标题： Lambert$W$函数的广义凸性 显示英文标题

标题： Generalized convexity of the Lambert $W$ function

摘要： 本文研究了Lambert$W$函数的广义凸性性质，该函数定义为$W(z)e^{W(z)}=z$的解。 重点关注与Hölder均值相关的$H_{p,q}$-凸性和凹性，我们推导了$W$在区间$(0,+\infty)$上表现出严格$H_{p,q}$-凸性或凹性的必要充分条件。 主要结果以特定参数区域$(p,q)$平面来描述这些性质。 涉及调和平均、几何平均和算术平均的不等式被建立，等号仅在$x=y$时成立。 显示英文摘要

摘要： This paper investigates the generalized convexity properties of the Lambert $W$ function, defined as the solution to $W(z)e^{W(z)}=z$. Focusing on $H_{p,q}$-convexity and concavity with respect to H\"older means, we derive necessary and sufficient conditions for $W$ to exhibit strict $H_{p,q}$-convexity or concavity on the interval $(0,+\infty)$. The main result characterizes these properties in terms of specific parameter regions $(p,q)$-plane. Inequalities involving harmonic, geometric, and arithmetic means are established, with equalities holding only when $x=y$.