On -convex functions of order with -fold symmetry.
On convex functions of order and type .
A note on certain subclass of close-to-convex functions.
Hyperbolically convex functions
We investigate univalent holomorphic functions f defined on the unit disk 𝔻 such that f(𝔻) is a hyperbolically convex subset of 𝔻; there are a number of analogies with the classical theory of (euclidean) convex univalent functions. A subregion Ω of 𝔻 is called hyperbolically convex (relative to hyperbolic geometry on 𝔻) if for all points a,b in Ω the arc of the hyperbolic geodesic in 𝔻 connecting a and b (the arc of the circle joining a and b which is orthogonal to the unit circle) lies in...
Uniformly convex functions II
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 . The series expansion for converges when , where depends on f. The sharp bounds on 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 and all extremal functions for n = 4, 5,...
Uniformly convex functions
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...
A remark on certain close-to-convex functions
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