In this paper, some new Jensen and Hermite-Hadamard inequalities for h-convex functions on fractal sets are obtained. Results proved in this paper may stimulate further research in this area.

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Background The actual effects of oxygen therapy on patients who have suffered a stroke are still unknown, and its recommendation as a routine measure in emergency services remains controversial. The aim of this study is to determine the effect of hyperoxia in functional recovery in patients with ischemic stroke who underwent intra-arterial mechanical thrombectomy (IAMT). Methods A prospective observational cohort study that included all adult patients consecutively admitted to the intensive care unit (ICU) due to an ischemic stroke in the anterior cerebral circulation and following an IAMT intervention, between 2010 and 2015. All patients were intubated and connected to mechanical ventilation for the intra-arterial therapy, receiving supplementary oxygen to achieve saturations above 94%. Two groups were established regarding oxygen partial pressure (paO 2 ) reached. It was based on a single ICU admission blood gas analysis. The hyperoxia group was defined as paO 2 > 120 mmHg. We measured functional recovery in each of the groups according to the modified Rankin scale after 90 days. Results For the analysis, a total of 333 patients were included. High levels of paO 2 were mostly related to higher scores in modified Rankin scale (mRS) after 90 days. There were 60.6% cases with mRS ≥ 4 and 70.6% with mRS ≥ 3 in the hyperoxia group, compared to 43.0% and 56.1% in the paO 2 ≤ 120 group, p < 0.01, respectively. Mortality was higher in the hyperoxia group, 28.6% vs 18.7%, p = 0.04. After regression adjustment by confounding factors, poor functional outcome was still significantly higher in the hyperoxia group, for both mRS ≥ 4 and mRS ≥ 3: OR 2.2.7, IC 95%, 1.22–4.23, p = 0.01 and OR 2.07, IC 95%, 1.05–4.029, p = 0.04, respectively. Both the National Institute of Health Stroke Scale Score (NIHSS) values at 24 h after the IAMT and the days of ICU stay were significantly higher in the hyperoxia group. Conclusions In patients with ischemic stroke in the anterior cerebral circulation treated with IAMT, we found an association between admission PaO2 > 120 mmHg and worse functional outcome 90 days after ischemic stroke, but this association needs further confirmation by other studies. Electronic supplementary material The online version of this article (10.1186/s13613-019-0533-8) contains supplementary material, which is available to authorized users.

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We establish some Hermite-Hadamard and Fejér type inequalities for the class of strongly reciprocally convex functions.

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