A q -Rogers–Ramanujan-based characterization of  q -bi-bounded turning functions

This study investigates the second Hankel determinant for a specific class of functions associated with the q -Rogers–Ramanujan polynomial. To accomplish this, we introduce a new subclass of q -bounded turning functions related to the q -Rogers–Ramanujan polynomial, denoted as <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="script">S</m:mi> <m:msub> <m:mrow> <m:mi mathvariant="script">R</m:mi> </m:mrow> <m:mrow> <m:mi mathvariant="normal">Σ</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>q</m:mi> <m:mo>,</m:mo> <m:mi>l</m:mi> <m:mo>,</m:mo> <m:mi>g</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> SR_Σ(q,l,g) . In complex analysis, establishing precise coefficient bounds for bi-univalent functions is crucial, as these coefficients influence the core properties of conformal mappings. In this work, we derive coefficient bounds for functions within the subclass <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="script">S</m:mi> <m:msub> <m:mrow> <m:mi mathvariant="script">R</m:mi> </m:mrow> <m:mrow> <m:mi mathvariant="normal">Σ</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>q</m:mi> <m:mo>,</m:mo> <m:mi>l</m:mi> <m:mo>,</m:mo> <m:mi>g</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> SR_Σ(q,l,g) , with most of the identified bounds being sharp. This research may stimulate further studies on sharp bounds for analytic functions associated with novel image domains.