Xuexiao You

This paper develops and extends Hermite–Hadamard-type inequalities for log-harmonically [Formula: see text]-convex functions using multiplicative Riemann–Liouville fractional integrals within the context of multiplicative calculus. In particular, we establish new inequalities for the products and quotients of log-harmonically [Formula: see text]-convex functions. Furthermore, new inequalities involving multiplicative Riemann–Liouville fractional integrals are developed for combinations of log-harmonically [Formula: see text]-convex and harmonically convex functions, including their products and quotients. Numerical examples and graphical illustrations confirm the validity and practical relevance of the results.

The acquisition and transmission of magnetic resonance (MR) images are frequently affected by random noise pollution, which hampers the diagnosis of diseases by doctors or automated systems. Hence, the search for advanced denoising methods is of great research interest, particularly in magnetic resonance imaging (MRI) denoising models, which are based on deep learning networks and achieve satisfactory results. However, the mining of noisy contextual information and effective information-guided transfer are often neglected in the denoising process, which leads to poor extraction of information at different scales, and poor retention of details. This greatly hinders the further development of MR image denoising methods. Here, we propose a denoising method, MSDRA-Net, for the mining and exploitation of different hierarchical noise features and construct a multi-scale dilated residual (MSDR) structure to transfer and retain noise information at different levels across the layer. Next, a contextual guidance attention (CGA) module guides and transfers contextual information, utilizing the features learnt from different layers of the model as weights. A reconstruction refinement block (RRB) is utilized to construct clean images from the obtained noise bias and the given noisy images. Experiments on simulated and clinical MRI data validated the effectiveness of our method, which demonstrated a superior performance compared to several state-of-the-art methods.

We introduce the forward (backward) gH-difference operator of interval sequences, and establish some new discrete Opial type inequalities for interval-valued functions. Further, we obtain generalizations of classical discrete Opial type inequalities. Some examples are presented to illustrate our results.

In 2021, Ullah et al., introduced a new approach for the derivation of results for Jensen’s inequality. The purpose of this article, is to use the same technique and to derive improvements of Slater’s inequality. The planned improvements are demonstrated in both discrete as well as in integral versions. The quoted results allow us to provide relationships for the power means. Moreover, with the help of established results, we present some estimates for the Csiszár and Kullback–Leibler divergences, Shannon entropy, and Bhattacharyya coefficient. In addition, we discuss some additional applications of the main results for the Zipf–Mandelbrot entropy.

In this work, two generalized quantum integral identities are proved by using some parameters. By utilizing these equalities, we present several parameterized quantum inequalities for convex mappings. These quantum inequalities generalize many of the important inequalities that exist in the literature, such as quantum trapezoid inequalities, quantum Simpson’s inequalities, and quantum Newton’s inequalities. We also give some new midpoint-type inequalities as special cases. The results in this work naturally generalize the results for the Riemann integral.

In this paper, we prove Hermite–Hadamard–Mercer inequalities, which is a new version of the Hermite–Hadamard inequalities for harmonically convex functions. We also prove Hermite–Hadamard–Mercer-type inequalities for functions whose first derivatives in absolute value are harmonically convex. Finally, we discuss how special means can be used to address newly discovered inequalities.

We introduce and investigate the Henstock-Kurzweil (HK) integral for Riesz-space-valued functions on time scales. Some basic properties of the HK delta integral for Riesz-space-valued functions are proved. Further, we prove uniform and monotone convergence theorems.