Convexity of ratios of the modified Bessel functions of the first kind with applications.

Let I ν x be the modified Bessel function of the first kind of order ν . Motivated by a conjecture on the convexity of the ratio W ν x = x I ν x / I ν + 1 x for ν > - 2 , using the monotonicity rules for a ratio of two power series and an elementary technique, we present fully the convexity of the functions W ν x , W ν x - x 2 / 2 ν + 4 and W ν x 1 / θ for θ ≥ 2 on 0 , ∞ in different value ranges of ν , which give an answer to the conjecture and extend known results. As consequences, some monotonicity results and new functional inequalities for W ν x are established. As applications, an open problem and a conjectures are settled. Finally, a conjecture on the complete monotonicity of W ν x 1 / θ for θ ≥ 2 is proposed.

Acute Myocardial Infarction in Chinese Medicine Hospitals in China from 2006 to 2013: An Analysis of 2311 Patients from Hospital Data.

OBJECTIVE: To assess the trends in characteristics, treatments, and outcomes of acute myocardial infarction (AMI) patients in tertiary Chinese medicine (CM) hospitals in China between 2006 and 2013. METHODS: This retrospective study was based on two nationwide epidemiological surveys of AMI in tertiary CM hospitals during 2 years (2006 and 2013). Patients admitted to the hospital for AMI were enrolled. Hospital records were used as the data source. Case data were derived regarding baseline characteristics, treatments, and outcomes of patients to assess changes from 2006 to 2013. Logistic regression was used to analyze the relationship between prognosis, general influencing factors of disease, and various treatment measures. RESULTS: Totally 26 tertiary CM hospitals in 2006 and 29 tertiary CM hospitals in 2013 (18 were repetitive) were surveyed. A total of 2,311 patients with AMI were enrolled (1,094 cases in 2006 and 1,217 cases in 2013). From 2006 to 2013, the mean age did not significantly change, but the proportion of patients younger than 65 years increased. The prevalence of risk factors such as hypertension, diabetes, and hyperlipidemia also increased. Significant increases were observed in primary percutaneous coronary intervention [20.48% (2006) vs. 24.90% (2013)] and revascularization [36.11% (2006) vs. 52.42% (2013)]. In-hospital mortality decreased from 11.15% in 2006 to 10.60% in 2013. A mortality logistic regression analysis identified reperfusion therapy [odds ratio (OR), 0.222; 95% confidence interval (CI), 0.106-0.464], Chinese patent medicines (OR, 0.394; 95% CI, 0.213-0.727), and CM decoctions (OR, 0.196; 95% CI, 0.109-0.353) as protective factors. CONCLUSION: Reperfusion and revascularization capabilities of tertiary CM hospitals have improved significantly, but in-hospital mortality has not significantly decreased. Efforts are needed to improve medical awareness of AMI and expand the use of CM to reduce in-hospital mortality in China.

Windschitl type approximation formulas for the gamma function.

In this paper, we present four new Windschitl type approximation formulas for the gamma function. By some unique ideas and techniques, we prove that four functions combined with the gamma function and Windschitl type approximation formulas have good properties, such as monotonicity and convexity. These not only yield some new inequalities for the gamma and factorial functions, but also provide a new proof of known inequalities and strengthen known results.

Triple Diamond-Alpha integral and Hölder-type inequalities.

In this paper, we first introduce the definition of triple Diamond-Alpha integral for functions of three variables. Therefore, we present the Hölder and reverse Hölder inequalities for the triple Diamond-Alpha integral on time scales, and then we obtain some new generalizations of the Hölder and reverse Hölder inequalities for the triple Diamond-Alpha integral. Moreover, using the obtained results, we give a new generalization of the Minkowski inequality for the triple Diamond-Alpha integral on time scales.

An accurate approximation formula for gamma function.

In this paper, we present a very accurate approximation for the gamma function: [Formula: see text] as [Formula: see text], and we prove that the function [Formula: see text] is strictly decreasing and convex from [Formula: see text] onto [Formula: see text], where [Formula: see text].

[Clinical features of acute myocardial infarction inpatients in 26 level three class A Chinese medicine hospitals in China and the investigation of treatment].

OBJECTIVE: To study the therapeutic state of acute myocardial infarction (AMI) inpatients in 26 level three class A Chinese medicine (CM) hospitals in China. METHODS: The case report form (CRF) was designed and used in this study. Totally 1 094 AMI patients were recruited from 26 level three class A CM hospitals from January 2006 to December 2006. The onset, the heart function, complications, previous history, and CM syndrome types were observed in AMI patients. Treatment by Western medicine (WM), CM intravenous preparation, CM decoction, and Chinese patent medicine were also observed in AMI patients. RESULTS: Blood stasis syndrome (854 cases, 78.06%) and stagnant phlegm syndrome (470 cases, 42.96%) were dominated in the CM sthenia syndrome. Qi deficiency syndrome (683 cases, 62.43%) and Xin-yin deficiency syndrome (231 cases, 21.12%) were dominated in the CM asthenia syndrome. Totally 355 patients (32.45%) received reperfusion. Of them, 224 (20.48%) received percutaneous coronary intervention (PCI). The top 5 often used Western drugs covered aspirin (1001 cases, 91.50%), low molecular heparin (917 cases, 83.82%), blood lipids regulators (833 cases, 76.14%), ACEI/ARB (822 cases, 75.14%), and nitrates (773 cases, 70.66%). Totally 946 patients (86.47%) used CM intravenous preparations. The CM intravenous preparations with the use frequency more than 5% were sequenced as Shenmai Injection (520 times, 54.97%), Salvia miltiorrhizae preparations (305 times, 32.24%), sanchi preparations (185 times, 19.56%), Shenfu Injection (68 times, 7.19%), Scutellarin Injection (64 times, 6.77%), and Acanthopanax Injection (29 times, 3.07%). Totally 575 patients (52.56%) used CM decoction. The main therapeutic methods covered activating blood circulation therapy (477 cases), qi benefiting therapy (332 cases), and phlegm resolving therapy (303 cases). Commonly used recipes covered Shengmai Powder, Taohong Siwu Decoction, Gualou Xiebai Baijiu Decoction, Erchen Decoction, and Sijunzi Decoction. Totally 394 patients (36.01%) used Chinese patent medicines. CONCLUSIONS: Qi deficiency syndrome and blood stasis syndrome were dominated in AMI patients. Great achievements have taken place in AMI reperfusion treatment and standardized treatment in CM hospitals. CM are extensively used. Clinical studies on treating AMI by CM should be further strengthened.