Coefficient functionals for Sakaguchi-type-Starlike functions subordinated to the three-leaf function

Abstract A challenging part of studying geometric function theory is figuring out the sharp boundaries for coefficient-related problems that crop up in the Taylor-Maclaurin series of univalent functions. The objective of this article is to define the families of Sakaguchi-type starlike functions with respect to symmetric points based on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>q</m:mi> </m:math> q -operator and to investigate the precise boundaries for a range of issues, including the first three initial coefficient estimates, Fekete-Szegö type and the Zalcman inequalities by subordinating to the function of the three leaves. Additionally, we discussed initial coefficients and Fekete-Szegö type inequalities for functions of the form <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi class="MJX-tex-caligraphic" mathvariant="script">ℱ</m:mi> </m:mrow> <m:mrow> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> </m:math> {{\mathcal{ {\mathcal F} }}}^{-1} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mfrac> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mrow> <m:mi class="MJX-tex-caligraphic" mathvariant="script">ℱ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:mfrac> </m:math> \frac{z}{{\mathcal{ {\mathcal F} }}\left(z)} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mfrac> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:mfrac> <m:mi>log</m:mi> <m:mfenced open="(" close=")"> <m:mrow> <m:mfrac> <m:mrow> <m:mi class="MJX-tex-caligraphic" mathvariant="script">ℱ</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mi>z</m:mi> </m:mrow> </m:mfrac> </m:mrow> </m:mfenced> </m:math> \frac{1}{2}\log \left(\phantom{\rule[-0.75em]{}{0ex}},\frac{{\mathcal{ {\mathcal F} }}(z)}{z}\right) linked with the function of the three leaves.