Advances in Difference Equations ( IF 3.1 ) Pub Date : 2020-09-18 , DOI: 10.1186/s13662-020-02977-3 Eze R. Nwaeze , Muhammad Adil Khan , Yu-Ming Chu
The notion of m-polynomial convex interval-valued function \(\Psi =[\psi ^{-}, \psi ^{+}]\) is hereby proposed. We point out a relationship that exists between Ψ and its component real-valued functions \(\psi ^{-}\) and \(\psi ^{+}\). For this class of functions, we establish loads of new set inclusions of the Hermite–Hadamard type involving the ρ-Riemann–Liouville fractional integral operators. In particular, we prove, among other things, that if a set-valued function Ψ defined on a convex set S is m-polynomial convex, \(\rho,\epsilon >0\) and \(\zeta,\eta \in {\mathbf{S}}\), then
$$\begin{aligned} \frac{m}{m+2^{-m}-1}\Psi \biggl(\frac{\zeta +\eta }{2} \biggr)& \supseteq \frac{\Gamma _{\rho }(\epsilon +\rho )}{(\eta -\zeta )^{\frac{\epsilon }{\rho }}} \bigl[{_{\rho }{\mathcal{J}}}_{\zeta ^{+}}^{\epsilon } \Psi (\eta )+_{ \rho }{\mathcal{J}}_{\eta ^{-}}^{\epsilon }\Psi (\zeta ) \bigr] \\ & \supseteq \frac{\Psi (\zeta )+\Psi (\eta )}{m}\sum_{p=1}^{m}S_{p}( \epsilon;\rho ), \end{aligned}$$where Ψ is Lebesgue integrable on \([\zeta,\eta ]\), \(S_{p}(\epsilon;\rho )=2-\frac{\epsilon }{\epsilon +\rho p}- \frac{\epsilon }{\rho }\mathcal{B} (\frac{\epsilon }{\rho }, p+1 )\) and \(\mathcal{B}\) is the beta function. We extend, generalize, and complement existing results in the literature. By taking \(m\geq 2\), we derive loads of new and interesting inclusions. We hope that the idea and results obtained herein will be a catalyst towards further investigation.
中文翻译：
m多项式凸区间值函数的Hermite-Hadamard类型的分数包含
因此，提出了m多项式凸区间值函数\（\ Psi = [\ psi ^ {-}，\ psi ^ {+}] \的概念。我们指出了Ψ及其组件实值函数\（\ psi ^ {-} \）和\（\ psi ^ {+} \）之间存在的关系。对于此类函数，我们建立了包含ρ -Riemann-Liouville分数积分算子的Hermite-Hadamard类型的新集合包含的负载。特别地，我们证明，除其他事项外，如果在凸集S上定义的集值函数is是m-多项式凸，则\（\ rho，\ epsilon> 0 \）和\（\ zeta，\ eta \在{\ mathbf {S}} \）中，然后
$$ \ begin {aligned} \ frac {m} {m + 2 ^ {-m} -1} \ Psi \ biggl（\ frac {\ zeta + \ eta} {2} \ biggr）＆\ supseteq \ frac { \ Gamma _ {\ rho} {\ epsilon + \ rho}} {（\ eta-\ zeta）^ {\ frac {\ epsilon} {\ rho}}} \ bigl [{_ {\ rho} {\ mathcal { J}}} _ {\ zeta ^ {+}} ^ {\ epsilon} \ Psi（\ eta）+ _ {\ rho} {\ mathcal {J}} _ {\ eta ^ {-}} ^ {\ epsilon } \ Psi（\ zeta）\ bigr] \\＆\ supseteq \ frac {\ Psi（\ zeta）+ \ Psi（\ eta）} {m} \ sum_ {p = 1} ^ {m} S_ {p} （\ epsilon; \ rho），\ end {aligned} $$Ψ是Lebesgue可积在\（[\ zeta，\ eta] \），\（S_ {p}（\ epsilon; \ rho）= 2- \ frac {\ epsilon} {\ epsilon + \ rho p}- frac {\ epsilon} {\ rho} \ mathcal {B}（\ frac {\ epsilon} {\ rho}，p + 1）\）和\（\ mathcal {B} \）是beta函数。我们扩展，归纳和补充文献中的现有结果。通过取\（m \ geq 2 \），我们得出了许多有趣的新包含物。我们希望本文获得的想法和结果将成为进一步研究的催化剂。
京公网安备 11010802027423号