Abstract
We establish new quantum Hermite–Hadamard and midpoint types inequalities via a parameter $\mu \in [0,1]$
μ
∈
[
0
,
1
]
for a function F whose $|{}_{\alpha }D_{q}F|^{u}$
|
α
D
q
F
|
u
is η-quasiconvex on $[\alpha ,\beta ]$
[
α
,
β
]
with $u\geq 1$
u
≥
1
. Results obtained in this paper generalize, sharpen, and extend some results in the literature. For example, see (Noor et al. in Appl. Math. Comput. 251:675–679, 2015; Alp et al. in J. King Saud Univ., Sci. 30:193–203, 2018) and (Kunt et al. in Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 112:969–992, 2018). By choosing different values of μ, loads of novel estimates can be deduced. We also present some illustrative examples to show how some consequences of our results may be applied to derive more quantum inequalities.
The existence, uniqueness and asymptotic estimates of solutions for third-order full nonlinear singularly perturbed vector boundary value problems
