Abstract
The notion of m-polynomial convex interval-valued function $\Psi =[\psi ^{-}, \psi ^{+}]$
Ψ
=
[
ψ
−
,
ψ
+
]
is hereby proposed. We point out a relationship that exists between Ψ and its component real-valued functions $\psi ^{-}$
ψ
−
and $\psi ^{+}$
ψ
+
. 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, $\rho,\epsilon >0$
ρ
,
ϵ
>
0
and $\zeta,\eta \in {\mathbf{S}}$
ζ
,
η
∈
S
, then $$\begin{aligned} \frac{m}{m+2^{-m}-1}\Psi \biggl(\frac{\zeta +\eta }{2} \biggr)& \supseteq \frac{\Gamma _{\rho }(\epsilon +\rho )}{(\eta -\zeta )^{\frac{\epsilon }{\rho }}} \bigl[{_{\rho }{\mathcal{J}}}_{\zeta ^{+}}^{\epsilon } \Psi (\eta )+_{ \rho }{\mathcal{J}}_{\eta ^{-}}^{\epsilon }\Psi (\zeta ) \bigr] \\ & \supseteq \frac{\Psi (\zeta )+\Psi (\eta )}{m}\sum_{p=1}^{m}S_{p}( \epsilon;\rho ), \end{aligned}$$
m
m
+
2
−
m
−
1
Ψ
(
ζ
+
η
2
)
⊇
Γ
ρ
(
ϵ
+
ρ
)
(
η
−
ζ
)
ϵ
ρ
[
ρ
J
ζ
+
ϵ
Ψ
(
η
)
+
ρ
J
η
−
ϵ
Ψ
(
ζ
)
]
⊇
Ψ
(
ζ
)
+
Ψ
(
η
)
m
∑
p
=
1
m
S
p
(
ϵ
;
ρ
)
,
where Ψ is Lebesgue integrable on $[\zeta,\eta ]$
[
ζ
,
η
]
, $S_{p}(\epsilon;\rho )=2-\frac{\epsilon }{\epsilon +\rho p}- \frac{\epsilon }{\rho }\mathcal{B} (\frac{\epsilon }{\rho }, p+1 )$
S
p
(
ϵ
;
ρ
)
=
2
−
ϵ
ϵ
+
ρ
p
−
ϵ
ρ
B
(
ϵ
ρ
,
p
+
1
)
and $\mathcal{B}$
B
is the beta function. We extend, generalize, and complement existing results in the literature. By taking $m\geq 2$
m
≥
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.
Hermite-Hadamard-Type Fractional Inclusions for Interval-Valued Preinvex Functions
