Properties and Riemann-Liouville fractional Hermite-Hadamard inequalities for the generalized ( α , m ) (α,m)(\alpha,m) -preinvex functions

Abstract The authors first introduce the concepts of generalized ( α , m ) $(\alpha,m)$ -preinvex function, generalized quasi m-preinvex function and explicitly ( α , m ) $(\alpha, m)$ -preinvex function, and then provide some interesting properties for the newly introduced functions. The more important point is that we give a necessary and sufficient condition respecting the relationship between the generalized ( α , m ) $(\alpha, m)$ -preinvex function and the generalized quasi m-preinvex function. Second, a new Riemann-Liouville fractional integral identity involving twice differentiable function on m-invex is found. By using this identity, we establish the right-sided new Hermite-Hadamard-type inequalities via Riemann-Liouville fractional integrals for generalized ( α , m ) $(\alpha,m)$ -preinvex mappings. These inequalities can be viewed as generalization of several previously known results

Association between glycosylated hemoglobin A1c and bone biochemical markers in type 2 diabetic postmenopausal women: a cross-sectional study

Abstract Background type 2 diabetes mellitus (T2DM) is a complicated disease that can affect bone health, but the change in bone biochemical markers caused by T2DM was controversial, so the aim of this study was to investigate whether there was a discrepancy in the levels of bone biochemical markers between postmenopausal women with T2DM and non-diabetic women and to explore the relationship between the level of glycosylated hemoglobin A1c (HbA1c) and bone biochemical markers in these subjects. Methods A total of 237 type 2 diabetic postmenopausal women visiting the First Affiliated Hospital of Anhui Medical University from January 2017 to October 2018 and 93 healthy postmenopausal women were retrospectively enrolled. The differences in the levels of bone biochemical markers between patients and controls were analyzed by one-way ANOVA or chi-square test. The relationship between HbA1c and bone biochemical markers was analyzed by multivariate regression, forest plot and fitted curve. Results Bone formation markers including N-MID osteocalcin and procollagen type 1 amino-terminal pro-peptide (PINP) were decreased in postmenopausal women with T2DM compared to controls (17.42 ± 9.50 vs 23.67 ± 7.58, p  = 0.74 ng/ml) showed a significantly negative with HbA1c. No significant association was seen between HbA1c and other biochemical markers. Conclusions Our study found that bone formation was inhibited in postmenopausal women with T2DM, but bone resorption was not affected, and poor glycemic control was related to lower levels of bone formation, may increase the risk of bone fracture in postmenopausal women with T2DM