In this manuscript, we established new (Hermite–Hadamard) $ \mathbb{HH} $-type inequalities for multiplicatively superquadratic functions, along with corresponding inequalities for the product and quotient of multiplicatively superquadratic, subquadratic, convex, and concave functions via Katugampola fractional integral operators. A notable advantage of these operators is their flexibility: By setting the parameter $ \rho = 1 $, the derived results naturally reduce to those involving the (Riemann–Liouville) $ \mathbb{RL} $ fractional integral operators. The validity and efficiency of the proposed results were demonstrated through graphical and numerical verifications. Furthermore, we explored applications of the established inequalities to special means, functions, and moments of random variables associated with the uniform probability distribution function (pdf). The developed $ \mathbb{HH} $-type inequalities offer precise bounds for fractional integral expressions that frequently appear in nonlinear boundary value problems (BVPs). These bounds help control the behavior of solution operators. Consequently, they support the analysis of existence, uniqueness, and stability of solutions in both classical and fractional BVPs.
Citation: Saad Ihsan Butt, Ghulam Jallani, Dawood Khan, Mohammed Alammar, Youngsoo Seol. Generalized multiplicatively fractional Hermite–Hadamard-type inequalities with applications to special functions and probability theory[J]. AIMS Mathematics, 2026, 11(10): 35366-35402. doi: 10.3934/math.20261373
In this manuscript, we established new (Hermite–Hadamard) $ \mathbb{HH} $-type inequalities for multiplicatively superquadratic functions, along with corresponding inequalities for the product and quotient of multiplicatively superquadratic, subquadratic, convex, and concave functions via Katugampola fractional integral operators. A notable advantage of these operators is their flexibility: By setting the parameter $ \rho = 1 $, the derived results naturally reduce to those involving the (Riemann–Liouville) $ \mathbb{RL} $ fractional integral operators. The validity and efficiency of the proposed results were demonstrated through graphical and numerical verifications. Furthermore, we explored applications of the established inequalities to special means, functions, and moments of random variables associated with the uniform probability distribution function (pdf). The developed $ \mathbb{HH} $-type inequalities offer precise bounds for fractional integral expressions that frequently appear in nonlinear boundary value problems (BVPs). These bounds help control the behavior of solution operators. Consequently, they support the analysis of existence, uniqueness, and stability of solutions in both classical and fractional BVPs.
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Figures(9) / Tables(6)
Saad Ihsan Butt, Ghulam Jallani, Dawood Khan, Mohammed Alammar, Youngsoo Seol. Generalized multiplicatively fractional Hermite–Hadamard-type inequalities with applications to special functions and probability theory[J]. AIMS Mathematics, 2026, 11(10): 35366-35402. doi: 10.3934/math.20261373
Graphical depiction of Theorem 3.1 via $ \Psi(\mathcal{W}) = \exp\{\mathcal{W}^{4}\} $ for $ {\vartheta} = 0 $, $ {\upsilon} = 1 $, $ \rho = 0.5 $ and $ \alpha\in(0, 1) $
Graphical depiction of Theorem 3.1 via $ \Psi(\mathcal{W}) = \exp\{\mathcal{W}^{4}\} $ for $ {\vartheta} = 0 $, $ {\upsilon} = 1 $, $ \rho = 0.6, $ and $ \alpha\in(0, 1) $
Graphical depiction of Theorem 3.1 via $ \Psi(\mathcal{W}) = \exp\{\mathcal{W}^{4}\} $ for $ {\vartheta} = 0 $, $ {\upsilon} = 1 $, $ \rho = 0.9, $ and $ \alpha\in(0, 1) $
Graphical depiction of Theorem 3.1 via $ \Psi(\mathcal{W}) = \exp\{\mathcal{W}^{4}\} $ for $ {\vartheta} = 0 $, $ {\upsilon} = 1 $, $ \alpha = 0.2, $ and $ \rho\in(0, 1) $
Graphical depiction of Theorem 3.1 via $ \Psi(\mathcal{W}) = \exp\{\mathcal{W}^{4}\} $ for $ {\vartheta} = 0 $, $ {\upsilon} = 1 $, $ \alpha = 0.6, $ and $ \rho\in(0, 1) $
Graphical depiction of Theorem 3.1 via $ \Psi(\mathcal{W}) = \exp\{\mathcal{W}^{4}\} $ for $ {\vartheta} = 0 $, $ {\upsilon} = 1 $, $ \alpha = 0.9, $ and $ \rho\in(0, 1) $
Graphical depiction of Theorem 3.1 for $ \Psi(\mathcal{W}) = \exp\{\mathcal{W}^{4}\} $, $ {\vartheta} = 0 $, $ {\upsilon} = 1 $, $ \alpha\in(0, 1), $ and $ \rho\in(0, 1) $
Graphical description of Theorem 3.13 for $ \Psi(\mathcal{W}) = exp\{\mathcal{W}^{4}\} $, $ {\varphi}(\mathcal{W}) = \exp\{\mathcal{W}^{6}\} $, $ {\vartheta} = 0 $, $ {\upsilon} = 1 $, $ \alpha\in(0, 1), $ and $ \rho\in(0, 1) $
Graphical description of Theorem 3.18 for $ {\varphi}(\mathcal{W}) = \exp\{-\mathcal{W}^{4}\} $, $ \Psi(\mathcal{W}) = \exp\{\mathcal{W}^{6}\} $, $ {\vartheta} = 0 $, $ {\upsilon} = 1 $, $ \alpha\in(0, 1), $ and $ \rho\in(0, 1) $
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