INEQUALITIES OF J-P-S-F TYPE

By means of the theory of majorization and under the proper hypotheses, the following inequalities of Jensen-Pečarić-Svrtan-Fan (Abbreviated as J-P-S-F) type are established

The optimization for the inequalities of power means

Let be the th power mean of a sequence of positive real numbers, where , and . In this paper, we will state the important background and meaning of the inequality ; a necessary and sufficient condition and another interesting sufficient condition that the foregoing inequality holds are obtained; an open problem posed by Wang et al. in 2004 is solved and generalized; a rulable criterion of the semipositivity of homogeneous symmetrical polynomial is also obtained. Our methods used are the procedure of descending dimension and theory of majorization; and apply techniques of mathematical analysis and permanents in algebra.</p

The risk prediction of intergenerational transmission of overweight and obesity between mothers and infants during pregnancy

Abstract Background Overweight and obesity in mothers before pregnancy lead to overweight and obesity in their offspring, which is the main form of intergenerational transmission of overweight and obesity in early life. Many factors, especially non-genetic factors, may influence intergenerational transmission, but little prediction research has been conducted. Therefore, we analyzed the status of intergenerational transmission in maternal and infant overweight and obesity. Second, we explored the factors during the pregnancy that might affect the the intergenerational transmission; According to the two application scenarios of pregnancy screen and self-management, risk prediction models for pregnant women were carried out. Methods Based on a prospective birth cohort, a total of 908 mothers and offspring were followed up during early life. Follow-up visits were performed at the first trimester, second trimester, third trimester, delivery, 42 days after delivery, and 6 months and 12 months of age. The investigation methods included questionnaire survey, physical examination, biological sample collection and clinical data collection. In terms of risk prediction, univariate analysis was used to screen candidate predictors. Second, multivariable Cox proportional hazard regression models were used to determine the final selected predictors. Third, the corresponding histogram models were drawn, and then the 10-fold cross-validation methods were used for internal verification. Results Regarding intergenerational transmission of overweight and obesity between mothers and infants during pregnancy, the risk prediction model for pregnancy screen was constructed. The model established: h(t|X) = h0(t)exp.(− 0.95 × (Bachelor Degree or above) + 0.75 × (Fasting blood glucose in the second trimester) + 0.89 × (Blood pressure in the third trimester) + 0.80 × (Cholesterol in third trimester) + 0.55 × (Abdominal circumference in third trimester))., with good discrimination (AUC = 0.82) and calibration (Hosmer–Lemeshow2 = 4.17). The risk prediction model for self-management was constructed. The model established: h(t|X) = h0(t)exp. (0.98 × (Sedentary >18METs) + 0.88 × (Sleep index≥8) + 0.81 × (Unhealthy eating patterns Q3/Q4) + 0.90 × (Unhealthy eating patterns Q4/Q4) + 0.85 × (Depression)), with good discrimination (AUC = 0.75) and calibration (Hosmer–Lemeshow2 = 3.81). Conclusions The risk predictions of intergenerational transmission of overweight and obesity between mothers and infants were performed for two populations and two application scenarios (pregnancy screening and home self-management). Further research needs to focus on infants and long-term risk prediction models

Optimal sublinear inequalities involving geometric and power means

summary:There are many relations involving the geometric means $G_n(x)$ and power means $[A_n(x^{\gamma })]^{1/\gamma }$ for positive $n$-vectors $x$. Some of them assume the form of inequalities involving parameters. There then is the question of sharpness, which is quite difficult in general. In this paper we are concerned with inequalities of the form $(1-\lambda )G_n^{\gamma }(x)+\lambda A_n^{\gamma }(x)\geq A_n(x^{\gamma })$ and $(1-\lambda )G_n^{\gamma }(x)+\lambda A_n^{\gamma }(x)\leq A_n(x^{\gamma })$ with parameters $\lambda \in \Bbb R$ and $\gamma \in (0,1).$ We obtain a necessary and sufficient condition for the former inequality, and a sharp condition for the latter. Several applications of our results are also demonstrated

Additional file 1 of The risk prediction of intergenerational transmission of overweight and obesity between mothers and infants during pregnancy

Additional file 1