Generalized convexity of the Lambert $W$ function
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866911119395782656 |
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| author | Wang, Gendi |
| author_facet | Wang, Gendi |
| contents | 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 Hölder 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 $x=y$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_17409 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generalized convexity of the Lambert $W$ function Wang, Gendi Classical Analysis and ODEs 33E20, 26D07 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 Hölder 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 $x=y$. |
| title | Generalized convexity of the Lambert $W$ function |
| topic | Classical Analysis and ODEs 33E20, 26D07 |
| url | https://arxiv.org/abs/2508.17409 |