In the paper, we introduce the generalized convex function on fractal sets (0 1)real line numbers and study the properties of the generalized convex function. Based on these properties, we establish the generalized Jensen's inequality and generalized Hermite-Hadamard's inequality. Furthermore, some applications are given.
The coordination of non-steroidal anti-inflammatory drugs (NSAIDs) to metal ions could improve the pharmaceutical efficacy of NSAIDs due to the unique characteristics of metal complexes. However, the structures of many metal-NSAID complexes are not well characterized; the functional mechanism and pharmaceutical effect of these complexes thus are not fully understood. In this work, three manganese-mefenamic acid (Mn-mef) complexes were synthesized and structurally characterized, and their pharmaceutical effect was investigated. We found that the three Mn-mef complexes exhibit higher lipoxygenase (LOX-1) inhibitory activity (IC50 values are 16.79, 38.63 and 28.06 μM, respectively) than the parent ligand mefenamic acid (78.67 μM). More importantly, the high inhibitory activity of the Mn-mef complexes is closely related to their spatial arrangements, which determine their interaction with LOX-1. Computer docking of the Mn-mef complexes with the LOX-1 confirms the experimental results: smaller Mn-mef complexes tend to bind competitively to LOX-1 at the substrate binding site, which is also analogous to the binding of the ligand mefenamic acid, while the bulky metal complexes inhibit the enzyme activity un-competitively. In addition, the Mn-mef complexes exhibit higher anti-oxidant activity than the ligand mefenamic acid. The higher anti-oxidant activity of the Mn-mef complexes apparently originated from the manganese centre of the complexes. We thus conclude that Mn-mef complexes enhance the anti-inflammatory activity of mefenamic acid by increasing their activity via changing their interaction mode with the enzymes, and/or by improving their anti-oxidant ability using metal ions. This work provides experimental evidence that with the unique spatial arrangements, metal-NSAID complexes could interact with the target enzymes more specifically and efficiently, which is superior to their parent NSAID ligand.
In this paper, we establish the Hermite-Hadamard-type inequalities for the generalized s-convex functions in the second sense on real linear fractal set R a ð0\a\1Þ:
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