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Generalized convexity and inequalities involving special functions. (English) Zbl 1132.52001

A function \(M:(0,\infty)\times (0,\infty)\rightarrow (0,\infty)\) is called a mean value if \(M(x,x)=x\), \(M(x,y)=M(y,x)\), \(x<M(x,y)<y\) for \(x<y\), and \(M(ax,ay)=aM(x,y)\) for all \(a>0\). For given mean values \(M\) and \(N\), the function \(f:[0,\infty)\rightarrow [0,\infty)\) is called \(MN\)-convex (concave) if \(f(M(x,y))\leq(\geq)N(f(x),f(y))\) for \(x,y\in [0,\infty)\).
The authors show a relation between above mentioned generalized convexity and super-multiplicative (sub-multiplicative) property (\(f(x)f(y)\leq(\geq)f(x,y)\) for all \(x,y\geq 1\)). The authors also discuss some generalized convexity and inequalities involving some special functions such as Gaussian hypergeometric function, the generalized \(\eta-\)distortion function \(\eta_K^a(x)\) and the generalized Grötzsch function \(\mu_a(r)\).

MSC:

52A01 Axiomatic and generalized convexity
26D07 Inequalities involving other types of functions
33C05 Classical hypergeometric functions, \({}_2F_1\)
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