On a class of analytic functions governed by subordination

The purpose of this paper is to introduce a class of functions ℱ λ , λ ∈ [0, 1], consisting of analytic functions f normalized by f(0) = f´(0) − 1 = 0 in the open unit disk U which satisfies the subordination condition that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mtext>z</m:mtext> <m:msup> <m:mi>f</m:mi> <m:mo>′</m:mo> </m:msup> <m:mrow> <m:mo>(</m:mo> <m:mtext>z</m:mtext> <m:mo>)</m:mo> </m:mrow> <m:mo>/</m:mo> <m:mrow> <m:mo>{</m:mo> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>-</m:mo> <m:mi>λ</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mtext>f</m:mtext> <m:mrow> <m:mo>(</m:mo> <m:mtext>z</m:mtext> <m:mo>)</m:mo> </m:mrow> <m:mo>+</m:mo> <m:mi>λ</m:mi> <m:mtext>z</m:mtext> </m:mrow> <m:mo>}</m:mo> </m:mrow> <m:mo>≺</m:mo> <m:mtext>q</m:mtext> <m:mrow> <m:mo>(</m:mo> <m:mtext>z</m:mtext> <m:mo>)</m:mo> </m:mrow> <m:mo>,</m:mo> <m:mi> </m:mi> <m:mi> </m:mi> <m:mi> </m:mi> <m:mi> </m:mi> <m:mi> </m:mi> <m:mtext>z</m:mtext> <m:mo>∈</m:mo> <m:mo>𝕌</m:mo> <m:mo>,</m:mo> </m:mrow> </m:math> $${z}f'( {z} )/\{ {( {1 - λ } ){f}( {z} ) + λ {z}} \} {q}( {z} ),\,\,\,\,\,{z} ∈ {{U,}}$$ where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mtext>q</m:mtext> <m:mrow> <m:mo>(</m:mo> <m:mtext>z</m:mtext> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:msqrt> <m:mrow> <m:mn>1</m:mn> <m:mo>+</m:mo> <m:msup> <m:mrow> <m:mtext>z</m:mtext> </m:mrow> <m:mn>2</m:mn> </m:msup> </m:mrow> </m:msqrt> <m:mo>+</m:mo> <m:mtext>z</m:mtext> </m:mrow> </m:math> q( z ) = √ 1 + z² + z . Some basic properties (including the radius of convexity) are obtained for this class of functions.