Characters of odd degree and Thompson’s character degree theorem

For a positive integer m and a finite group G, let u2′(G,m)=∑χ∈Irr2′(G)χ(1)m∑χ∈Irr2′(G)χ(1)m-1,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} u_{2'}(G,m)=\frac{\sum _{\chi \in \mathrm{Irr}_{2'}(G)}\chi (1)^{m}}{\sum _{\chi \in \mathrm{Irr}_{2'}(G)}\chi (1)^{m-1}}, \end{aligned}$$\end{document}where Irr2′(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{Irr}_{2'}(G)$$\end{document} denotes the set of all complex irreducible characters of G of odd degrees. The Thompson’s theorem on character degrees states that if u2′(G,m)=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_{2'}(G,m)=1$$\end{document}, then G is 2-nilpotent. In this paper, we prove that if u2′(G,m)<3+3m3+3m-1,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} u_{2'}(G,m)< \frac{3+3^{m}}{3+3^{m-1}}, \end{aligned}$$\end{document}then G is 2-nilpotent. This is a strengthened version of Thompson’s theorem in terms of u2′(G,m)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_{2'}(G,m)$$\end{document}.