Generalized convexity and inequalities

Abstract

Let ${\mathcal{R}}_{+}=(0,\infty )$ and let $\mathcal{M}$ be the family of all mean values of two numbers in ${\mathcal{R}}_{+}$ (some examples are the arithmetic, geometric, and harmonic means). Given $m_1,m_2\in \mathcal{M}$, we say that a function $f:{\mathcal{R}}_{+}\to {\mathcal{R}}_{+}$ is $(m_1,m_2)$-convex if $f(m_1(x,y))⩽m_2(f(x),f(y))$ for all $x,y\in {\mathcal{R}}_{+}$. The usual convexity is the special case when both mean values are arithmetic means. We study the dependence of $(m_1,m_2)$-convexity on $m_1$ and $m_2$ and give sufficient conditions for $(m_1,m_2)$-convexity of functions defined by Maclaurin series. The criteria involve the Maclaurin coefficients. Our results yield a class of new inequalities for several special functions such as the Gaussian hypergeometric function and a generalized Bessel function.