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Uniformly convex functions

Wancang Ma, David Minda (1992)

Annales Polonici Mathematici

Recently, A. W. Goodman introduced the geometrically defined class UCV of uniformly convex functions on the unit disk; he established some theorems and raised a number of interesting open problems for this class. We give a number of new results for this class. Our main theorem is a new characterization for the class UCV which enables us to obtain subordination results for the family. These subordination results immediately yield sharp growth, distortion, rotation and covering theorems plus sharp...

Uniformly convex functions II

Wancang Ma, David Minda (1993)

Annales Polonici Mathematici

Recently, A. W. Goodman introduced the class UCV of normalized uniformly convex functions. We present some sharp coefficient bounds for functions f(z) = z + a₂z² + a₃z³ + ... ∈ UCV and their inverses f - 1 ( w ) = w + d ₂ w ² + d ₃ w ³ + . . . . The series expansion for f - 1 ( w ) converges when | w | < ϱ f , where 0 < ϱ f depends on f. The sharp bounds on | a n | and all extremal functions were known for n = 2 and 3; the extremal functions consist of a certain function k ∈ UCV and its rotations. We obtain the sharp bounds on | a n | and all extremal functions for n = 4, 5,...

Uniformly convex spiral functions and uniformly spirallike functions associated with Pascal distribution series

Gangadharan Murugusundaramoorthy, Basem Aref Frasin, Tariq Al-Hawary (2022)

Mathematica Bohemica

The aim of this paper is to find the necessary and sufficient conditions and inclusion relations for Pascal distribution series to be in the classes 𝒮𝒫 p ( α , β ) and 𝒰𝒞𝒱 p ( α , β ) of uniformly spirallike functions. Further, we consider an integral operator related to Pascal distribution series. Several corollaries and consequences of the main results are also considered.

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