<p>In this paper, we introduce a new subclass of infinitesimal generator related to close-to-convex functions of order <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\alpha\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>. Infinitesimal generators play a pivotal role in analyzing the dynamics of one-parameter semigroups of holomorphic functions. Extending the classical Marx-Strohhacker type results that link function classes to infinitesimal generators, we investigate these relationships for the class of close-to-convex functions. Our work focuses on addressing an open problem posed by Tuneski et al. (2018), concerning the sharpest parameter <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\alpha\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> for which certain subclasses of analytic functions (specifically <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathcal {C}(\alpha ,\beta )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">C</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>) remain within the class of infinitesimal generators. Crucially, our study provides an improved and larger value for this parameter.</p>