Radius estimates for functions in the class 𝒰 <i> ^r </i> ( <i>λ</i> )

Abstract A normalized analytic function f defined on the open unit disc 𝔻 is called Ma-Minda starlike if the function zf′ / f is subordinate to the function φ. For 0 &lt; λ ≤ 1, let 𝒰(λ) denote the class of functions <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>f</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>z</m:mi> <m:mo stretchy="false">)</m:mo> <m:mo>=</m:mo> <m:mi>z</m:mi> <m:mo>+</m:mo> <m:mrow> <m:munderover> <m:mo movablelimits="false">∑</m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>=</m:mo> <m:mn>2</m:mn> </m:mrow> <m:mrow> <m:mi mathvariant="normal">∞</m:mi> </m:mrow> </m:munderover> </m:mrow> <m:msub> <m:mi>a</m:mi> <m:mi>n</m:mi> </m:msub> <m:msup> <m:mi>z</m:mi> <m:mi>n</m:mi> </m:msup> </m:math> $f(z)=z+\sum\limits_{n=2}^{\infty} a_n z^n$ analytic in the unit disc 𝔻 satisfying the condition |( z/f ( z )) 2 f′ ( z )−1| &lt; λ for z ∈ 𝔻. The class 𝒰 r (λ) consists of functions g defined by g ( z ) = (( z/f ( z )) − 1)/(− a 2 ), where f ∈ 𝒰(λ) and a 2 = f′′ (0)/2 ≠ 0. The radius of Janowski starlikeness and in particular, the radius of starlikeness of order γ are derived. Also, the sharp radii of Ma-Minda starlikeness associated with nephroid, lune, lemniscate of Bernoulli, cardioid, rational function, and so on, are estimated. The inclusion conditions based on λ are derived as specific cases so that functions in the class 𝒰 r (λ) are included in various Ma-Minda starlike classes.