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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...

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П.И. Сижук (1975)

Sibirskij matematiceskij zurnal

Variability regions of close-to-convex functions

Takao Kato, Toshiyuki Sugawa, Li-Mei Wang (2014)

Annales Polonici Mathematici

M. Biernacki gave in 1936 concrete forms of the variability regions of z/f(z) and zf'(z)/f(z) of close-to-convex functions f for a fixed z with |z|<1. The forms are, however, not necessarily convenient to determine the shape of the full variability region of zf'(z)/f(z) over all close-to-convex functions f and all points z with |z|<1. We propose a couple of other forms of the variability regions and see that the full variability region of zf'(z)/f(z) is indeed the complex plane minus the origin....

Verification of Brannan and Clunie's conjecture for certain subclasses of bi-univalent functions

S. Sivasubramanian, R. Sivakumar, S. Kanas, Seong-A Kim (2015)

Annales Polonici Mathematici

Let σ denote the class of bi-univalent functions f, that is, both f(z) = z + a₂z² + ⋯ and its inverse f - 1 are analytic and univalent on the unit disk. We consider the classes of strongly bi-close-to-convex functions of order α and of bi-close-to-convex functions of order β, which turn out to be subclasses of σ. We obtain upper bounds for |a₂| and |a₃| for those classes. Moreover, we verify Brannan and Clunie’s conjecture |a₂| ≤ √2 for some of our classes. In addition, we obtain the Fekete-Szegö relation...

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