Starlikeness associated with certain strongly functions

Let κ be an analytic function in the open unit disc <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>U</mml:mi></mml:math> in the complex plane with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>κ</mml:mi> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn></mml:math> . Then <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>κ</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>K</mml:mi> <mml:mo>(</mml:mo> <mml:mi>λ</mml:mi> <mml:mo>)</mml:mo></mml:math> if and only if <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>κ</mml:mi> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> <mml:mo>≺</mml:mo> <mml:msub><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow> <mml:mrow><mml:mi>λ</mml:mi></mml:mrow> </mml:msub> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo></mml:math> , where ≺ is a subordination relation and the function <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow> <mml:mrow><mml:mi>λ</mml:mi></mml:mrow> </mml:msub> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:msup><mml:mrow><mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo></mml:mrow> <mml:mrow><mml:mi>λ</mml:mi></mml:mrow> </mml:msup> </mml:math> , for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>0</mml:mn> <mml:mo><</mml:mo> <mml:mi>λ</mml:mi> <mml:mo><</mml:mo> <mml:mn>1</mml:mn></mml:math> , maps <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>U</mml:mi></mml:math> to a symmetric domain about the real axis in the right-half plane. This article finds the sharp radii of starlikeness for three classes of normalized analytic functions in the open unit disc. It also finds the solutions to first-order differential equations and utilizes these results together with the differential subordination developed by Miller and Mocanu to obtain the conditions so that some first-order differential relations associated with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow> <mml:mrow><mml:mi>λ</mml:mi></mml:mrow> </mml:msub> </mml:math> are subordinate to certain Ma and Minda functions. Consequently, it provides sufficient conditions for the normalized analytic functions to belong to some established subclasses of starlike functions.