On q-ANALOGUE of Differential Subordination Associated with Lemniscate of Bernoulli

This article comprises the study of differential subordination with analogue of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"> <a:mi>q</a:mi> </a:math> -derivative. It includes the sufficient condition on <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" id="M2"> <c:mi>γ</c:mi> </c:math> for <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" id="M3"> <e:mn>1</e:mn> <e:mo>+</e:mo> <e:mfenced open="(" close=")" separators="|"> <e:mrow> <e:mrow> <e:mrow> <e:mi>γ</e:mi> <e:mo>∂</e:mo> <e:msub> <e:mrow> <e:mi>z</e:mi> </e:mrow> <e:mrow> <e:mi>q</e:mi> </e:mrow> </e:msub> <e:mi>h</e:mi> <e:mrow> <e:mfenced open="(" close=")" separators="|"> <e:mrow> <e:mi>z</e:mi> </e:mrow> </e:mfenced> </e:mrow> </e:mrow> <e:mo>/</e:mo> <e:mrow> <e:msup> <e:mrow> <e:mi>h</e:mi> </e:mrow> <e:mrow> <e:mi>n</e:mi> </e:mrow> </e:msup> <e:mfenced open="(" close=")" separators="|"> <e:mrow> <e:mi>z</e:mi> </e:mrow> </e:mfenced> </e:mrow> </e:mrow> </e:mrow> </e:mfenced> </e:math> to be subordinated by <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" id="M4"> <p:mfenced open="(" close=")" separators="|"> <p:mrow> <p:mrow> <p:mrow> <p:mn>1</p:mn> <p:mo>+</p:mo> <p:mi>A</p:mi> <p:mi>z</p:mi> </p:mrow> <p:mo>/</p:mo> <p:mrow> <p:mn>1</p:mn> <p:mo>+</p:mo> <p:mi>B</p:mi> <p:mi>z</p:mi> </p:mrow> </p:mrow> </p:mrow> </p:mfenced> </p:math> , <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" id="M5"> <u:mo>−</u:mo> <u:mn>1</u:mn> <u:mo>≤</u:mo> <u:mi>B</u:mi> <u:mo><</u:mo> <u:mi>A</u:mi> <u:mo>≤</u:mo> <u:mn>1</u:mn> </u:math> , and implies that <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" id="M6"> <w:mi>h</w:mi> <w:mfenced open="(" close=")" separators="|"> <w:mrow> <w:mi>z</w:mi> </w:mrow> </w:mfenced> <w:mo>≺</w:mo> <w:msqrt> <w:mrow> <w:mn>1</w:mn> <w:mo>+</w:mo> <w:mi>z</w:mi> </w:mrow> </w:msqrt> </w:math> , where <bb:math xmlns:bb="http://www.w3.org/1998/Math/MathML" id="M7"> <bb:mi>h</bb:mi> <bb:mfenced open="(" close=")" separators="|"> <bb:mrow> <bb:mi>z</bb:mi> </bb:mrow> </bb:mfenced> </bb:math> is the analytic function in the open unit disk. Moreover, certain sufficient conditions for <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" id="M8"> <gb:mi>q</gb:mi> </gb:math> -starlikeness of analytic functions related with lemniscate of Bernoulli are determined.