In this paper, we take into account the notion of strongly multiplicative convex function and derive integral inequalities of Hermite-Hadamard ( $H.H$) type for such a function in the frame of multiplicative calculus. We also develop integral inequalities of $H.H$ type for product and quotient of strongly multiplicative convex and strongly multiplicative concave functions via multiplicative calculus. All the results of the theorems are verified graphically by taking into account some reasonable examples. Additionally, we establish the inequalities of the Milne type for strongly multiplicative convex functions.

We consider time-inhomogeneous Hawkes processes with an exponential kernel, and we analyze some properties of the model. Time-inhomogeneity for the Hawkes process is indispensable for short rate models or for other calibration purposes, while financial applications for the time-homogeneous case already well known. Distributional properties for such a model generate computational tractability for a financial application. In this paper, moments and the Laplace transform of time-inhomogeneous Hawkes processes are obtained from the distributional properties of the underlying processes. As an applications to finance, we investigate the pricing formula for zero-coupon bonds when short-term interest rates are governed by the time-inhomogeneous Hawkes process. Numerical illustrations are also provided. As an illustrative example, we apply the derived moments and Laplace transform of time-inhomogeneous Hawkes processes to the pricing of zero-coupon bonds within a financial context. By considering the short-term interest rate as driven by inhomogeneous Hawkes processes, we develop explicit formulae for valuing zero-coupon bonds. This application is particularly relevant for modeling interest rate dynamics in real-world scenarios, allowing for a more nuanced understanding of pricing dynamics. Through numerical illustrations, we demonstrate the computational tractability of our approach, showcasing its practical utility for financial practitioners and providing insights into the intricate interplay between time-inhomogeneous Hawkes processes and bond pricing in dynamic markets.

In this paper, we introduced a novel class of superquadraticity, termed multiplicatively (superquadratic interval-valued functions) superquadratic $\mathcal{I}\mathcal{V}\mathcal{F}$s and investigated their unique properties using interval order relations and (Riemann-Liouville) $\mathcal{R}\mathcal{.}\mathcal{L}$-fractional integral operators. By employing the framework of multiplicative calculus, we established new fractional integral inequalities specifically of Hermite-Hadamard ( $\mathcal{H}\mathcal{.}\mathcal{H}$) type, for these superquadratic $\mathcal{I}\mathcal{V}\mathcal{F}$. Furthermore, we extended our analysis to derive fractional inequalities for the product and quotient of multiplicatively superquadratic and subquadratic $\mathcal{I}\mathcal{V}\mathcal{F}$ functions within the same calculus setting. By setting $\ell =1$, the results naturally reduced to their corresponding integer-order forms for multiplicatively superquadratic $\mathcal{I}\mathcal{V}\mathcal{F}$. To validate our theoretical findings, we present numerical computations and graphical illustrations based on several illustrative examples, showcasing the practical utility and robustness of the results. In addition, we explored potential applications of these inequalities, particularly in the context of linear combinations of special means. This provides a fresh perspective on superquadratic $\mathcal{I}\mathcal{V}\mathcal{F}s$ and expands the scope of multiplicative convex analysis. The results presented in this work are entirely new within the framework of fractional multiplicative calculus and have not been previously reported in the literature. We believe that this study will pave the way for future research, offering a deeper understanding of convexity phenomena and powerful tools for mathematical modeling involving $\mathcal{I}\mathcal{V}\mathcal{F}s$.

Hahn multiplicative calculus is the generalization of quantum multiplicative ( $\mathit{q}$-multiplicative) calculus. In this manuscript, we defined novel definitions for derivative and definite integral called left Hahn multiplicative derivative and definite integral in the Hahn multiplicative calculus. In addition, we derived fundamental results for this newly defined integral. Furthermore, we constructed left Hahn multiplicative Hermite-Hadamard inequalities. Additionally, we defined new definitions for the derivative and definite integral in the Hahn calculus, which enabled us to define further definitions in Hahn multiplicative calculus called the right Hahn multiplicative derivative and definite integral. Moreover, we construct the power rule of the newly defined definite integral in the Hahn calculus, which assisted us in deriving the right Hahn multiplicative Hermite-Hadamard inequalities. Finally, we gave an application of the newly established Hermite-Hadamard inequalities through an example wherein it could be seen that these inequalities were crucial for finding the lower and upper bounds of the range of those functions whose Hahn multiplicative definite integrals were very difficult to find.

The goal of this study is to improve some known results related to Boole's type inequalities that use five points (Boole's rule). We first prove an important auxiliary identity connected to these inequalities. Using this auxiliary identity, we develop new Boole's type inequalities by applying a differentiable convex function within the setting of local fractional calculus. In this work, we study different types of functions, including convex, bounded, and Lipschitz functions over fractal sets. Additionally, a feedforward Artificial Neural Network (ANN) was used to approximate the left-hand side and right-hand side of fractal Boole-type inequalities. The model takes two input values and passes them through hidden layers to produce two outputs as predictions. This type of ANN is widely used because it can learn complex relationships from data without needing any fixed formulas. In this work, we apply an ANN model for the first time to predict the bounds of inequalities in fractal dimensions, which is an important outcome of our study. The ReLU activation function was applied to help the model learn nonlinear patterns, while training was carried out using the Mean Squared Error (MSE) loss and the Adam optimizer for stable and efficient learning. The network was trained for 500 epochs, and its performance was evaluated using loss curves. Finally, 3-dimensional surface plots were created to compare the predicted and actual inequality values. We also present examples and applications to show the usefulness of our main results.

Superquadraticity is a generalization of convexity that yields more refined results compared to those obtained through convexity alone. In this work, we established, for the first time, a class of superquadratic stochastic processes and explored their fundamental properties. Based on these properties, we derived Jensen's and (Hermite-Hadamard) $\mathbb{H}\mathbb{H}$'s type inequalities, along with their fractional counterparts, in the context of mean-square stochastic (Riemann-Liouville) $\mathbb{R}\mathbb{.}\mathbb{L}$ fractional integrals. The validity of our findings was supported by graphical illustrations using suitable examples. Furthermore, we extended the applicability of our results to information theory by introducing several stochastic divergence measures.