Functional Inequalities Involving Numerical Differentiation Formulas of Order Two

We write expressions connected with numerical differentiation formulas of order 2 in the form of Stieltjes integral, then we use Ohlin lemma and Levin–Stechkin theorem to study inequalities connected with these expressions. In particular, we present a new proof of the inequality f(x+y2)≤1(y-x)2∫xy∫xyf(s+t2)dsdt≤1y-x∫xyf(t)dtsatisfied by every convex function f:R→R and we obtain extensions of this inequality. Then we deal with non-symmetric inequalities of a similar form

Heritability of a skeletal biomarker of biological aging

Changes in the skeletal system, which include age-related bone and joint remodeling, can potentially be used as a biomarker of biological aging. The aim of the present study was to investigate the extent and mode of inheritance of skeletal biomarker of biological aging—osseographic score (OSS), in a large sample of ethnically homogeneous pedigrees. The investigated cohort comprised 359 Chuvashian families and included 787 men aged 18–89 years (mean 46.9) and 723 women aged 18–90 years (mean 48.5). The TOSS - transformed OSS standardized in 5-year age groups for each sex, was analyzed as a BA index. We evaluated familial correlations and performed segregation analysis. Results of our study suggest the familial aggregations of TOSS variation in the Chuvashian pedigrees. In a segregation analysis we found a significant major gene (MG) effect in the individual’s TOSS with a dominant most parsimonious model (H2 = 0.32). Genetic factors (MG genotypes) explained 47% of the residual OSS variance after age adjustment and after including sex-genotype interaction, they explained 52% of the residual variance. Results of our study also indicated that the inherited difference in the skeletal aging pattern in men lies mostly in the rate of aging, but in women in the age of the onset of the period of visible skeletal changes

Ohlin’s lemma and some inequalities of the Hermite–Hadamard type

Using the Ohlin lemma on convex stochastic ordering we prove inequalities of the Hermite–Hadamard type. Namely, we determine all numbers a,α,β∈[0,1] such that for all convex functions f the inequality af(αx+(1−α)y)+(1−a)f(βx+(1−β)y)≤1y−x∫xyf(t)dt is satisfied and all a,b,c,α∈(0,1) with a + b + c = 1 for which we have af(x)+bf(αx+(1−α)y)+cf(y)≥1y−x∫xyf(t)d