Starlikeness of certain non-univalent functions

Abstract

Abstract We consider three classes of functions defined using the class $${\mathcal {P}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>P</mml:mi></mml:math> of all analytic functions $$p(z)=1+cz+\cdots $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>p</mml:mi><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>c</mml:mi><mml:mi>z</mml:mi><mml:mo>+</mml:mo><mml:mo>⋯</mml:mo></mml:mrow></mml:math> on the open unit disk having positive real part and study several radius problems for these classes. The first class consists of all normalized analytic functions f with $$f/g\in {\mathcal {P}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>f</mml:mi><mml:mo>/</mml:mo><mml:mi>g</mml:mi><mml:mo>∈</mml:mo><mml:mi>P</mml:mi></mml:mrow></mml:math> and $$g/(zp)\in {\mathcal {P}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>g</mml:mi><mml:mo>/</mml:mo><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mi>p</mml:mi><mml:mo>)</mml:mo><mml:mo>∈</mml:mo><mml:mi>P</mml:mi></mml:mrow></mml:math> for some normalized analytic function g and $$p\in {\mathcal {P}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>p</mml:mi><mml:mo>∈</mml:mo><mml:mi>P</mml:mi></mml:mrow></mml:math> . The second class is defined by replacing the condition $$f/g\in {\mathcal {P}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>f</mml:mi><mml:mo>/</mml:mo><mml:mi>g</mml:mi><mml:mo>∈</mml:mo><mml:mi>P</mml:mi></mml:mrow></mml:math> by $$|(f/g)-1|&lt;1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>|</mml:mo><mml:mo>(</mml:mo><mml:mi>f</mml:mi><mml:mo>/</mml:mo><mml:mi>g</mml:mi><mml:mo>)</mml:mo><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo>|</mml:mo><mml:mo>&lt;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math> while the other class consists of normalized analytic functions f with $$f/(zp)\in {\mathcal {P}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>f</mml:mi><mml:mo>/</mml:mo><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mi>p</mml:mi><mml:mo>)</mml:mo><mml:mo>∈</mml:mo><mml:mi>P</mml:mi></mml:mrow></mml:math> for some $$p\in {\mathcal {P}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>p</mml:mi><mml:mo>∈</mml:mo><mml:mi>P</mml:mi></mml:mrow></mml:math> . We have determined radii so that the functions in these classes to belong to various subclasses of starlike functions. These subclasses includes the classes of starlike functions of order $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>α</mml:mi></mml:math> , parabolic starlike functions, as well as the classes of starlike functions associated with lemniscate of Bernoulli, reverse lemniscate, sine function, a rational function, cardioid, lune, nephroid and modified sigmoid function.

Locations

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