@article{Chen_2014, title={A Note on a Bound of Adan-Bante}, volume={66}, url={https://umj.imath.kiev.ua/index.php/umj/article/view/2195}, abstractNote={Let <em class="a-plus-plus">G</em> be a finite solvable group and let <em class="a-plus-plus">χ</em> be a nonlinear irreducible (complex) character of <em class="a-plus-plus">G.</em> Also let <span class="a-plus-plus inline-equation id-i-eq1"> <span class="a-plus-plus equation-source format-t-e-x">\( \eta \)</span> </span>(<em class="a-plus-plus">χ</em>) be the number of nonprincipal irreducible constituents of <span class="a-plus-plus inline-equation id-i-eq2"> <span class="a-plus-plus equation-source format-t-e-x">\( \upchi \overline{\upchi} \)</span> </span> <em class="a-plus-plus">,</em> where <span class="a-plus-plus inline-equation id-i-eq3"> <span class="a-plus-plus equation-source format-t-e-x">\( \overline{\upchi} \)</span> </span> denotes the complex conjugate of <em class="a-plus-plus">χ.</em> Adan-Bante proved that there exist constants <em class="a-plus-plus">C</em> and <em class="a-plus-plus">D</em> such that dl (<em class="a-plus-plus">G/</em> ker <em class="a-plus-plus">χ</em>) <em class="a-plus-plus">≤ C</em> <span class="a-plus-plus inline-equation id-i-eq4"> <span class="a-plus-plus equation-source format-t-e-x">\( \eta \)</span> </span>(<em class="a-plus-plus">χ</em>) +<em class="a-plus-plus">D.</em> In the present work, we establish a bound lower than the Adan-Bante bound for <span class="a-plus-plus inline-equation id-i-eq5"> <span class="a-plus-plus equation-source format-t-e-x">\( \eta \)</span> </span>(<em class="a-plus-plus">χ</em>) <em class="a-plus-plus">></em&gt; 2}, number={7}, journal={Ukrains’kyi Matematychnyi Zhurnal}, author={Chen, Xiaoyou}, year={2014}, month={Jul.}, pages={1006–1008} }