Ma-Minda φ-classes studied in this paper:
Abstract
A normalized analytic function defined on the open unit disc D is
called Ma-Minda starlike if zf ′(z)/f(z) is subordinate to the function ϕ. For a
normalized convex function f defined on D and α > 0, we determine the radius
of Ma-Minda starlikeness of the function g defined as g(z) = (zf ′(z)/f(z))α f(z)
for certain choices of ϕ. In particular, we investigate the radius of Janowski
starlikeness of the function g.
Mathematics Subject Classification (2010): 30C80, 30C45.
Results & Lemmas (3)
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Lemma 1.1.
Lemma 1.1. [16] If p ∈P[A, B], then p(z) −1 −ABr2 1 −B2r2 ⩽(A −B)r 1 −B2r2 (|z| ⩽r < 1). The class of functions f ∈A with the property that…
Lemma 1.1. [16] If p ∈P[A, B], then p(z) −1 −ABr2 1 −B2r2 ⩽(A −B)r 1 −B2r2 (|z| ⩽r < 1). The class of functions f ∈A with the property that zf ′(z)/f(z)/ ∈P[A, B] is denoted by ST [A, B]. In this manuscript, we are interested in the class J α 1 defined as follows: J α 1 := g ∈A : g(z) = zf ′(z)
Theorem 2.1.
Theorem 2.1. The ST [A, B] radius of the class J α 1, α > 0, is given by RST [A,B] = A −B 1 + α + |A + αB|.
Theorem 2.1. The ST [A, B] radius of the class J α 1 , α > 0, is given by RST [A,B] = A −B 1 + α + |A + αB|.
Theorem 2.2.
Theorem 2.2. Let α > 0. For the class J α 1, the following radius results hold: 1. The ST e radius is given by RST e = ( e−1 eα+1 if α ⩾1…
Theorem 2.2. Let α > 0. For the class J α 1 , the following radius results hold: 1. The ST e radius is given by RST e = ( e−1 eα+1 if α ⩾1 e−1 e+α if α ⩽1.
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