Radius Problems for Starlike Functions Associated with the Tan Hyperbolic Function
Khalil Ullah, Saira Zainab, Muhammad Arif, Maslina Darus, Meshal Shutaywi
Abstract
The aim of this particular article is at studying a holomorphic function <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"> <a:mi>f</a:mi> </a:math> defined on the open-unit disc <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" id="M2"> <c:mi mathvariant="fraktur">D</c:mi> <c:mo>=</c:mo> <c:mfenced open="{" close="}"> <c:mrow> <c:mi mathvariant="fraktur">z</c:mi> <c:mo>∈</c:mo> <c:mi>ℂ</c:mi> <c:mo>:</c:mo> <c:mfenced open="|" close="|"> <c:mrow> <c:mi mathvariant="fraktur">z</c:mi> </c:mrow> </c:mfenced> <c:mo><</c:mo> <c:mn>1</c:mn> </c:mrow> </c:mfenced> </c:math> for which the below subordination relation holds <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" id="M3"> <l:mi mathvariant="fraktur">z</l:mi> <l:msup> <l:mrow> <l:mi>f</l:mi> </l:mrow> <l:mrow> <l:mo>′</l:mo> </l:mrow> </l:msup> <l:mfenced open="(" close=")"> <l:mrow> <l:mi mathvariant="fraktur">z</l:mi> </l:mrow> </l:mfenced> <l:mo>/</l:mo> <l:mi>f</l:mi> <l:mfenced open="(" close=")"> <l:mrow> <l:mi mathvariant="fraktur">z</l:mi> </l:mrow> </l:mfenced> <l:mo>≺</l:mo> <l:msub> <l:mrow> <l:mi>q</l:mi> </l:mrow> <l:mrow> <l:mn>0</l:mn> </l:mrow> </l:msub> <l:mfenced open="(" close=")"> <l:mrow> <l:mi mathvariant="fraktur">z</l:mi> </l:mrow> </l:mfenced> <l:mo>=</l:mo> <l:mn>1</l:mn> <l:mo>+</l:mo> <l:mi mathvariant="normal">tan</l:mi> <l:mi>h</l:mi> <l:mfenced open="(" close=")"> <l:mrow> <l:mi mathvariant="fraktur">z</l:mi> </l:mrow> </l:mfenced> <l:mo>.</l:mo> </l:math> The class of such functions is denoted by <bb:math xmlns:bb="http://www.w3.org/1998/Math/MathML" id="M4"> <bb:msubsup> <bb:mrow> <bb:mi mathvariant="fraktur">S</bb:mi> </bb:mrow> <bb:mrow> <bb:mi mathvariant="normal">tan</bb:mi> <bb:mi>h</bb:mi> </bb:mrow> <bb:mrow> <bb:mo>∗</bb:mo> </bb:mrow> </bb:msubsup> <bb:mo>.</bb:mo> </bb:math> The radius constants of such functions are estimated to conform to the classes of starlike and convex functions of order <fb:math xmlns:fb="http://www.w3.org/1998/Math/MathML" id="M5"> <fb:mi>β</fb:mi> </fb:math> and Janowski starlike functions, as well as the classes of starlike functions associated with some familiar functions.