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Univalence, strong starlikeness and integral transforms

M. Obradović, S. Ponnusamy, P. Vasundhra (2005)

Annales Polonici Mathematici

Let represent the class of all normalized analytic functions f in the unit disc Δ. In the present work, we first obtain a necessary condition for convex functions in Δ. Conditions are established for a certain combination of functions to be starlike or convex in Δ. Also, using the Hadamard product as a tool, we obtain sufficient conditions for functions to be in the class of functions whose real part is positive. Moreover, we derive conditions on f and μ so that the non-linear integral transform...

Univalent harmonic mappings

Albert E. Livingston (1992)

Annales Polonici Mathematici

Let a < 0, Ω = ℂ -(-∞, a] and U = z: |z| < 1. We consider the class S H ( U , Ω ) of functions f which are univalent, harmonic and sense preserving with f(U) = Ω and satisfy f(0) = 0, f z ( 0 ) > 0 and f z ̅ ( 0 ) = 0 . We describe the closure S H ( U , Ω ) ¯ of S H ( U , Ω ) and determine the extreme points of S H ( U , Ω ) ¯ .

Univalent harmonic mappings II

Albert E. Livingston (1997)

Annales Polonici Mathematici

Let a < 0 < b and Ω(a,b) = ℂ - ((-∞, a] ∪ [b,+∞)) and U= z: |z| < 1. We consider the class S H ( U , Ω ( a , b ) ) of functions f which are univalent, harmonic and sense-preserving with f(U) = Ω and satisfying f(0) = 0, f z ( 0 ) > 0 and f z ̅ ( 0 ) = 0 .

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