On the generalized convexity and concavity

A function $f:\mathbb{R}_+\to \mathbb{R}_+$ is $(m_1,m_2)$-convex (concave) if $f(m_1(x,y))\leq\thinspace(\geq)\thinspace m_2(f(x),f(y))$ for all $x,y\in \mathbb{R}_+=(0,\infty)$ and $m_1$ and $m_2$ are two mean functions. Anderson et al. [1] studies the dependence of $(m_1,m_2)$-convexity (concavity) on $m_1$ and $m_2$ and gave the sufficient conditions of $(m_1,m_2)$-convexity and concavity of a function defined by Maclaurin series. In this paper, we make a contribution to the topic and study the $(m_1,m_2)$-convexity and concavity of a function where $m_1$ and $m_2$ are identric and Alzer mean. As well, we prove a conjecture posed by Bruce Ebanks in [2].