Convexity with respect to Hölder mean involving zero-balanced hypergeometric functions
In this note a \(H_{p,q}\)-convexity of the zero-balanced Gaussian hypergeometric function \(F(a,b;a+b;t)\) for some \(p\) and \(q\) is investigated. Let \(\varphi:I\to\mathbb R\) be a strictly monotonic function, then the function \(M_\varphi:I^2\to I\) defined by \(M_\varphi(r,s)=\varphi^{-1}((\varphi(r)+\varphi(s))/2)\) is called the quasi-arithmetic mean associated to \(\varphi\). In the special case when the function \(\varphi_p(r):(0,\infty)\to\mathbb R\) is defined by \(\varphi_p(r)=r^p\) for \(p\neq0\) and by \(\varphi_0(r)= \log r\) then \(M_{\varphi_p}(r,s)\equiv H_p(r,s)\) is the Hölder mean. For any two quasi-arithmetic means \(M_\varphi\), \(M_\psi\) associated to \(\varphi\), \(\psi\) defined on intervals \(I\), \(J\), respectively, a function \(f:I\to J\) is called \(M_{\varphi, \psi}\)-convex if \(f(M^{(\lambda)}_\varphi(r,s))\leq M^{(\lambda)}_\psi(f(r),f(s))\) for all \(r,s\in I\) and \(\lambda \in[0,1]\), where \(M^{(\lambda)}_\varphi(r,s)=\varphi^{-1}((1- \lambda)\varphi(r)+\lambda\varphi(s))\). In Theorem 1 it is proved that for all \(a\), \(b>0\), \(p\in(-\infty,1]\) and \(q\in[0,\infty)\) the hypergeometric function \(F(a,b;a+b;t)\), where \[ F(a,b;c;t)=\sum_{n=0}^\infty\frac{(a,n)(b,n)}{(c,n)}\frac{t^n}{n!}, \quad t\in(-1,1), \] is \(H_{p,q}\)-convex on \((0,1)\) with respect to the Hölder means \(H_p\) and \(H_q\). In Theorem 4 sufficient conditions for a differentiable log-convex function to be \(H_{p,q}\)-convex with respect to the Hölder means \(H_p\) and \(H_q\) are given.
scientific article; zbMATH DE number 5549425