In Hilbert space, we develop a novel framework to study for two new classes of convex function depending on arbitrary non-negative function, which is called a predominating ℏ-convex function and predominating quasiconvex function, with respect to η , are presented. To ensure the symmetry of data segmentation and with the discussion of special cases, it is shown that these classes capture other classes of η -convex functions, η -quasiconvex functions, strongly ℏ-convex functions of higher-order and strongly quasiconvex functions of a higher order, etc. Meanwhile, an auxiliary result is proved in the sense of κ -fractional integral operator to generate novel variants related to the Hermite–Hadamard type for p t h -order differentiability. It is hoped that this research study will open new doors for in-depth investigation in convexity theory frameworks of a varying nature.

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The main objective of this paper is to develop a novel framework to study a new fractional operator depending on a parameter K > 0, known as the generalized K-fractional integral operator. To ensure appropriate selection and with the discussion of special cases, it is shown that the generalized K-fractional integral operator generates other operators. Meanwhile, we derived notable generalizations of the reverse Minkowski inequality and some associated variants by utilizing generalized K-fractional integrals. Moreover, two novel results correlate with this inequality, and other variants associated with generalized K-fractional integrals are established. Additionally, this newly defined integral operator has the ability to be utilized for the evaluation of many numerical problems.

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In this article, the investigation is centered around the quantum estimates by utilizing quantum Hahn integral operator via the quantum shift operator ${}_{\eta}\psi_{\mathfrak{q}}(\zeta)=\mathfrak{q}\zeta+(1-\mathfrak{q})\eta$ ψ q η ( ζ ) = q ζ + ( 1 − q ) η , $\zeta\in[\mu,\nu]$ ζ ∈ [ μ , ν ] , $\eta=\mu+\frac{\omega}{(1-\mathfrak{q})}$ η = μ + ω ( 1 − q ) , $0<\mathfrak{q}<1$ 0 < q < 1 , $\omega\geq0$ ω ≥ 0 . Our strategy includes fractional calculus, Jackson’s $\mathfrak{q}$ q -integral, the main ideas of quantum calculus, and a generalization used in the frame of convex functions. We presented, in general, three types of fractional quantum integral inequalities that can be utilized to explain orthogonal polynomials, and exploring some estimation problems with shifting estimations of fractional order $\varrho_{1}$ ϱ 1 and the $\mathfrak{q}$ q -numbers have yielded fascinating outcomes. As an application viewpoint, an illustrative example shows the effectiveness of $\mathfrak{q}$ q , ω-derivative for boundary value problem.