s\alpha \right)\Gamma \left(1+\alpha \right)}{\Gamma \left(1+\left(s+1\right)\alpha \right)}\underset{i=1}{\overset{n}{\sum }}{w}_{i}^{s\alpha }f\left({a}_{i}\right)\right]\end{array}$ (14)

${\text{B}}^{\alpha }\left(x,y\right)=\frac{1}{\Gamma \left(1+\alpha \right)}{\int }_{0}^{1}{t}^{\left(x-1\right)\alpha }{\left(1-t\right)}^{\left(y-1\right)\alpha }{\left(\text{d}t\right)}^{\alpha }$$x>0$$y>0$

$\begin{array}{l}\frac{1}{\Gamma \left(1+\alpha \right)}{\int }_{0}^{1}\underset{i=1}{\overset{n}{\sum }}{w}_{i}^{s\alpha }\left({\left(1-t\right)}^{s\alpha }+{t}^{s\alpha }\right)\left(\underset{i=1}{\overset{n}{\sum }}{w}_{i}^{s\alpha }{\left(1-t\right)}^{s\alpha }+{t}^{s\alpha }\right){\left(\text{d}t\right)}^{\alpha }\\ =\frac{1}{\Gamma \left(1+\alpha \right)}{\int }_{0}^{1}\left[\underset{i=1}{\overset{n}{\sum }}{w}_{i}^{s\alpha }\underset{i=1}{\overset{n}{\sum }}{w}_{i}^{s\alpha }{\left(1-t\right)}^{2s\alpha }+\underset{i=1}{\overset{n}{\sum }}{w}_{i}^{s\alpha }{t}^{2s\alpha }\right]{\left(\text{d}t\right)}^{\alpha }\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }+\frac{1}{\Gamma \left(1+\alpha \right)}{\int }_{0}^{1}\left[\underset{i=1}{\overset{n}{\sum }}{w}_{i}^{s\alpha }\underset{i=1}{\overset{n}{\sum }}{w}_{i}^{s\alpha }{\left(1-t\right)}^{s\alpha }{t}^{s\alpha }+\underset{i=1}{\overset{n}{\sum }}{w}_{i}^{s\alpha }{\left(1-t\right)}^{s\alpha }{t}^{s\alpha }\right]{\left(\text{d}t\right)}^{\alpha }\\ =\left(\underset{i=1}{\overset{n}{\sum }}{w}_{i}^{s\alpha }\underset{i=1}{\overset{n}{\sum }}{w}_{i}^{s\alpha }+\underset{i=1}{\overset{n}{\sum }}{w}_{i}^{s\alpha }\right)\left(\frac{\Gamma \left(1+2s\alpha \right)}{\Gamma \left(1+\left(2s+1\right)\alpha \right)}+{\text{B}}^{\alpha }\left(s+1,s+1\right)\right)\end{array}$

$\frac{1}{\Gamma \left(1+\alpha \right)}{\int }_{0}^{1}{\left(1-t\right)}^{s\alpha }{\left(\text{d}t\right)}^{\alpha }=\frac{1}{\Gamma \left(1+\alpha \right)}{\int }_{0}^{1}{t}^{s\alpha }{\left(\text{d}t\right)}^{\alpha }=\frac{\Gamma \left(1+s\alpha \right)}{\Gamma \left(1+\left(s+1\right)\alpha \right)}$

$\begin{array}{l}\frac{1}{\Gamma \left(1+\alpha \right)}{\int }_{0}^{1}f\left({\left[\frac{1}{{a}^{p}}+\frac{1}{{b}^{p}}-\left(\frac{1-t}{{\stackrel{¯}{a}}^{p}}+\frac{t}{{a}_{i}^{p}}\right)\right]}^{-\text{\hspace{0.17em}}\frac{1}{p}}\right){\left(\text{d}t\right)}^{\alpha }\\ =\frac{1}{\Gamma \left(1+\alpha \right)}{\int }_{0}^{1}f\left({\left[\left(1-t\right)\left(\frac{1}{{a}^{p}}+\frac{1}{{b}^{p}}-\frac{1}{{\stackrel{¯}{a}}^{p}}\right)+t\left(\frac{1}{{a}^{p}}+\frac{1}{{b}^{p}}-\frac{1}{{a}_{i}^{p}}\right)\right]}^{-\text{\hspace{0.17em}}\frac{1}{p}}\right){\left(\text{d}t\right)}^{\alpha }\\ =\frac{{p}^{\alpha }}{{\left(\frac{1}{{a}_{i}^{p}}-\frac{1}{{\stackrel{¯}{a}}^{p}}\right)}^{\alpha }}\frac{1}{\Gamma \left(1+\alpha \right)}{\int }_{{\left[\frac{1}{{a}^{p}}+\frac{1}{{b}^{p}}-\frac{1}{{\stackrel{¯}{a}}^{p}}\right]}^{-\text{\hspace{0.17em}}\frac{1}{p}}}^{{\left[\frac{1}{{a}^{p}}+\frac{1}{{b}^{p}}-\frac{1}{{a}_{i}^{p}}\right]}^{-\text{\hspace{0.17em}}\frac{1}{p}}}\frac{f\left(u\right)}{{u}^{\left(p+1\right)\alpha }}{\left(\text{d}u\right)}^{\alpha }\\ =\frac{{p}^{\alpha }}{{\left(\frac{1}{{a}_{i}^{p}}-\frac{1}{{\stackrel{¯}{a}}^{p}}\right)}^{\alpha }}{}_{{\left[\frac{1}{{a}^{p}}+\frac{1}{{b}^{p}}-\frac{1}{{\stackrel{¯}{a}}^{p}}\right]}^{-\text{\hspace{0.17em}}\frac{1}{p}}}I{}_{{\left[\frac{1}{{a}^{p}}+\frac{1}{{b}^{p}}-\frac{1}{{a}_{i}^{p}}\right]}^{-\text{\hspace{0.17em}}\frac{1}{p}}}^{\left(\alpha \right)}\frac{f\left(x\right)}{{x}^{\left(p+1\right)\alpha }}\end{array}$

$\begin{array}{c}f\left({\left[\frac{1}{{a}^{p}}+\frac{1}{{b}^{p}}-\frac{1}{{\stackrel{¯}{a}}^{p}}\right]}^{-\text{\hspace{0.17em}}\frac{1}{p}}\right)\le \underset{i=1}{\overset{n}{\sum }}\frac{{w}_{i}^{\alpha }\Gamma \left(1+\alpha \right){p}^{\alpha }}{{\left(\frac{1}{{a}_{i}^{p}}-\frac{1}{{\stackrel{¯}{a}}^{p}}\right)}^{\alpha }}{}_{{\left[\frac{1}{{a}^{p}}+\frac{1}{{b}^{p}}-\frac{1}{{\stackrel{¯}{a}}^{p}}\right]}^{-\text{\hspace{0.17em}}\frac{1}{p}}}I{}_{{\left[\frac{1}{{a}^{p}}+\frac{1}{{b}^{p}}-\frac{1}{{a}_{i}^{p}}\right]}^{-\text{\hspace{0.17em}}\frac{1}{p}}}^{\left(\alpha \right)}\frac{f\left(x\right)}{{x}^{\left(p+1\right)\alpha }}\\ \le \frac{{2}^{\alpha }{\Gamma }^{2}\left(1+\alpha \right)}{\Gamma \left(1+2\alpha \right)}\left(f\left(a\right)+f\left(b\right)-\underset{i=1}{\overset{n}{\sum }}{w}_{i}^{\alpha }f\left({a}_{i}\right)\right)\end{array}$ (15)

$\begin{array}{c}f\left(\frac{a+b}{2}\right)\le \frac{\Gamma \left(1+\alpha \right)}{{\left(b-a\right)}^{\alpha }}{}_{a}I{}_{b}^{\left(\alpha \right)}f\left(x\right)\\ \le \frac{{\Gamma }^{2}\left(1+\alpha \right)}{\Gamma \left(1+2\alpha \right)}\left[f\left(a\right)+f\left(b\right)\right]\end{array}$

$\begin{array}{c}f\left({\left[\frac{1}{{a}^{p}}+\frac{1}{{b}^{p}}-\frac{1}{{\stackrel{¯}{a}}^{p}}\right]}^{-\text{\hspace{0.17em}}\frac{1}{p}}\right)\le \underset{i=1}{\overset{n}{\sum }}{w}_{i}^{\alpha }f\left({\left[\frac{1}{{a}^{p}}+\frac{1}{{b}^{p}}-\frac{1}{2}\left(\frac{1}{{\stackrel{¯}{a}}^{p}}+\frac{1}{{a}_{i}^{p}}\right)\right]}^{-\text{\hspace{0.17em}}\frac{1}{p}}\right)\\ \le \frac{\underset{i=1}{\overset{n}{\sum }}{w}_{i}^{\alpha }{p}^{\alpha }\Gamma \left(1+\alpha \right)}{{\left(\frac{1}{{a}_{i}^{p}}-\frac{1}{{\stackrel{¯}{a}}^{p}}\right)}^{\alpha }}{}_{{\left[\frac{1}{{a}^{p}}+\frac{1}{{b}^{p}}-\frac{1}{{\stackrel{¯}{a}}^{p}}\right]}^{-\text{\hspace{0.17em}}\frac{1}{p}}}I{}_{{\left[\frac{1}{{a}^{p}}+\frac{1}{{b}^{p}}-\frac{1}{{a}_{i}^{p}}\right]}^{-\text{\hspace{0.17em}}\frac{1}{p}}}^{\left(\alpha \right)}\frac{f\left(x\right)}{{x}^{\left(p+1\right)\alpha }}\\ \le \frac{{2}^{\alpha }{\Gamma }^{2}\left(1+\alpha \right)}{\Gamma \left(1+2\alpha \right)}\left[f\left(a\right)+f\left(b\right)-\underset{i=1}{\overset{n}{\sum }}{w}_{i}^{\alpha }f\left({a}_{i}\right)\right]\end{array}$

$\begin{array}{c}f\left({\left[\frac{1}{{a}^{p}}+\frac{1}{{b}^{p}}-\frac{1}{{\stackrel{¯}{a}}^{p}}\right]}^{-\text{\hspace{0.17em}}\frac{1}{p}}\right)=f\left({\left[\underset{i=1}{\overset{n}{\sum }}{w}_{i}\left(\frac{1}{{a}^{p}}+\frac{1}{{b}^{p}}-\frac{1}{2}\left(\frac{1}{{\stackrel{¯}{a}}^{p}}+\frac{1}{{a}_{i}^{p}}\right)\right)\right]}^{-\text{\hspace{0.17em}}\frac{1}{p}}\right)\\ \le \underset{i=1}{\overset{n}{\sum }}{w}_{i}^{\alpha }f\left({\left[\frac{1}{{a}^{p}}+\frac{1}{{b}^{p}}-\frac{1}{2}\left(\frac{1}{{\stackrel{¯}{a}}^{p}}+\frac{1}{{a}_{i}^{p}}\right)\right]}^{-\text{\hspace{0.17em}}\frac{1}{p}}\right)\end{array}$ (16)

$\begin{array}{l}f\left({\left[\frac{1}{{a}^{p}}+\frac{1}{{b}^{p}}-\frac{1}{2}\left(\frac{1}{{\stackrel{¯}{a}}^{p}}+\frac{1}{{a}_{i}^{p}}\right)\right]}^{-\text{\hspace{0.17em}}\frac{1}{p}}\right)\\ =f\left({\left[\frac{1}{2}\left(\frac{1}{{a}^{p}}+\frac{1}{{b}^{p}}-\left(\frac{t}{{\stackrel{¯}{a}}^{p}}+\frac{1-t}{{a}_{i}^{p}}\right)\right)+\frac{1}{2}\left(\frac{1}{{a}^{p}}+\frac{1}{{b}^{p}}-\left(\frac{1-t}{{\stackrel{¯}{a}}^{p}}+\frac{t}{{a}_{i}^{p}}\right)\right)\right]}^{-\text{\hspace{0.17em}}\frac{1}{p}}\right)\\ \le {\left(\frac{1}{2}\right)}^{\alpha }\left[f\left({\left[\frac{1}{{a}^{p}}+\frac{1}{{b}^{p}}-\left(\frac{t}{{\stackrel{¯}{a}}^{p}}+\frac{1-t}{{a}_{i}^{p}}\right)\right]}^{-\text{\hspace{0.17em}}\frac{1}{p}}\right)+f\left({\left[\frac{1}{{a}^{p}}+\frac{1}{{b}^{p}}-\left(\frac{1-t}{{\stackrel{¯}{a}}^{p}}+\frac{t}{{a}_{i}^{p}}\right)\right]}^{-\text{\hspace{0.17em}}\frac{1}{p}}\right)\right]\end{array}$ (17)

$\frac{1}{\Gamma \left(1+\alpha \right)}f\left({\left[\frac{1}{{a}^{p}}+\frac{1}{{b}^{p}}-\frac{1}{{\stackrel{¯}{a}}^{p}}\right]}^{-\text{\hspace{0.17em}}\frac{1}{p}}\right)\le \frac{\underset{i=1}{\overset{n}{\sum }}{w}_{i}^{\alpha }}{\Gamma \left(1+\alpha \right)}f\left({\left[\frac{1}{{a}^{p}}+\frac{1}{{b}^{p}}-\frac{1}{2}\left(\frac{1}{{\stackrel{¯}{a}}^{p}}+\frac{1}{{a}_{i}^{p}}\right)\right]}^{-\text{\hspace{0.17em}}\frac{1}{p}}\right)$ . (18)

$\begin{array}{l}\frac{\underset{i=1}{\overset{n}{\sum }}{w}_{i}^{\alpha }}{\Gamma \left(1+\alpha \right)}f\left({\left[\frac{1}{{a}^{p}}+\frac{1}{{b}^{p}}-\frac{1}{2}\left(\frac{1}{{\stackrel{¯}{a}}^{p}}+\frac{1}{{a}_{i}^{p}}\right)\right]}^{-\text{\hspace{0.17em}}\frac{1}{p}}\right)\\ \le \frac{\underset{i=1}{\overset{n}{\sum }}{w}_{i}^{\alpha }}{\Gamma \left(1+\alpha \right)}{\int }_{0}^{1}f\left({\left[\frac{1}{{a}^{p}}+\frac{1}{{b}^{p}}-\left(\frac{t}{{\stackrel{¯}{a}}^{p}}+\frac{1-t}{{a}_{i}^{p}}\right)\right]}^{-\text{\hspace{0.17em}}\frac{1}{p}}\right){\left(\text{d}t\right)}^{\alpha }\\ =\frac{\underset{i=1}{\overset{n}{\sum }}{w}_{i}^{\alpha }{p}^{\alpha }}{{\left(\frac{1}{{a}_{i}^{p}}-\frac{1}{{\stackrel{¯}{a}}^{p}}\right)}^{\alpha }}{}_{{\left[\frac{1}{{a}^{p}}+\frac{1}{{b}^{p}}-\frac{1}{{\stackrel{¯}{a}}^{p}}\right]}^{-\text{}\frac{1}{p}}}I{}_{{\left[\frac{1}{{a}^{p}}+\frac{1}{{b}^{p}}-\frac{1}{{a}_{i}^{p}}\right]}^{-\text{\hspace{0.17em}}\frac{1}{p}}}^{\left(\alpha \right)}\frac{f\left(x\right)}{{x}^{\left(p+1\right)\alpha }}\end{array}$ (19)

4. 概率方面的应用

$\chi$ 是一个随机变量， $\chi$ 的广义概率密度函数为 $f:\left[a,b\right]\to \left[{0}^{\alpha },{1}^{\alpha }\right]$ 且f为广义调和p-凸函数。在分形空间中，为研究概率问题，可给出如下定义 [15] ：

${P}_{\alpha }\left(\chi \le \frac{1}{{\left[\frac{1}{{a}^{p}}+\frac{1}{{b}^{p}}-\frac{{\xi }_{1}^{p}+{\xi }_{2}^{p}}{2{\xi }_{1}^{p}{\xi }_{2}^{p}}\right]}^{\frac{1}{p}}}\right)\le {P}_{\alpha }\left(\chi \le a\right)+{P}_{\alpha }\left(\chi \le b\right)-\frac{{P}_{\alpha }\left(\chi \le {\xi }_{1}\right)+{P}_{\alpha }\left(\chi \le {\xi }_{2}\right)}{{2}^{\alpha }}$ . (20)

${\left(\frac{1}{{a}^{p}}+\frac{1}{{b}^{p}}-\frac{{\xi }_{1}^{p}+{\xi }_{2}^{p}}{2{\xi }_{1}^{p}{\xi }_{2}^{p}}\right)}^{-\text{\hspace{0.17em}}\frac{p\text{+1}}{p}\alpha }\le {b}^{\left(p+1\right)\alpha }+{a}^{\left(p+1\right)\alpha }-\frac{{\xi }_{1}^{\left(p+1\right)\alpha }+{\xi }_{2}^{\left(p+1\right)\alpha }}{{2}^{\alpha }}$ . (21)

$\begin{array}{c}{P}_{\alpha }\left(\chi \le \frac{{2}^{\frac{1}{p}}ab}{{\left[{a}^{p}+{b}^{p}\right]}^{\frac{1}{p}}}\right)\le {\left(\frac{{a}^{p}{b}^{p}}{{b}^{p}-{a}^{p}}\right)}^{\alpha }\frac{{p}^{\alpha }\Gamma \left(1+\alpha \right)\Gamma \left(1-\left(p+1\right)\alpha \right)}{\Gamma \left(1-p\alpha \right)}\left[{\left(\frac{1}{{b}^{p}}\right)}^{\alpha }-{E}_{-p\alpha }\left(\chi \right)\right]\\ \le \frac{{\Gamma }^{2}\left(1+\alpha \right)}{\Gamma \left(1+2\alpha \right)}\left[{P}_{\alpha }\left(\chi \le a\right)+{P}_{\alpha }\left(\chi \le b\right)\right]\end{array}$

5. 总结