Sharp higher-order Schwarzian derivatives for Sakaguchi functions

In this paper, we investigate the higher-order Schwarzian derivatives σ n ( f ) for n = 3, 4, as defined by Schippers, for two significant subclasses of univalent functions related to the unified class of Sakaguchi functions. Specifically, we consider the classes <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msubsup> <m:mrow> <m:mi mathvariant="script">S</m:mi> </m:mrow> <m:mrow> <m:mi>S</m:mi> <m:mo>,</m:mo> <m:mi>e</m:mi> </m:mrow> <m:mrow> <m:mo>*</m:mo> </m:mrow> </m:msubsup> </m:math> SS,e and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msubsup> <m:mrow> <m:mi mathvariant="script">S</m:mi> </m:mrow> <m:mrow> <m:mi>S</m:mi> <m:mo>,</m:mo> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mo>*</m:mo> </m:mrow> </m:msubsup> </m:math> SS,L , which are characterized by the fact that the ratio <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mfrac> <m:mrow> <m:mn>2</m:mn> <m:mi>z</m:mi> <m:msup> <m:mrow> <m:mi>f</m:mi> </m:mrow> <m:mrow> <m:mo>′</m:mo> </m:mrow> </m:msup> <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>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:mo>−</m:mo> <m:mi>f</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mo>−</m:mo> <m:mi>z</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mfrac> </m:math> 2zf(z)f(z)-f(-z) is subordinate to the exponential function e z and the lemniscate of Bernoulli function <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msqrt> <m:mrow> <m:mn>1</m:mn> <m:mo>+</m:mo> <m:mi>z</m:mi> </m:mrow> </m:msqrt> </m:math> √1+z , respectively. Furthermore, we derive sharp bounds for the associated initial Grunsky coefficients for both classes.