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In this paper, we introduce a new convexity notion for inter-valued functions, known as Geometrically–Arithmetically Cr-h-convex functions (abbreviated as GA-Cr-h-CFs) and explore its properties. The family of GA-Cr-h-CFs simultaneously covers the family of GA-CFs, GA-h-CFs and GA-Cr-CFs. On the other hand, we demonstrate its equivalence with convex functions (CFs), h-CFs, Cr-CFs and Cr-h-CFs via exponential function. We also investigate necessary and sufficient conditions for GA-Cr-h-CFs via two corresponding real-valued GA-h-CFs. Due to significance and applications of mathematical inequalities, we apply our findings and propose several inequalities such as Jensen–Mercer inequality (JMI), Hermite–Hadamard inequality (HHI) and weighted Hermite–Hadamard-type inequalities (wHHIs) for GA-Cr-h-CFs. Through our results, we re-capture many known results and inequalities for subclasses of GA-Cr-h-CFs studied in the recent literature. [ABSTRACT FROM AUTHOR]

In the present study, we investigate the dynamic behavior of the fractional-order Bonhoeffer–van der Pol (BVP) oscillator. Previous studies on the integer-order BVP have shown that it exhibits mixed-mode oscillations (MMOs) with respect to the frequency of external forcing. We explore the effect of fractional-order on these MMOs and observe interesting phenomena. For fractional-order q 1 , we find that as we vary the frequency of external forcing, the system exhibits increasingly small amplitude oscillations. Eventually, as q 1 decreases, the MMOs disappear entirely, indicating that lower fractional orders eliminate the presence of MMOs in the BVP oscillator. On the other hand, for the fractional-order q 2 , we observe more complex MMOs compared to q 1. However, we find that the elimination of MMOs occurs with less variation from the integer order 1. Intriguingly, as we change q 2 , the fractional-order BVP oscillator undergoes a phenomenon known as a crisis, where the attractor expands and extreme events occur. Overall, our study highlights the rich dynamics of the fractional-order BVP oscillator and its ability to display various modes of oscillations and crises as the order is changed. [ABSTRACT FROM AUTHOR]