Coefficient problems for the classes connected with a lune-shaped domain

The study of coefficient problems in geometric function theory has a rich history. The challenging and interesting task is to determine sharp bounds for various inequalities related to different classes of univalent functions. In this article, we derive sharp bounds for the Hankel determinant involving logarithmic coefficients, the Zalcman functional <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>J</m:mi> <m:mrow> <m:mn>2</m:mn> <m:mo>,</m:mo> <m:mn>3</m:mn> </m:mrow> </m:msub> <m:mo>:=</m:mo> <m:mrow> <m:mrow> <m:msub> <m:mi>a</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>⁢</m:mo> <m:msub> <m:mi>a</m:mi> <m:mn>3</m:mn> </m:msub> </m:mrow> <m:mo>-</m:mo> <m:msub> <m:mi>a</m:mi> <m:mn>4</m:mn> </m:msub> </m:mrow> </m:mrow> </m:math> {J2,3:=a₂a₃-a₄} , the Fekete–Szegö inequality and the moduli differences of the logarithmic coefficients as well as the logarithmic coefficients of the inverse function for the classes <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mi mathvariant="script">𝒮</m:mi> <m:mi mathvariant="normal">☾</m:mi> <m:mo>*</m:mo> </m:msubsup> </m:math> {S*-1.422638pt} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="script">𝒞</m:mi> <m:mi mathvariant="normal">☾</m:mi> </m:msub> </m:math> {C-1.422638pt} , respectively.