Generalized convexity and inequalities (Q2642186)

Let \[ F(x):=1+\sum^\infty_{n=1}\frac{a(a+1)\dots (a+n-1)b(b+1)\dots (b+n-1)}{c(c+1)\dots (c+n-1)n!}x^n\quad (| x|<1;\;a,b,c>0) \] be the Gauss hypergeometric function. One of the three main results offered is that, if \(a+b\geq c>2ab\), \(2c\geq 2a+2b-1,\) then \(1/F\) is concave (in the paper ``on \((0,\infty)\)'' even though \(F\) has been defined only for \(| x| <1\)). To achieve proofs the authors define means (actually, symmetric means, homogeneous of degree 1) and \(MN\)-convexity in the usual way (\(f[M(x,y)]\leq N[f(x),f(y)]\)), give several known examples of \(MN\)-convexity and, in the main part of the paper, \(MN\)-convexity results for functions defined by their Maclaurin series.