Geometric investigations of a certain subclass of univalent functions

Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="script">H</m:mi> </m:math> H be the space of all functions that are analytic in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="double-struck">D</m:mi> </m:math> D . Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="script">A</m:mi> </m:math> A denote the family of all functions <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>f</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant="script">H</m:mi> </m:math> f∈ H and normalized by the conditions f (0) = 0 = f ′(0) − 1. Obradović and Ponnusamy have introduced the class <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="script">M</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>λ</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> M(λ ) such that the functions in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="script">M</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>λ</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> M(λ ) are univalent in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="double-struck">D</m:mi> </m:math> D whenever 0 < λ ≤ 1. The main objective of this paper is to examine the largest disks with sharp radius for which the quotient of analytic functions belong to the class <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="script">M</m:mi> </m:math> M , where the analytic functions belongs to some suitable subclasses of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="script">S</m:mi> </m:math> S , the class of univalent functions from <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="script">A</m:mi> </m:math> A .